title:

Network AnalysisSage UniversityPapers Series.QuantitativeApplications inthe SocialSciences ; No.07-028

author:Knoke, David.;Kuklinski, JamesH.

publisher: SagePublications, Inc.

isbn10 |asin: 080391914X

print isbn13: 9780803919143ebook

isbn13: 9780585217024language: English

subject Social sciences--Network analysis.

publicationdate: 1982

lcc: H61.K6331982eb

ddc: 300/.28/51

subject: Social sciences--Network analysis.

Network Analysis

SAGE UNIVERSITY PAPERSSeries: Quantitative

Applications in the SocialSciences

Series Editor: Michael S.Lewis-Beck, University of

IowaEditorial Consultants

Richard A. Berk, Sociology,University of California, Los

AngelesWilliam D. Berry, Political

Science, Florida StateUniversity

Kenneth A. Bollen,Sociology, University of

North Carolina, Chapel HillLinda B. Bourque, Public

Health, University ofCalifornia, Los AngelesJacques A. Hagenaars,Social Sciences, Tilburg

UniversitySally Jackson,

Communications, Universityof Arizona

Richard M. Jaeger,

Education, University ofNorth Carolina, GreensboroGary King, Department of

Government, HarvardUniversity

Roger E. Kirk, Psychology,Baylor University

Helena Chmura Kraemer,Psychiatry and Behavioral

Sciences, StanfordUniversity

Peter Marsden, Sociology,Harvard University

Helmut Norpoth, PoliticalScience, SUNY, Stony Brook

Frank L. Schmidt,Management and

Organization, University ofIowa

Herbert Weisberg, PoliticalScience, The Ohio State

UniversityPublisher

Sara Miller McCune, SagePublications, Inc.

INSTRUCTIONS TOPOTENTIAL

CONTRIBUTORS

For guidelines onsubmission of a monograph

proposal to this series,please write

Michael S. Lewis-Beck,Editor

Sage QASS SeriesDepartment of Political

ScienceUniversity of Iowa

Iowa City, IA 52242

Page 1

Series / Number 07-028

Network Analysis

David KnokeJames H. KuklinskiIndiana University

SAGE PublicationsThe International Professional

PublishersNewbury Park London NewDelhi

Copyright© 1982 by Sage Publications, Inc.All rights reserved. No part of this book maybe reproduced or utilized in any form or byany means, electronic or mechanical,including photocopying, recording, or by anyinformation storage and retrieval system,without permission in writing from thepublisher.For information address:

SAGE Publications, Inc.

2455 Teller RoadNewbury Park, California 91320E-mail: order@sagepub.comSAGE Publications Ltd.6 Bonhill StreetLondon EC2A 4PUUnited KingdomSAGE Publications India Pvt. Ltd.M-32 MarketGreater Kailash INew Delhi 110 048 IndiaPrinted in the United States of AmericaInternational Standard Book Number 0-

8039-1914-XLibrary of Congress Catalog Card No. L.C. 82-04262299 00 01 02 16 15 14 13 12 11When citing a university paper, please usethe proper form. Remember to cite the SageUniversity Paper series title and include thepaper number. One of the following formatscan be adapted (depending on the stylemanual used):(1) Iversen, G.R., & Norpoth, H. (1976).Analysis of variance (Sage University Paperseries on Quantitative Applications in the

Social Sciences, No. 07-001). Beverly Hills,CA: Sage.OR(2) Iversen, G.R., & Norpoth, H. 1976.Analysis of variance. Sage University Paperseries on Quantitative Applications in theSocial Sciences, series no. 07-001. BeverlyHills, CA: Sage.

Contents

Series Editor's Introduction1. Introduction2. Basic Concepts

Attributes and RelationsNetworksResearch Design ElementsStructure in Complete Networks

3. Data CollectionBoundary SpecificationSampling of Networks in Large PopulationsGeneric Types of Measures and ReliabilityMissing Data

4. Methods and ModelsAn Interorganizational Relations ExampleVisual DisplaysMatrix RepresentationIndices for Actors and Networks

Clique DetectionStructural EquivalenceMatrix Permutation and ImagesHypothesis TestingAnalysis of Structural CorrelatesOther Issues

References

Page 5

Series Editor'sIntroductionWhat is described in thisinteresting new volumerepresents a differentapproach to the world thanthat of most social sciencedata analysis. As theauthors point out inChapter 2, the social worldis most often characterized

by the attributes ofindividuals. But individualsare also characterized bytheir relationships to oneanother. It is the study ofthese relationships that isdescribed in NetworkAnalysis.Networks of relationshipsbetween individuals,objects, or events, may bethose of friendship,dominance,communications, and so on.

The relations may be one-directional or mutual, andthey may be characterizedby different levels ofintensity or involvement.These differences in thecontent and form of therelationship help define thekind of analysis performed,as explained in the text. Inaddition, the analysis maybe done at several levels,concentrating on individualsand their relationships withspecific other individuals or,

at the highest level, on thecomplete network or systemof relationships.Chapter 2 covers the basicterms and concepts thatare necessary tounderstand networkanalysis. Chapter 3discusses important pointsregarding data collectionsuch as the problem ofmaking inferences aboutnetworks when what onesamples is individuals.

Chapter 4 is an introductionto a variety of analyticalmeans of displaying andstudying networks. Thisincludes another look at thedifferent levels of analysis,including specification ofindices for individual actors,the definition andmeasurement of cliques,and measures of theequivalence of parts ofsystems and even entirenetworks.

While an occasionalequation may lookforbidding because ofdouble summations andtriple subscripts, thismonograph requires nomore than high schoolalgebra. Moreover, byjudicious use of examples,the authors have takengreat care to show exactlyhow the formulas areactually evaluated. Withjust a little care, the readershould come away from the

volume with a goodunderstanding of the scopeof network analysis, thoseaspects of data collectionthat are

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important and oftenoverlooked, and the majorapproaches to the analysisof networks.

RICHARD G.NIEMISERIES CO-EDITOR

Page 7They say you are not youexcept in terms of relation toother people. If there weren'tany other people therewouldn't be any you becausewhat you do, which is whatyou are, only has meaning inrelation to other people.Robert Penn WarrenAll the King's Men

1.Introduction

Network concepts andmethods of social researchhave undergone dramaticgrowth during the pastdecade, particularly insociology and anthropology.Their rapid introduction toaudiences in politicalscience, education, andrelated fields can beanticipated. A sign thatnetwork analysis had comeof age was the founding in1978 of the International

AUTHORS' NOTE: NationalScience Foundation grantsto both authors and aresearch scientistdevelopment award fromthe National Institute ofMental Health to Knoke(KP2MH00131) facilitatedwork on this monograph.Kuklinski gratefullyacknowledges hisintellectual debt to HeinzEulau, who has led theway in bringing socialrelations into the study ofpolitics. Our deepestgratitude to Ronald S.

Burt and an anonymousreferee for their manyuseful comments on anearlier draft.

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Network for Social NetworkAnalysis (INSNA) and itstwo journals, Connectionsand Social Networks. Acritical mass of active,proselytizing networkscholars has formed, andtheir ideas will percolatethrough many disciplines aspublications and coursespresenting these ideasproliferate.To date, however, few

comprehensive basictreatments of networks areavailable to students andprofessionals, unlike thevast literature on suchtraditional statisticaltechniques as regression,analysis of variance, andlog-linear models. Much ofthe published literature iswritten by and for insiderswho already speak thelanguage of networkanalysis, leaving interestedand curious outsiders to

scramble as best they canbe decipher the plethora ofterms, concepts, andtechniques. Thismonograph is a basicprimer designed to guidethe interested user throughthese topics. Wesystematically inventory thecentral features of networkanalysis, cite originalsources to be consulted forgreater detail, and suggestdiverse applications tosocial scientific research.

Our treatment assumes noprior exposure to thesubject and presumes nomore than a solidbackground in basicstatistics.The monograph consists ofthree parts. In Chapter 2,we introduce the idea ofrelational data andnetworks, illustrating theirapplication to a variety ofsubstantive topics, units ofobservation, and types of

data. Taking a structuralapproach, we emphasizethe value of networkanalysis for uncovering thepatterns of order underlyingempirical observations. InChapter 3, we devoteattention to problems indelineating sets of actors towhich network analysisapplies, the difficulties inmaking inferences fromsamples, the basicrequisites for collecting dataabout relationships, and

various special topics in themeasurement of ties. InChapter 4, we discuss bothvisual (graphic) and matrixrepresentations of somesimple networks and variouselementary measures ofnetwork properties. Wepresent at length varioustechniques for thequantitative analysis ofdifferent types of networkdata.Because network analysis is

such a broad and rapidlychanging methodology, wecan do little more in thisbrief space than provide ageneral overview. But if wedo our task well, we hopethat by the end readers willbe convinced of the meritsof the methods and willcontinue to pursueadvanced topics on theirown. If this primer helps toform a foundation thatmakes such studiespossible, we will be well

satisfied.

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2.Basic ConceptsReally, universally, relationsstop nowhere, and theexquisite problem of theartist is eternally but to draw,by a geometry of his own,the circle within which theyshall happily appear to do so.Henry JamesRoderick Hudson

This chapter presents

concepts essential forunderstanding networkanalysis procedures. Usageamong practitioners is notentirely consensual, and wehave made no effort to beterminologically exhaustive.We have, however,attempted to adhere mostclosely to the labels anddefinitions that strike us asbelonging in themainstream of sociologicalapproaches to networkanalysis. Throughout, we

attempt to illustrate thebasic concepts withcitations to recentliterature, which the readermay wish to consult tounderstand further theirapplication to substantiveproblems.To appreciate fully thedistinctive theoreticalunderpinnings of networkapproaches to socialphenomena, a comparisonwith more traditional,

individualistic approachesmay be useful. In theatomistic perspectivestypically assumed byeconomics and psychology,individual actors aredepicted as making choicesand acting without regardto the behavior of the otheractors. Whether analyzedas purposive action basedon rational calculations ofutility maximization, or asdrive-reduction motivationbased on causal

antecedents, suchindividualistic explanationsgenerally ignore the socialcontexts within which thesocial actor is embedded.In contrast, networkanalysis incorporates twosignificant assumptionsabout social behavior. Itsfirst essential insight is thatany actor typicallyparticipates in a socialsystem involving manyother actors, who are

significant reference pointsin one another's decisions.The nature of therelationships a given actorhas with other systemmembers thus may affectthat focal actor'sperceptions, beliefs, andactions. But networkanalysis does not stop withan account of the socialbehavior of individuals. Itssecond essential insight liesin the importance ofelucidating the various

levels of structure in a socialsystem, where

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structure consists of''regularities in the patternsof relations among concreteentities" (White et al.,1976). In the individualisticapproach, social structure isseldom an explicit focus ofinquiry, to the extent that itis even considered at all.Network analysis, byemphasizing relations thatconnect the social positionswithin a system, offers a

powerful brush for paintinga systematic picture ofglobal social structures andtheir components. Theorganization of socialrelations thus becomes acentral concept in analyzingthe structural properties ofthe networks within whichindividual actors areembedded, and fordetecting emergent socialphenomena that have noexistence at the level of theindividual actor. This

dualistic quality of networkanalysisits capacity toilluminate entire socialstructures and tocomprehend particularelements within thestructureprobably accountsfor its rapidly increasingpopularity among socialtheorists and researcherswho have found the olderindividualistic traditionwanting as a framework forunderstanding socialphenomena.

Attributes and RelationsTwo basic approaches toviewing and classifying thevarious aspects of the socialworldaccording to theirattributes or theirrelationshipsare oftentreated as antithetical andeven irreconcilable. As thismonograph must deal atgreat length with relationaldata, we need to makeclear from the outset howthese two approaches to

measurement differ. Weshall also point out thatneither perspective by itselfyields satisfactoryunderstandings of socialphenomena.Attributes are intrinsiccharacteristics of people,objects, or events. Whenwe think of explainingvariance among such unitsof observation, we almostnaturally resort to attributemeasures, those qualities

that inherently belong to aunit apart from its relationswith other units or thespecific context withinwhich it is observed.Various types of attributescan be measured: anoccupation's averageincome, a nation's grossnational product, a riot'sduration, a birth cohort'smean formal schooling, aperson's opinion about thepresident.

Persons, objects, andevents may also be involvedin relationships, that is,actions or qualities thatexist only if two or moreentities are consideredtogether. A relation is notan intrinsic characteristic ofeither party taken inisolation, but is anemergent property of theconnection or linkagebetween units ofobservation. Whereattributes persist across the

various contexts in whichan actor is involved (e.g.,

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a person's age, sex,intelligence, income, andthe like remain unchangedwhether at home, at work,at church), relations arecontext specific and alter ordisappear upon an actor'sremoval from interactionwith the relevant otherparties (e.g., astudent/teacher relationdoes not exist outside aschool setting; a marital

relation vanishes upondeath or divorce of aspouse). A wide variety ofrelational properties can bemeasured: the strengths ofthe friendships amongpupils in a classroom, thekinship obligations amongfamily members, theeconomic exchangesbetween organizations. Asystematic classification ofrelationships will bepresented below.

Many aspects of socialbehavior can be treatedfrom both the attribute andthe relational perspectives,with only a slight alterationof conceptualization. Forexample, the value of goodsthat a nation imports inforeign trade each year isan attribute of the nation'seconomy, but the volume ofgoods exchanged betweeneach pair of nationsmeasures an exchangerelationship. Similarly, while

a college student's homestate is a personal attribute,a structural relationshipbetween colleges andstates could be measuredby the proportions ofenrolled students coming toeach college from eachstate. We could ask citizensabout their attendance atpolitical rallies and thusassign attribute scores forparticipation, but if wedesignate sets of citizenswho attended the same

rallies, we begin to probetheir cooccurrencerelationships. The point weare stressing is that, whileattributes and relationshipsare conceptually distinctapproaches to socialresearch, they should beseen as neither polar normutually exclusivemeasurement options.Undoubtedly, the vast bulkof social research todayrelies upon attributemeasures from surveys,

experiments, and fieldobservations of given unitsof analysis. Relationalmeasures capture emergentproperties of social systemsthat cannot be measuredby simply aggregating theattributes of individualmembers. Furthermore,such emergent propertiesmay significantly affect bothsystem performance andthe behavior of networkmembers. For example, thestructure of informal

friendships andantagonisms in formal workgroups can affect bothgroup and individualproductivity rates in waysnot predictable from suchpersonal attributes as age,experience, intelligence,and the like (Homans,1950). As another example,the structure ofcommunication amongmedical practitioners canshape the rate of diffusionof medical innovations in a

local community and candetermine which physiciansare likely to be early or lateadopters (Coleman et al.,1966). By ignoring thesocial-structural contextwithin which

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actors are located, a purelyattribute-based analysisloses much of theexplanatory potential thatrelational analysis can offer.The ultimate advance ofsocial scientific knowledgerequires combinations ofboth types of data and thecreation of measurementand analysis methodscapable of incorporatingthem.

NetworksRelations are the buildingblocks of network analysis.A network is generallydefined as a specific type ofrelation linking a defined setof persons, objects, orevents (see Mitchell, 1969).Different types of relationsidentify different networks,even when imposed on theidentical set of elements.For example, in a set ofemployees at a workplace,the advice-giving network is

unlikely to be the same asthe friendship network orthe formal authoritynetwork. The set ofpersons, objects, or eventson which a network isdefined may be called theactors or nodes. Theseelements possess someattribute(s) that identifythem as members of thesame equivalence class forpurposes of determining thenetwork of relations amongthem. For example, we

might stipulate that allpayroll employees at plantsix of the National WidgetCorp. comprise the set ofactors among whom anadvice-giving network issought. Additionalrestrictions on thepermissible actors could beimposed (e.g., only males inmanagerial jobs), indicatingthat delimiting networkboundaries depends to agreat extent upon aresearcher's purposes.

Our generic definition of anetwork may imply thatonly those linkages thatactually occur are part of anetwork. But networkanalysis must take intoaccount both the relationsthat occur and those thatdo not exist among theactors. For example,attending only to the gossipconnections in a communityand not to the structural"holes" that occur wherelinks are absent might

result in an inaccurateunderstanding of howrumors spread orevaporate. Theconfiguration of presentand absent ties among thenetwork actors reveals aspecific network structure.Structures vary dramaticallyin form, from the isolatedstructure in which no actoris connected to any otheractor, to the saturatedstructure in which everyactor is directly linked to

every other individual. Moretypical of real networks arevarious intermediatestructures in which someactors are more extensivelyconnected amongthemselves than are others.A discussion of some basictypes of network structuresappears below. A coretheoretical problem in

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network analysis is toexplain the occurrence ofdifferent structures and, atthe nodal level, to accountfor variation in linkages toother actors. The parallelempirical task in networkanalysis is to detect thepresence of such structuresin empirical network data.If network analysis werelimited just to a conceptualframework for identifying

how a set of actors is linkedtogether, it would not haveexcited much interest andeffort among socialresearchers. But networkanalysis contains a furtherexplicit premise of greatconsequence: The structureof relations among actorsand the location ofindividual actors in thenetwork have importantbehavioral, perceptual, andattitudinal consequencesboth for the individual units

and for the system as awhole. In Mitchell's (1969)felicitous terms, "Thepatterning of linkages canbe used to account forsome aspects of behavior ofthose involved". Forexample, a formalorganization with acentralized structure ofauthority among it variousdivisions and departmentsmay be most effective (e.g.,enjoying high growth andprofitability) in a relatively

placid environment, but in aturbulent, rapidly changingenvironment anorganization with a lesscentralized structure maybe more adaptable.Investigating thishypothesis requires astructural analysis of theinformation collection,processing, dissemination,decision making, andimplementation relationslinking organizational units.Network analysis offers a

means for bridging the gapbetween macro-and micro-level explanations and itholds out the promise ofsurpassing if not entirelysupplanting attribute-basedapproaches.To illustrate the potentialpower of a networkapproach, consider a varietyof contemporary socialscience problems: thesources of homophyly ofbeliefs within a power elite,

the adoption oftechnological innovations,the causes of corporateprofitability, the incomeearnings of occupationalgroups, the recruitmentprocesses of socialmovement organizations,the development ofnontraditional sex roles. Ineach of these and manyother substantive areas, alarge research literature canbe uncovered that attemptsto explain the phenomena

as a function of individual orgroup attributes. Yet inmany instances, suchcharacteristics may predictbehavior only because ofunderlying patterns ofrelations that are oftenassociated with theseattributes. For example,innovation diffusion studiesfrequently find that highlyeducated persons tend toadopt sooner, but, thisrelation may really reflectthe tendency for such

persons to be moreprominent in theirnetworks, thus giving themgreater access to the flowof information (e.g., Knokeand Burt, 1982). Networkapproaches can morefaithfully capture the

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context of social relationswithin which actorsparticipate and makebehavioral decisions.

Research Design ElementsNetwork analyses takemany forms to suitresearchers' diversetheoretical and substantiveconcerns. Four elements ofa research design inparticular shape the

measurement and analysisstrategies available to aresearcher: the choice ofsampling units, the form ofrelations, the relationalcontent, and the level ofdata analysis. Varyingcombinations of thesedesign elements havecreated a wide diversityamong network studies thatis evident in the researchliterature.Sampling Units

Before collecting data, anetwork researcher mustdecide the most relevanttype of social organizationand the units within thatsocial form that comprisethe network nodes. Orderedin a roughly increasing scaleof size and complexity are ahalf-dozen basic units fromwhich samples may bedrawn: individuals, groups(both formal and informal),complex formalorganizations, classes and

strata, communities, andnation-states. A typicaldesign involves somehigher-level system whosenetwork is to beinvestigated with one ormore lower-level units asthe nodes, for example, acorporation with itsdepartments and individualemployees as the actors, ora city with its firms,bureaus, and voluntaryassociations as the nodes.

The earliest and still mostprevalent network studiesselect small-scale socialorganizationsclassrooms,offices, gangs, social clubs,schools, villages, andartificially created laboratorygroupsand treat theirindividual members as thenodes (see Moreno, 1934;Homans, 1950; Leinhardt,1972; Hallinan, 1978;Kandel, 1978; Rogers andKincaid, 1981; Miller et al.,1981). Although these

settings have theconsiderable advantages ofsharply delineatedboundaries and enumeratedpopulations, nothingintrinsic to network analysisprevents applications tolarge-scale systems. Recentexamples from the literatureinclude: elite leadershipnetworks in communities(Perrucci and Pilisuk, 1970;Laumann and Pappi, 1976);interorganizational networksin communities

(Galaskiewicz, 1979; Knokeand Wood, 1981; Knoke,forthcoming);intercorporate networks inthe national economy(Levine, 1972; Sonquistand Koenig, 1975; Burt etal., 1980); scientificnetworks in a professionaldiscipline (Crane, 1969;Breiger, 1976;

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Mullins et al., 1977); andinternational networks inthe world system (Snyderand Kick, 1979).Form of RelationsThe relations among actorshave both content andform. Content refers to thesubstantive type of relationrepresented in theconnections (e.g.,supervising, helping,gossiping), and an

inventory of content typesis presented below.Relational form refers toproperties of theconnections between pairsof actors (dyads) that existindependently of specificcontents. Two basic aspectsof relational form are (a)the intensity or strength ofthe link between twoactors, and (b) the level ofjoint involvement in thesame activities (Burt,1982:22). Conceivably, two

relations that are quitedistinct in content mayexhibit identical or highlysimilar forms. For example,within a small communitythe social visits betweenresidents might occur withthe same frequency anddegree of reciprocation asdo their exchanges of minoreconomics assistance (e.g.,Hansen, 1981).Relational ContentIn conjunction with

choosing the appropriatesampling units, a networkanalyst must decide whatspecific network linkages toinvestigate. Networkcontent is frequentlydetermined by theoreticalconsiderations; for example,a study of psychologicalbalance theory (Heider,1946, 1979; Anderson,1979) calls for sentimentrelations. Thus no singletype of connection can bepriori designed as the

correct network for apopulation, or even themost important network forall research purposes. Insome cases, substantiveproblems indicate that morethan one analyticallydistinct type of relationshipshould be investigated, inwhich case a networkcompounded of two ormore types of linkages (i.e.,a multiplex network) maybe most appropriate(Boissevain, 1974;

Kapferer, 1969).Because researchers'capacities to conceptualizeand operationalize varioustypes of networks arealmost unlimited, we canonly list the more commontypes of relational content,citing some representativestudies:·Transaction relations:Actors exchange controlover physical or symbolicmedia, for example, in gift

giving or economics salesand purchases (Burt et al.,1980; Laumann et al.,1978).

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· Communication relations:Linkages between actorsare channels by whichmessage may betransmitted from one actorto another in a system(Marshall, 1971; Lin, 1975;Rogers and Kincaid, 1981).· Boundary penetrationrelations: The ties betweenactors consist of constituentsubcomponents held incommon, for example,

corporation boards ofdirectors with overlappingmembers (Levine, 1972;Allen, 1974; Mariolis, 1975;Sonquist and Koenig, 1975;Burt, 1982: ch. 8).· Instrumental relations:Actors contact one anotherin efforts to secure valuablegoods, services, orinformation, such as a job,an abortion, political advice,recruitment to a socialmovement (Granovetter,

1974; Boissevain, 1974).· Sentiment relations:Perhaps the mostfrequently investigatednetworks are those in whichindividuals express theirfeelings of affection,admiration, deference,loathing, or hostility towardeach other (Hunter, 1979;Hallinan, 1974; Sampson,1969).· Authority/power relations:These networks, usually

occurring in complex formalorganizations, indicate therights and obligations ofactors to issue and obeycommands (White, 1961;Cook and Emerson, 1978;Williamson, 1970; Lincolnand Miller, 1979).· Kinship and descentrelations: A special instanceof several preceding generictypes of networks, thesebonds indicate rolerelationships among family

members (Nadel, 1957;Bott, 1955; White, 1963).Levels of AnalysisAfter selecting the samplingunits and relational content,a network analyst will haveseveral alternative levels towhich to analyze the datacollected for a project.Appropriate techniques willbe described at length inChapter 3, but here weconsider four conceptuallydistinct levels of analysis at

which an investigation canfocus.The simplest level is theegocentric network,consisting of each individualnode, all others with whichit has relations, and therelations among thesenodes. If the sample size isN, there are N units ofanalysis at the ego-centered level. Each actorcan be desired by thenumber, the magnitude,

and other characteristics ofits linkages with the otheractors, for example, theproportion of reciprocatedlinkages or the density ofties among the actors inego's first "zone" (i.e., theset of actors directlyconnected to ego). In manyways, an egocentric level ofanalysis

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strongly resembles typicalattribute-based research,with the usualcharacteristicssupplemented by measuresderived from a node's directnetwork relations. Goodexamples of ego-centerednetwork research includeLaumann's (1973) analysisof friendship among urbanmen and Granovetter's(1974) investigation of job

information transmission.At the next highest level ofanalysis is the dyad, formedby a pair of nodes. If thesample size is N, there are(N2 N)/2 distinct units ofanalysis at the dyadic level.The basic question about adyad is whether or not adirect tie exists betweenthe two actors, or whetherindirect connections mightexist via other actors in thesystem to which they are

connected. Typical dyadicanalyses seek to explainvariation in dyadic relationsas a function of jointcharacteristics of the pair,for example, the degree ofsimilarity of their attributeprofiles. Laumann,Verbrugge, and Pappi(1976: 145-161), forinstance, measured theproximities among pairs ofelites in a German town'scommunity affairsdiscussion network. Using

causal modeling methods,they found values andfriendships to mediate mostof the effects of partyaffiliation, religion, formalpositions in government,and business/professionalsimilarities on distancesbetween the dyads.Not surprisingly, the thirdlevel of analysis consists oftriads. If N is the samplesize, there are distincttriads formed by selecting

each possible subset ofthree nodes and theirlinkages. Research usingtriads has largelyconcentrated on the localstructure of sentiment tiesamong individual actors,with a particular concern fordetermining transitivityrelations (i.e., if A choosesB and B chooses C, does Atend to choose C?). Triadsreceived quite elaborateand elegant statisticaltreatment at the hands of

Holland, Leinhardt, andDavis (see Holland andLeinhardt, 1975, 1978;Davis, 1979); we lack thespace to review them in thisvolume.Beyond the triadic level, themost important level ofanalysis is that of thecomplete network, orsystem. In these analyses,a researcher uses thecomplete information aboutpatterning of ties among all

actors to ascertain theexistence of distinctpositions or roles within thesystem and to describe thenature of relations amongthese positions. Althoughthe sample may consist of Nnodes and (N2 N) possibledyadic ties of a given type,these elements altogetheradd up to only a singlesystem. Thus to testhypotheses about thecauses or consequences ofvariation in complete

network configurationstypically requires severaldistinct systems, which maytax a researcher'sresources. Nevertheless, thecomplete network hasbecome one of the mostpopular levels of analysis

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in recent years with thegrowth of new methods andtools to handle its particularproblems. Because the bulkof this book focuses on thistype of data, the followingsection examines conceptsuseful in analyzing thesocial structure of completenetworks.This brief overview of fourlevels of network analysisunderscores the important

emergence of structuralproperties that cannotsimply be induced fromlower-level phenomena. Forexample, transitivity ofchoice relations is asubstantively importantvariable in friendshipformation theories that canbe observed at the triadicbut not at the dyadic oregocentric level. As anotherillustration, consider twoscientific researchcommunities with roughly

similar egocentric, dyadic,or even triadic structures intheir scientific discussionnetworks. However, if thefirst community's completenetwork consists offragmented subgroups suchthat many individualscannot easily reach oneanother directly orindirectly, while the secondcommunity's completenetwork exhibits a highdegree of integrationamong subgroups, we may

well anticipate a freer flowof information and greaterinnovation in the lattercommunity. This proteancapacity of network analysisto address problems atmultiple levels of analysis byencompassing emergentstructural properties liesbehind its increasingpopularity as a frame-workfor guiding empiricalresearch.

Structure in Complete

NetworksOne major use of networkanalysis in sociology andanthropology has been touncover the social structureof a total system. Systemsmay be as small as anelementary schoolclassroom and a nativevillage, or as large as anational industry and theworld system of nation-states. But for any system,an important step in a

structural analysis toidentify the significantpositions within a givennetwork of relations thatlink the system actors. Theobservable actorsbe theypupils, organizations, ornational governmentsarenot the social structure. Theregular pattern of relationsamong the positionscomposed of concreteactors constitutes the socialstructure of the system.Hence identification of

positions is a necessary butincomplete prelude incomplete network analysis,which requires thesubsequent appraisal of therelations connectingpositions one to another.Positions, or social roles, aresubgroups within a networkdefined by the pattern ofrelations (which representreal observable behaviors)that connect the empiricalactors to each other.

Theorists of social

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structure, such as Linton(1936) and Nadel (1957),usually reserve the term"status" for a position orrole that carries specialrights and duties defined bythe pattern of relations. Allstatuses are positions, butnot all positions arestatuses. By occupyingpositions in a networkstructure, individual actorshave certain connections to

other actors, who in turnalso occupy uniquestructural positions.Although empirical actorsand their observablelinkages provide the datafor identifying positions, anetwork's positions areconceptually distinct fromany specific incumbents.For example, in a hospitalsystem the positionsdefined by patterns ofrelations among actorsgiven

such conventional labels asdoctor, patient, nurse,administrator,paraprofessional, and soforthpersist despitefrequent changes in theunique individualsoccupying these positions.New positions may becreated when an actor(s)establishes a unique set ofties to the pre-existingpositions, for example,when data processingspecialists are hired to

manage the diagnostic andadministrative informationflow of the hospital. Thepoint we are making is thata structural analysis of acomplete network seeks touncover fundamental socialpositions, as defined byobserved relations amongsocial actors. The networkanalyst's task is to use thenetwork relations to mapthe empirical actors into thelatent positions. In theprocess, the complexity of

the network is typicallysimplified, reducing a largenumber of N actors into asmaller number of Mpositions, since typicallyseveral empirical actorsoccupy the same position(e.g., many doctors, manynurses, many patients, andso on).In deciding the basis onwhich to identify thepositions in a completenetwork and to determine

which actors jointly occupyeach position, the networkanalyst has two basicalternatives (Burt, 1978).The first criterion is socialcohesion. Actors areaggregated together into aposition to the degree thatthey are connected directlyto each other by cohesivebonds. Positions soidentified are called"cliques" if every actor isdirectly tied to every otheractor in the position (i.e.,

maximal connection), or"social circles" if the analystpermits a less stringentfrequency of direct contact,say, to example, that anactor need have direct tiesto only 80% of the positionmembers to be included(Alba, 1973; Alba andKadushin, 1976; Kadushin,1968). Note that theweaker social cohesioncriterion will actuallyidentify multiple positions,for example, leaders and

followers.The second criterion foridentifying networkpositions is structuralequivalence (Lorrain andWhite, 1971; White et al.,1976; Sailer, 1978).

Figure 1A Hypothetical Medical Practice Network

Actors are aggregated into a jointly

occupied position or role to theextent that they have a commonset of linkages to the others actorsin the system. No requirement isimposed that the actors in aposition have direct ties to eachother. Thus, a structurallyequivalent position may or may notbe a clique or circle, whereas asocially cohesive position maycontain actors with quite distinctpatterns of ties to the otherpositions.A simple hypothetical example

should make these conceptualdistinctions clearer. Figure 1portrays a fictional medical practicenetwork. The lines connecting theactors represent ''frequent contactson medical matters" (the diagramis an unrealistic representation, butuseful for illustrative purposes). Asocial cohesion criterion identifiestwo distinct cliques, a small oneinvolving just the two receptionistsand a large one containing all threenurses and both physicians. Butusing structural equivalencecriteria, four distinct positions

would emerge, corresponding tothe four roles labeled in thediagram. Nurses and doctors areno longer aggregated because theydiffer in their patterns of contactswith the other actors (i.e., thedoctors are linked to the patientsbut the nurses are not,undoubtedly untrue in a realsystem).

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There of these structurallyequivalent positions are alsocliques, but the patientposition is not a cliquebecause its occupants donot discuss medical mattersamong themselves. Thepoint of this exercise is thatdifferent criteria foridentifying structuralpositions in networks can,and usually do, yielddifferent results. The choice

of methods for locatingpositions in an empiricalnetwork ultimatelydepends, as in theapplication of any method,on the substantive andtheoretical problem theanalyst is addressing. Forsome purposes a cliqueapproach will be preferred,while in other situations astructural equivalenceprocedure will be moreuseful. To state a definitiverule about which one to

choose that would cover allsituations is impossible.Burt (1978: 209) presentsseveral reasons why thestructural equivalencecriterion is usually preferredas the basis for identifyingstructural positions:

Relative to subgroupsbased on cohesion, thosebased on structuralequivalence: (1) include abroader range of types ofsubgroups, (2) extend thescope of types of

subgroups in whichhomogeneity of attitudesand behaviors can beexpected, (3) have amore consistent meaningas operationalized byavailable computeralgorithms, (4) can besubjected to statisticaltests of goodness-of-fit,(5) are accordingly morerobust over random errorin relations, and (6)provide a basis forsampling populationnetworks in largesystems.

He further argues thatcliques can be consideredspecial cases of structurallyequivalent positions if anassumption is made ofsimilar types of links toexternal actors. We shallreturn to the importantconcept of network positionin Chapter 4 when wediscuss methods fordetecting cliques andstructurally equivalentsubgroups in empiricaldata. Without prejudging

the relative importance ofthe two criteria, weunderscore the centrality ofthe structural positionconcept to the wholenetwork enterprise.

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3.Data CollectionAmid the seeming confusionof our mysterious world,individuals are so nicelyadjusted to a system, andsystems to one another, that,by stepping aside for amoment, a man exposeshimself to a fearful risk oflosing his place forever.Nathaniel HawthorneWakefield

In any empirical research,the investigator mustattend to two matters:sampling andmeasurement. Bothrepresent potentially acuteproblems when theintention is to use networkanalysis. Boundaryspecification presents yet athrid, and unique, problem.Because setting theboundaries is the startingpoint in the collection ofnetwork data, we begin our

discussion with this topic.

Boundary SpecificationThe matter of boundaryspecification (or "delimitingthe graph") can be simplyput: Where does one setthe limits when collectingdata on social networks thatin reality may have noobvious limits (Barnes,1979: 414)? Laumann et al.(1982) presented a cogentdiscussion of the centralissues, organized around a

primary dimension of realistversus nominalist views ofsocial phenomena. In arealist approach toboundary specification, thenetwork analyst adopts thepresumed subjectiveperceptions of systemactors themselves, definingthe boundaries of a socialentity as the limits that areconsciously experienced byall or most of the actorsthat are members of theentity (e.g., a family,

corporation, socialmovement). In a nominalistperspective, networkclosure is imposed by theresearcher's conceptualframework that serves ananalytic purpose, forexample, defining a socialclass as all workers having acommon relation to a modeof production. The extent towhich subjective awarenessand analytic impositionproduce coincidentboundaries is, of course,

always an empiricalquestion.To illustrate the problematicnature of networkboundaries, consider theflow of political informationamong citizens during anelection campaign. This isnot a new researchquestion: Lazarsfeld and hiscolleagues (1948) tried toidentify social influences onpolitical attitudes

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in the 1940 presidentialcampaign. Because theysimply asked their randomlychosen respondents toname the people fromwhom they receivedpolitical information, theywere able to distinguish thepersonal attributes of"opinion leaders" from"opinion followers," but theycould not identify theunderlying social structure

through which conversationpresumably took place (forexcellent critiques, seeEulau, 1980; Sheingold,1973).Network analysis representsa potentially more powerfulstrategy for understandingthe process by whichpolitical ideas move amongvoters. It entails initiallychoosing a random sampleof citizens and asking themto indicate the persons from

whom they receive politicalinformation and to whomthey give it. (Althoughnetwork analysis focuses onrelations, such linkagescannot be sampled directly,so the usual procedure is toidentify the relevantpopulation of actors, draw asample of actors, and eitherask about or directlyobserve their relations.)"Snowball sampling"outward (Goodman, 1961),the researcher might then

interview each of thenamed persons ("first zone"respondents) as well asthose persons the first zonerespondents name. Butwhat about third, fourth,and tenth zone persons?Where does a researcherstop? Unfortunately, thereis no criterion for decidinghow many zones tosnowball outward from theoriginal sample. And nomatter where theinvestigator stops, complete

enumeration almostcertainly will not have beenachieved. The decisionabout where to draw theboundary must ultimatelybe set by social-theoreticconsiderations of thephenomenon underinvestigation. For someproblems such as politicalinformation a two-zonesnowball might beconsidered adequate, whilefor other problems such asemployment opportunities a

four-or five-zone extensionmight be judged relevant.In any research problem,the failure to attain acomplete enumeration ofthe relevant network canproduce significantdistortions of the resultingpicture of social structure,so it is always advisable toerr on the liberal ratherthan the conservative sidein setting boundaries.The problem of delineating

a political communicationstructure is exemplified insmall-world phenomena,captured by the commonlyused expression, "It's asmall world." When askedto transmit a message to aspecified target, even aclerk in Nebraska can reachan unknown stock broker inMassachusetts with anaverage of only 5.5intermediaries (Milgram,1967; Travers and Milgram,1969; also Lin et al., 1977;

1978). In short, networkscentering on individuals aresimultaneously everexpanding and overlapping,so that persons can reachalmost anyone

Figure 2Communication Patterns from Snowball Samplesin large social systems in very few steps(unless the individual is an isolate, i.e., heor she neither gives nor receives politicalinformation).The tendency of real networks to ramifyendlessly can pose serious obstacles toreaching substantive conclusions fromnetwork data. Assume, for example, that aresearcher identifies the structure ofpolitical communication on the basis of aninitial sample and its first two zones. Asshown in Figure 2, actor 1 is one of theinitial respondents that has been asked to

name other actors to and from whom heor she directly sends and receivesinformation about politics. In turn, theprimary contacts of actor 1 (that is, actors2 through 5) were asked to name theircontacts, thereby adding actors 6 and 7 tothe network. If the snowballing isterminated at this point, actor 4 wouldemerge

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as an opinion leader, sincehe or she receivesinformation from threenetwork members.Suppose, as shown inFigure 2b, the snowballingwere continued by askingactors 6 and 7 to nametheir political discussionpartners. Now, actor 6appears to be the leader,since he or she has addedthree new sources and is

thus the recipient of fourcontacts. Subsequentsnowballing involving theseadditional actors couldfurther alter the networkstructure. Theconsequences of truncatingempirical networksgenerally will not be asdramatic as this contrivedexample, but networkresearchers must besensitive to the possibility.Researchers using a

nominalist approachcommonly study laboratorygroups or formally boundednatural groups, such asclassrooms or organizationalunits, as a means of settinglimits on networks (seeHallinan, 1978; Homans,1950; Miller et al., 1981).This strategy, though usefulfor many purposes, suffersfrom two potentialshortcomings. First,investigation is restricted torelatively small groups,

which precludes the kind oflarge-scale research thatcharacterizes contemporarysocial science. Second,small-scale settings maysolve the boundary problemin appearance only. Ananalyst who examines theeffect of school childrens'classroom interactions ontheir academicperformance, for example,arbitrarily omitsrelationships that occuroutside the classroom (e.g.,

playground activities). Ifthese latter interactionssomehow conditionacademic performance,conclusions based on in-classroom observationsalone may be incorrect.Ultimately, the researcher'stheoretical concern mustdictate which relationshiphe or she observes.A realist approach tospecifying the boundarieswould apply the criterion of

mutual relevance (Laumannet al., 1982). This criterionsays that only actors whoare relevant to each other(as defined by thesubstantive question)should be included in thesocial network. Actorswhose actions or potentialactions are inconsequential,either because they haveno interest in thesubstantive area or becausetheir significance is trivial,are excluded. Application of

the consequentialitycriterion can be problematicin situations in which actortriviality is difficult todetermine beforehand.However, for somepopulations of actors apriori evidence is available,for example, publishedstatistics on the major firmsin an industry or experts'opinions about the leadingcontenders for a politicaloffice.

Knoke and Laumann (1982)used the mutual relevancecriterion to delimit theactors in two national policydomains: energy andhealth. They ignored actorswith trivial capacity to affectthe behavior of elites

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within the domain, butrecognized that inactionitself is insufficient reasonto exclude an actor, sinceother members of thedomain may anticipate thereactions of those notdirectly involved. To applythe theoretical criterion ofmutual relevance, theydrew on four types ofempirical evidence: (1)positional (i.e., formal

organizations that haveprima facie functions orinterests in the domain),(2) decisional (organizationsor groups appearing atcongressional hearings orwhose actions andstatements are reported inthe national press), (3)reputational (groups andorganizations judged to beinfluential by a panel ofexperts), and (4) relational(groups and organizationsnamed during the

interviews withrepresentatives oforganizations obtained bythe first three criteria).Though the mutualrelevance criterion does notalways set precise anddefinite boundaries, itshould be increasinglyvaluable for the study oflarge-scale systems,especially those consistingof policymakers.We have discussed

boundary specification first,and for good reason. Muchsocial science research isbased on random samplesand thus does not require aprecise statement aboutwho or what is the focus ofinvestigation. The collectionof network data is not"business as usual." Not tospecify carefully theboundaries of a socialnetwork before datacollection can lead to direand costly consequences.

Sampling of Networks inLarge PopulationsOften it is necessary tomake statements aboutmembers of a largepopulation, such as anation's votes or a largecommunity's citizens. Insuch instances, of course, acomplete enumeration ofthe network is impossibleand sampling necessary.Incomplete enumerationalways runs the risk of

inaccurate measurement ofnetwork structure.Unfortunately, the samplingof network data isproblematic. Burt stated theproblem well (1981: 314-315):

The network data ignoredin a random survey of k%of a population is roughly(1k)% so that a randomsurvey of 10% of apopulation ignores 90% ofits network data, arandom sample of 25%

ignores 75% and so on.Those lost data aresignificant. To whatextent do the nonsampledpersons reciprocaterelations directed to themfrom the sampledrespondents? Arerespondents the object ofstrong relations from thesystem as a whole or arethey relatively isolatedfrom the system? How arethe nonsampled

Page 27persons interconnected,apart from therespondents? Answers tothese questions define thenetwork context of therandom sample, but thetypical survey researchdesign obliterates thatcontext from our view.

Since the number ofpotential (symmetric) ties ina population of size N isN(N1)/2, even estimatingthe acquaintance volume

(number of people known)in a small town of 10,000poses insurmountabledifficulties, given 50 millionpotential acquaintance pairsto be investigated.In short, the problem isthis: Network concepts andmethods of analysis usuallyrequire data on therelations among all actors inthe system. When thesystem is too large to studyin its entirety, the only

alternative is to estimatethose relations of interest asaccurately as possible bymeans of an appropriatenetwork sampling strategy.No completely satisfactorystrategy currently exists.On the positive side,scholars interested innetwork sampling havemade great strides in recentyears. Ove Frank (1971,1978, 1979), a Swedishmathematician, deserves

credit for much of theprogress in networksampling theory (see alsoCapobianco, 1970; Proctor,1979). His workconvincingly demonstratedthe possibility of estimatingthe number of ties in apopulation of size N given asample size of n.Because our purpose is toprovide an overview oftopics for potentialcollectors of network data,

we cannot review Frank'sabstruse work. It hasinfluenced others' efforts tobring the collection ofnetwork data within therealm of standard datacollection procedures.Granovetter (1976)proposed that a researcherdraw a random sample ofactors, make a list of therespondents' names, andask each respondent whomhe or she knows on the list.The interviewer may simply

ask the respondent toindicate acquaintance withthe persons on the list ormay seek more detailedinformation such as thenature of the relation, itsintensity, duration, and soforth. The observed densityof ties betweenrespondents is an unbiasedestimate of the density ofthe entire network fromwhich the sample wasdrawn. Erickson et al.(1981) demonstrated the

feasibility of such samplingand measurementprocedures in a moderatelylarge population (a 400-member bridge club).A potential user of thissampling procedure shouldbe aware of its limitations(Morgan and Rytina, 1977;Granovetter, 1977). It doesnot produce data that canbe used at the level of theindividual. The procedure'svalue lies principally in

obtaining an estimate of thedensity of relations in theentire population of actors,using a random

sample drawn from a list of names (forexample, from a voluntary association'smembership roster or from a commerciallypublished directory of a city as large as100,000 or more residents). It may alsoprove useful in estimating density withinand between subgroups (blacks andwhites, for example), although largersamples will probably be necessary to getstable parameters. Moreover, because theprocedure consists of asking therespondent about every other person inthe sample, the investigator must haveaccess to actual names. Simply identifying

access to actual names. Simply identifyinghouseholds, which is all that standardrandom surveys require, is inadequate forcollecting the necessary networkinformation. Other problems, such aspossible interviewee fatigue and missingdata, apply more generally to networkanalysis, and thus we postpone adiscussion of them until the followingsections.A second suggested sampling procedurebuilds on standard random surveys(Beniger, 1976; Burt, 1981). Traditionalsurvey research assumes that selectedpersonal attributes (race, education, age)adequately capture social stratification;

adequately capture social stratification;i.e., they are good surrogates for actualrelations. The essence of the samplingprocedure is to collect information fromeach respondent on the attributes ofpersons to whom he or she goes for eachtype of relation studied. The researcherthen tries to identify combinations ofattributes (including those of therespondents) so that individuals fallinginto each category have similar relationswith others.Assume, for example, that the researcherbegins with two theoretically importantattributes: race (white and black) and age

attributes: race (white and black) and age(young and old). The two attributes givefour possible combinations:

White(w)

Young(y)Old(o)

Let zj be the total number of persons thatrespondent j cites during his or herinterview, and let fj,wy be the number ofcitations to persons who are white andyoung (to take the first combination).Then the proportion of respondent j'srelations that involve persons who arewhite and young is given as

white and young is given aszj,wy = fj,wy/rj

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Assume, furthermore, thatrespondent j is black andold. It is possible to sumrelations to white andyoung people across allrespondents who are blackand old:

where dj,bo is a dummyvariable equal to zerounless respondent j is blackand old. If no black and old

respondent cites someonewho is white and young,f(bo,wy) equals zero.Dividing by nbo, thenumber of black and oldrespondents, givesz(do,wy) = f(bo,wy)/nbo

the mean percentage ofblack and old respondents'citations that are directedtoward people who arewhite and young.Both respondent j's

relations to persons witheach of the four attributecombinations and therelations to persons withrespondent j's attributesfrom all other respondentsare of interest. The totalvector of relations forrespondent j, in otherwords, iszj = (zj,wy, zj,by, zj,wo,zj,bo, zwy,j, zby,j, zwo,j,zbo,j)where the relations from

him or her are given by thefirst four elements, andrelations to him or her bythe last four. Note thatzj,wyzj,bo are based on tiesinvolving respondent jpersonally, while zwy,jzbo,jare aggregate relations tohis or her combination ofattributes rather than tohim or her personally.Individuals with identicalattributes are assumed tobe the recipients ofidentical relations.

What is important to thisprocedure, then, is not theindividual per se, but his orher combination ofattributes. The task is toidentify combinations suchthat people with eachcombination shareequivalent relations withothers, the others alsobeing defined in terms ofattribute combinations.Chapter 4 defines, inpreciseterms, structurallyequivalent relationships.

The most crucial step in thissampling procedure isidentification of all relevantattributes that presumablystratify the population (ourexample is simplified,needless to say). To theextent that relevantattributes are omitted,people will be defined asequivalent when in factthey are not. As Burt(1981: 331) puts it, "[A]good deal of iterative dataanalysis will be required in

order to specify theparameter set."

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Investigators commonlycollect network datawithout serious attention tosampling and then simplymake whatever limitedinferences they can fromthe data. While pastreliance on this willy-nillyapproach isunderstandable, given thestate of the art, it willundoubtedly become lessacceptable as developments

in network sampling theoryand applications continue.Finally, a brief observationabout the use of statisticalinference in networkanalysis. Conventionalstatistical analyses of socialdata are permised on therandom sampling of units ofobservation, for example, incross-sectional surveys of apopulation of persons.Network data, consisting ofrelations among

interdependent actors,clearly violate the randomsampling assumption andmake problematic theapplication of conventionalstatistical procedures. Atpresent, the bases forstatistical inferences fromnetwork data are poorlyunderstood, and we canonly advise cautious useand hope that the problemswill eventually be resolved.

Generic Types of Measures

and ReliabilityAs mentioned in Chapter 2,relations can vary in formand content. Distinctionsamong relational forms at ageneral level can bemeasured as the strengthof the link between pairs ofactors and their level ofjoint involvement in thesame activities. Strengthmight be measured as asimple dichotomy (presentversus absent connection)

or on a finely gradedquantitative scale (e.g., thenumber of interactions overa specified time). Mutualinvolvement may also bemeasured with varyingdegrees of specificity (e.g.,the frequency with whichcontacts are initiated byone actor toward another,such as visiting, helping,instructing).Assuming that a researcherhas determined the forms

of relations and theircontents, any of severaldata collection proceduresmight be employed. Onemethod, of which theadvantages anddisadvantages are wellknown, is directobservation. On the positiveside, the data will almostcertainly be valid, unlessthe investigator'sparticipation changes thebehavior of the individualsbeing studied. Similarly, the

researcher will understandprecisely what the data canmean: Misinterpretationand misrepresentation areunlikely. On the negativeside, direct observation isfeasible only when thesystem is small; those whouse the laboratory or studyrelatively contained units(such as a bureau in anorganization or a primitivevillage) will find the methodmore valuable

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than those who study largecommunities or nation-states. Moreover, thebehavior about which theresearcher collects datamay occur infrequently.frequently. Thus oneanthropologist, in an effortto explain why a birthcontrol method diffusedmore slowly in an Indianvillage than did a variety ofwheat, had to live in the

village for more than a year(Marshall, 1971).Archival records representthe opposite extreme fromdirect observation. Theresearcher is completelyremoved from the behavioror event being studied, andthe data are usuallyrecorded for purposes otherthan scientific research.Unobtrusive measuresobtained from archivalrecords have two unique

advantages. First, they canprovide network data thatwould otherwise not beobtainable because peoplehave died, potentialrespondents refuseinterviews, or organizationsdissolve themselves.Second, archival data may,under the best ofcircumstances, span anextended period. Burt(1975; Burt and Lin, 1977),for example, was able toexamine

individual/corporationrelationships from 1877 to1972 by content analyzingthe New York Times.Sociologists often usearchival data to identifyinterlocking directorates(i.e., common membershipon boards of directors); theunderlying assumption isthat individuals who sit onthe same boards sharecommon interests (Allen,1974; Burt, 1979; Levine,1972; Mariolis, 1975).

Given the general popularityof survey data among socialscientists, it is hardlysurprising that self-reportsin surveys are the mostcommon source ofinformation on socialnetworks, Self-reports alsopose some of the mostsevere measurementproblems. One potentialdifficulty is recall. Considerthe matter of asking peopleto identify those with whomthey ''discuss politics."

Because politics is not verycentral to people's lives, itwould be unrealistic toexpect perfect recall ofpolitical conversations. Theproblem is not confined torelatively nonsalient topicssuch as politics. In a seriesof articles, Bernard andKillworth (1977, 1978;Killworth and Bernard,1976, 1979) argue that therelations in which peoplesay they are involvedscarcely resemble those in

which they are actuallyinvolved. Contrastingindividuals' reports ofinteractions with directobservations of behavior,the authors conclude(Bernard et al., 1980: 28):

We are now convincedthat cognitive data aboutcommunication cannot beused as a proxy for theequivalent behavioraldata. This onefundamental conclusionhas occurred

systematically with

Page 32a variety of treatments,all as kind to the data aspossible. We musttherefore recommendunreservedly that anyconclusion drawn from thedata gathered by thequestion "who do you talkto" are of no use inunderstanding the socialstructure ofcommunication.

This strong indictment, iftaken literally, questions the

validity of survey-basednetwork data. There are,however, good reasons tobelieve that Bernard andKillworth's conclusion is nottotally warranted. Thegroups they study arehardly typical: (1) 60 blindpersons in the Washington,D.C., areas who were linkedby teletype; (2) 40employees of a small socialscience research office; (3)34 persons in a graduateprogram at West Virginia

University; (4) 58 residentsin a West Virginia fraternity;and (5) 44 amateur radiooperators who belonged tothe Monongalia WirelessAssociation. The level ofinteraction among allmembers of such groups islikely to be uncommonlyhigh; asking individuals toseparate very frequent fromfrequent interactions maybe splitting hairs. Moreover,in several instances theform of direct observation

consisted of a projectresearcher walking amongthe subjects every 15minutes and coding thefrequency with whichpeople contacted eachother. This technique,which is used to challengethe validity of surveynetwork data, is itself opento question. To whatextent, for example, did the"unobtrusive" observerdisrupt ongoing activities?How accurately was he able

to record interactions? Andwhat about interactionsduring the unobservedtimes (Rogers and Kincaid,1981: 120-122)? Finally,Burt and Bittner (1981)challenged the analysis thatBernard and Killworthperformed on some of theirdata. The telling talenonetheless remains:Inaccuracy in the naming ofrelevant contactsrepresents a potentiallyserious source of

measurement error.There are ways to reducethis type of measurementerror. The most effective isto be as precise and specificas possible when definingthe content of the networkbeing elicited (McCallisterand Fischer, 1978).Consider the matter ofasking respondents to namethose persons with whomthey "discuss politics."Political discussion means

different things to differentpeople. Some may define itonly in terms of candidatesand elections, others just interms of corruption, and stillothers may include allfacets of government (butwhich level?). The key tomeasuring equivalent ties isto ask explicitly aboutcandidates and election,and corruption, and wellgovernmental problems.

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Because recall is difficult,especially in an interviewsituation, the interviewershould provides cues tohelp the respondent. Whenasking about people withwhom the respondent talksabout candidates, forexample, the interviewermight ask specifically aboutpeople at work, at church,or with whom therespondent spends leisure

time.A final consideration in thecollection of survey networkdata is interviewee fatigue,which might become aproblem if a respondent hasan extensive network forthe particular relation (ormultiple relations) underinvestigation. Faced withnaming and providinginformation on scores ofpeople, the respondent maytire quickly, although

practical experiencesuggests that manyrespondents will patientlycomply for hours on endwith researchers' repetitivequestioning. Indeed,interviewer fatigue may setin before some respondentscall it quits! One frequentlyused remedy is to ask therespondent to provide twoor three names (seeLaumann, 1973). This"solution" no longer hascredibility among network

analysis for the obviousreason that it can severelytruncate the networks ofactors who are extensivelyinvolved with others.Fischer (1982; McCallisterand Fischer, 1978) hasrefined a technique,developed by Laumann(1966, 1973), that providesdetailed networkinformation on up to thirtyindividuals within 20minutes of interview time.

The first step is to ask therespondent how often(usually, sometimes, orhardly ever) he or she talksabout personal matters withsomeone, and to give thefirst names of theseindividuals. The purpose ofthe second step is toidentify network membersfrom a variety of socialcontexts (work,neighborhood, recreation,and so on). Among the tenname-eliciting questions

Fischer uses are: whowould care for therespondents' Homes if theywent out of town? If theywork, with whom do theytalk about work decisions?With whom do they talkabout hobbies? Throughoutthe first two steps, theinterviewer records thenames (no more than tenper question). As part ofthe third step, theinterviewer hands the list tothe respondent and asks

him or her if there is anyoneelse important to him or herwho is not on the list. Withcompleted list in hand, theinterviewer then elicitsadditional information oneach named person, suchas sex, the role relation tothe respondent, andwhether the person liveswithin a 5-mile radius of therespondent. In the finalstep, the interviewer selectsa subsample of the elicitednames (between three and

five) and has therespondent fill out a self-administered questionnairethat asks more detailedinformation (employmentand marital status, forexample). For each

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pair of names, theinterviewer also askswhether the two "knoweach other well," whichprovides a rough measureof the respondent's networkdensity.Fischer's general format(though perhaps not hisspecific questions) will beespecially valuable to thosewho intend to begin with arandom sample of

individuals and obtaindetailed information ontheir networks, as seen bythe respondentsthemselves. Itcomplements, for example,the sampling procedurethat Burt proposes. On theother hand, the format willbe less useful toinvestigators who intend tosnowball the sampleoutward since it does notelicit last names oraddresses.

Fatigue represents an evengreater potential problemwhen the interviewer givesthe respondent a long list ofnames and asks him or herto indicate which people onthe list of or she knows. Yetas we noted earlier, thislong and boring task maybe necessary if networksampling theory is to be putinto practice. Based on theirstudy of competitive bridgeplayers in Toronto, Ericksonet al. (1981: 135)

concluded that "one cansafely use network samplingwith lists of at least 130names. In an interviewsetting, lists of 150 shouldbe feasible, but lists as longas 200 may well be toogreat a burden even forwilling respondents."However, Knoke andLaumann (1982) found thatnational energy policydomain elite informantswillingly responded toseveral repeated uses of

lists involving more than250 organization names,preclassified intosubstantive types (e.g., oilindustry, electric utilities,congressionalsubcommittees). Erickson etal. (1981) also madeseveral recommendations toincrease respondentenjoyment and responserates:

(1) Make abrief.introductorystatement about the

purpose of the task in therespondents' own terms(i.e., "we are interested inknowing how people in acommunity get to knoweach other").

(2) Include therespondent's name in thename list.

(3) Pretest the relationalquestions to ensure thatthey are unambiguous(i.e., "does the person livein Jonestown" rather than"does the person livearound here").

Given a list of 130 names,the investigator should allotat least 15 minutes ofinterview time to completeit.

Missing DataInvestigators should besensitive to the problem ofmissing data. As in anyresearch method,respondents may refuse tobe interviewed

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or may not know theanswers to certain queries.In conventional cross-sectional surveys,population parameters canfrequently be estimatedwith great accuracy despitea large number of missingcases, assuming that thereasons for thenonresponse are unrelatedto the variables understudy. For network analysis,

however, the consequencesof each missing case aremore severe, becauseeliminating a case alsoremoves the N 1 possiblerelationships involving othernetwork actors. Obviously,such estimates as networkdensity can be distorted ifeven a handful of cases ismissing. For example, in thenational energy policydomain two federalorganizationsthe EconomicRegulatory Administration

and the Federal EnergyRegulatoryCommissionturned out to bemajor brokers in the flow ofpolicy information. Failureto include these two unitsamong the 200 interviewedin the project could haveresulted in a much morefragmented picture of thenetwork than actuallyemerged.No failsafe solution to themissing data problem

exists. Often one can askrespondents not only abouttheir behavior but aboutother actors' behaviortoward them, for example,not only to whom they giveassistance but also fromwhich actors they receive it.Then at least a portion ofthe missing cases' relationscan be reconstructed fromothers' reports. Thepotential damage of missingdata argues for great careand extraordinary effort in

convincing respondents ofthe importance ofparticipating in theresearch. Elite andorganizational networkstudies, for instance, haveoften exceeding 90%response rates,substantially higher thantypical household surveys,by using personal letters,telephone contacts,reassurances of confidentialtreatment, and therespondent's own network

contacts as go-between(Laumann et al., 1977;Knoke and Wood, 1981).

4.Methods and ModelsOnly connect!E.M. ForsterHoward's End

This part is concerned withtechnical matters. Wepresent a variety ofprocedures for describing

and analyzing networkdata. Two excellent similarreviews are Burt (1980,1982: ch. 2). The former isorganized

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around a sixfold typologyformed by crossing twoanalytic approaches(relational versus positional)with three levels of actoraggregation into units ofanalysis (single actors,subgroups, and structuredsystems). Although thistypology is insightful, ourpresent approach is lesscomplex. We begin with anexposition of the

elementary terms for visualand algebraicrepresentations of networkdata and progress throughincreasingly moresophisticated and complexmethods for describingrelational systems andtesting hypotheses aboutnetwork structure. Weconclude with a look atunresolved issues.Throughout this chapter,the various methods andmodels are illustrated with a

basic set of data, describedin the following section.

An InterorganizationalRelations ExampleThe technical procedureswill become moremeaningful to many readersby applying methods to anillustrative data set. For thispurpose, we chose aportion of the networkinformation collected from95 Indianapolis formalorganizations as part of a

larger study of voluntaryassociation behavior (Knokeand Wood, 1981; Knoke,1981, forthcoming). In1978, questionnaires weresent to leaders of a set ofprivate firms, governmentagencies, and voluntaryassociations. Therespondents were asked tocheck off on lists oforganizations the ones withwhich their organizationshad engaged in thirteendifferent types of

relationships during thepast two years.For the present monograph,we selected the subset often organizations shown inTable 1, and combined fourof the types of relations intotwo networks: (1) to andfrom which organizationsdid the respondent'sorganization send or receive"information aboutcommunity affairs"; and (2)to and from which

organizations did therespondent's organizationsend or receive "money orother material resources." Ifeither organization in a pairreported that a type oftransaction occurred, it wastreated as an establishedexchange relationship. Therelations in these twonetworks are simply"present'' or "absent":Neither the level nor thefrequency of a transactionis given. Hence the money

and the informationnetworks are both binary inform. Although as discussedin Chapter 3 networkrelations can beoperationalized in manyways, binary measuresprovide the most basicform, and many of theanalyses described in thischapter can be mostusefully applied to this typeof relational data.

Indianapolis Organizations Used in Network ExamplesOrganization NameCity/County CouncilChamber of CommerceBoard of EducationLocal IndustriesMayor's OfficeWomen's Rights OrganizationStar-NewsUnited WayWelfare DepartmentWestend Organization*GOV = government; VOL = voluntary; PVT = private profit making

The example is chosen mainly to clarify the use of variousanalytical techniques and to illustrate how the can beinterpreted in a real situation. The data do not constitute ameaningful, closed system of network actors and no substantiveconclusions should be drawn. Rather, the analyses reportedbelow are to be regarded as purely illustrative, based on adeliberately simplified set of actors selected for the sake ofclarity.Table 1 also reports each organization's influence reputationscore. A panel of 24 community informants was asked toevaluate all organizations in the project on a 7-point scaleaccording to "how much influence and organization has inachieving its own objectives, and not how widespread itsinfluence is in the entire community." Thus informants wereasked to judge both the intentions and the actual success of

asked to judge both the intentions and the actual success ofthe 95 organizations. For the ten organizations in the presentexample, the median ratings by the judges are reported,ranging from 6.40 for the newspaper to 2.50 for the Women'sRights Organization (a pseudonym).

Visual DisplaysMany terms in network analysis suggest spatial or geometricrepresentations, for example, centrality, periphery, boundary,distance,

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isolation. Moreno's (1934)pioneering sociometrictechniques emphasizedconstruction of thesociogram, a two-dimensional diagramdisplaying the relationsobserved among actors in asystem (e.g., pupils in anelementary schoolclassroom). Although, asnoted below, networkdiagrams have limited

usefulness, a didacticadvantage can be gainedby discussing some basicnetwork terms indiagrammatic format beforeproceeding to morepowerful matrixrepresentations.In a sociogram, actors arerepresented by a set ofpoints, often labeled byidentifying names, letters,or numbers, and the set ofrelations linking them are

represented by lines (also,arcs or edges) drawnbetween pairs of pointshaving direct connections.These two primitiveelements also define agraph (Harary, 1959), thusallowing the application ofmany terms and theoremsfrom graph theory (seeHarary et al., 1965; Harary,1969; Flament, 1963;Behzad and Chartrand,1971).

Well-constructed visualdisplays of network relationsoften have a dramaticimpact on viewers and canconvey an intuitive feel forthe structure of a system.Unfortunately, a virtuallylimitless number ofdiagrams can be drawn thatcontain the same relationalinformation but impartstarkly differentimpressions. Furthercomplicating the situation,as the number of actors

and the number ofconnections increase,parsimony andinterpretability of diagramsrapidly dwindle. Sociogramconstruction is essentiallyan art form, despitesporadic and unsuccessfulefforts to specify invariantprocedures. (However, seeKlovdahl, 1981, for aproposal to adapt acrystallographic data-plotting program to producethree-dimensional,

computer-drawnrepresentations of networkscapable of rotation.) Forthese reason, ourillustration focuses only onthe ten-organizationdiagram of the moneyexchange network, shownin Figure 3. Although onlyabout a quarter of the 90possible linkages arepresent in this network,crossing three lines wasunavoidable. An attempt todiagram the information

exchange network, whichhad twice as manyconnections among the tenactors, proved to be futile.A superficial inspection ofFigure 3 suggests that thecity council, local industries,and the mayor are the mainsources of funds, while theUnited Way, board ofeducation, and welfaredepartment are the mainrecipients. Additional detailsof the network structure willbe revealed in the following

discussion of technicalterms.If all N2 N possible linesbetween the set of N pointsare present, a graph iscomplete. Obviously, thegraph of the moneyexchange network is farfrom complete. Two pointsare adjacent if a linedirectly

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Figure 3Sociogram of Money Exchange

connects them, althoughrestriction of the diagram totwo dimensions frequentlyrequires adjacent points tobe located quite far apart(e.g., the mayor andchamber of commerce). Inthe example, every point isadjacent to at least oneother point, except for theWomen's RightsOrganization (WRO), whichis unconnected.

The money exchangenetwork is a special type ofgraph, a digraph (directedgraph), consisting of the Npoints linked by a set ofdirected lines. The directionis indicated by thearrowheads; that is, anarrow emerges from theactor initiating the relationand terminates at the actorreceiving the relation. In adigraph, three types of linesoccur between the (N2 -

N)/2 pairs of points: (1)mutual, with both pointsdirecting lines toward eachother, shown by two-headed arrows (e.g.,

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the welfare department andUnited Way); (2)asymmetric, in which oneactor directs a line towardanother that is notreciprocated (e.g., all otherlines in Figure 3); and (3)null, in which no line ineither direction existsbetween a pair of points(e.g., the nine null relationsof the WRO with the otherorganizations). In all, Figure

3 shows 1 mutual, 20asymmetric, and 24 nulllinkages.A path exists between twopoints if a sequence of lineslinks them through anintervening set of points.That is, all points in theintervening set must besequentially adjacent. Forexample, a path betweenpoints A and D mightinvolve the sequence oflines AB, BC, and CD. When

a graph does not indicatethe direction of therelations, any suchsequence of lines can beused to trace a pathbetween the pairs of points.But if a graph is a digraph,the network researcher mayrestrict analysis only todirected paths, thosesequences in which thedirection of the arrowsnever reverses. Forexample, in Figure 3 theonly directed path from the

chamber of commerce tothe Westend Organization isthe sequence COMM-EDUC,EDUC-UWAY, UWAY-WEST.A cycle is a closed path,one that can begin and endwith any of its points (e.g.,the cycle involving UWAY,WELF, and EDUC).Path distance is theminimum number ofsequential lines that mustbe traversed to link twopoints, i.e., the length of

the shortest path. The pathdistance between thechamber of commerce andWestend is thus three. Anonzero path distancebetween two points meansthat one actor is reachablefrom another. For example,in a communication networkreachability means that achannel exists by which amessage can betransmitted from a givenorigin to a givendestination. In Figure 3, the

reachable set for moneycontributed by localindustries includes six otheractors (NEWS, COMM,EDUC, WELF, UWAY,WEST), while the reachableset for the mayor includesthese same organizationsexcept for the newspaper.Connectedness is acharacteristic of pairs ofpoints, indicating how twopoints may be linked bydirected lines: 0-connected

points have no directedlines joining them in eitherdirection (i.e., a pair of 0-connected points is notreachable), 1-connectedpoints are joined by linesdisregarding their direction:2-connected points arejoined by a path in onedirection but not inanother; and 3-connectedpoints are joined by pathsin both directions. In Figure3, the chamber ofcommerce is 1-connected

to the welfare departmentby a path distance of two(with EDUC or INDU as anintervening point), and is 2-connected by a path ofdistance 3 (EDUC andUWAY as interveningactors), but no

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3-connection occursbetween these two actors.Connectedness at the levelof the entire graph can becharacterized in similarterms: a graph is stonglyconnected if every pair ofpoints in it is 3-connected;it is unilaterally connected ifevery pair is 2-connected; itis weakly connected if allpairs are 1-connected; andit is a disconnected graph if

at least one point is 0-connected (unconnected)with all others. Figure 3 isobviously a disconnectedgraph, but if WRO isignored, it is only weaklyconnected.Two additional conceptsfrom graph theoryapplications to networksconcern the effects of singlechanges of elements.Removal of a point involvesdeleting a point and its

associated lines. If a point'sremoval results in adisconnected graph, thepoint represents a cut pointin the network, and theactor presumably plays aliaison or brokerage role inthe system. Removal of aline involves deletion of asingle connection betweentwo points. If adisconnected graph results,the line represents a bridgebetween system actors. Themoney-exchange network in

Figure 3 contains noinstances of cut points orbridges, but thehypothetical networksdiagrammed in Figure 4illustrate these situations.In network a, actor 4 is acut point, because his orher removal results indisaggregation of thenetwork into twodisconnected sub-graphs.In network b, the linkbetween actors 3 and 4 is a

bridge, since its removalagain would result in twodisconnected groups.Granovetter (1973)discussed the importance of"weak ties" as bridges overwhich information,influence, and resourcescan be transmitted incomplex networks.Graph theory contains manyother concepts andpropositions that we cannotbegin to consider in this

general survey of networkmethods. A large literaturein social psychology usessigned graphs, in which thelines have plus or minussigns attached to them torepresent positive ornegative affect betweenactors. Theorems aboutstructural balance in suchgraphs involve multiplyingthe signed lines in cycles(see Heider, 1946;Cartwright and Harary,1956; Flament, 1963).

Another large literatureuses numerical values onthe lines to indicate thestrength or type of relationbetween points, forexample physical distancesor costs of transportinggoods betweengeographical sites (Duffin etal., 1967). Theseapplications are lesspractical for manynoneconomic social sciencesituations in whichmeasures of relations are

less sophisticated. In anyevent, exposition of theseadvanced techniques isbeyond the scope of thepresent volume, althoughwe shall encounter graphtheory ideas below indiscussing clique-detection

Figure 4Hypothetical Networks Illustrating (a) Cut Point and (b) Bridge

procedures. Before then, we must consider how visualnetworks can be represented algebraically.

Matrix RepresentationAn algebraic representation of network relations canexpress all the information embedded in a sociogramand can do a great deal more with the data than ispossible with a visual format (Forsyth and Katz,

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1946; Festinger, 1949).The standard algebraictreatment of network datais a tabular display called amatrix, a rectangular arrayof elements arranged inrows and columns. A matrixis usually denoted by acapital letter marked, e.g.,A. In typical networkapplications of matrices, therows represent systemactors and the columns

represent the same set ofactors in the identicalsequence. Thus, matrix Q,displaying a specified typeof relation among a set ofactors, will usually be asquare matrix of order(N,N). Sometimesnonsquare matrices will beemployed in networkanalysis, with the Mcolumns standing for otherentities (e.g., attributes,events, locations) than theN actors in the rows.

Although we cannot devotespace in this monograph tosuch formats, the interestedreader is advised to consultAtkin (1974, 1977) andother works (Breiger, 1974;Doreian, 1980, 1981; Gouldand Gatrell, 1980).By convention, in networksof directed relations, theactors arrayed in the matrixrows are initiators of thespecified relation and theactors arrayed across the

columns are the recipientsof the relation. Thesubscripts i and j, whichtake integer values from 1to N (the sample size), areused to reference theelements appearing in theith row and jth column of amatrix. Because directedrelations are seldomperfectly reciprocated inempirical social data, mostmatrices of directedrelations will be asymmetric.That is, the element in row

i, column j need not beidentical to the element inthe jth row, ith column.However, if the type ofrelation is an undirectedone, the "from/to"convention is meaningless.In such cases, every tiefrom i to j is also a tie from jto i and the matrix form willbe symmetric. As noted inChapter 3, networkresearchers oftensymmetrize their empiricaldata when it is initially

asymmetric if the type of tiethey are investigating isconceptually reciprocal ormutual (e.g., acommunication channel or aco-worker relation).The matrix elements are N2numerical values thatindicate the nature of thelinkages between every pairof actors in the network.The simplest elements arebinary values, with a "1"standing for the occurrence

of a tie from actor i to actorj, and a "O" standing forthe absence of such a tiebetween the pair. Binarymatrices are also calledadjacency matrices,because they revealwhether or not any twopoints in a correspondingsociogram are adjacent.More complex matrixelements might be integervalues, for example, toreflect the frequency ofcontacts between actors, or

signed interval or ratiovalues, for example, toindicate strengths ormagnitudes of the specifiedrelation. In general, thevariable zijk represents thevalue of a relation from theith actor directed to the jthactor in the kth network.For square matrices, the

Money Exchange MC C E IO O D NU M U DN M C U

COUN 0 0 1 0COMM 0 0 1 0EDUC 0 0 0 0INDU 0 1 1 0MAYO 0 1 1 0WRO 0 0 0 0NEWS 0 1 0 0UWAY 0 0 0 0WELF 0 0 1 0

WELF 0 0 1 0WEST 0 0 0 0Information Exchange I

C C E IO O D NU M U DN M C U

COUN 0 1 0 0COMM 1 0 1 1EDUC 0 1 0 1INDU 1 1 0 0MAYO 1 1 1 1WRO 0 0 1 0NEWS 0 1 0 1UWAY 1 1 0 1WELF 0 1 0 0

WELF 0 1 0 0WEST 1 1 1 0

Figure 5Interorganizational Networks in Matrix Format

diagonal values (the N, zijk elements) correspond to self-directed relations. As network operationalizations oftendo not allow for meaningful self-choices, zmatrix representations.Figure 5 shows the binary matrix representations forboth example networks, the money exchange (M) andinformation exchange (I) systems among the tenorganizations. Note that the labels for the rows andcolumns are not part of the matrix, but are included forthe reader's

the reader's

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convenience. Matrix M isessentially an algebraictranslation of Figure 3. Forexample, reading across thefirst row, one can see thatthe city/county council gavemoney to five organizations,while reading down the firstcolumn, the councilreceived money from none.Similarly, the WRO'sisolation from all nine othergroups is evident in its row

and column of solid 0s.Turning to Matrix I, weobserve immediately that itcontains a larger number ofdirected ties and that everyactor sends and receivesinformation with at leastone other organization.Some simple descriptivestatistics are obtained bysumming across matrixentries. The degree of apoint is the integer count ornumber of other actors with

which a given actor hasdirect contact. Theoutdegree of actor i is thenumber (or proportion) ofrelations from that actor toall others, that is, the sumof 1s within actor i's row:

Similarly, actor j's indegreeis the number (orproportion) of relationsreceived by actor j from allothers, calculated as the

sum of 1s within actor j'scolumn:

Particularly in the case inwhich the relation is anaffective tie (e.g., liking,friendship), an actor'sindegree measures his orher popularity within thenetwork. In acorresponding sociogram,such network "stars" areevident as the recipients of

many directed arrows fromthe other points.Density, a characteristic ofthe entire network, is aproportion that is calculatedas the number of all tiesoccurring in the matrixdivided by the number of allpossible ties (N2 N, if self-directed relations are notpermissible):

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Density ranges between 0and 1.00, representing theextremes of a totallydisconnected or totallyconnected graph.In matrix I, the outdegreesrange from 3 for WRO to 8for the mayor, and theindegrees range from 1 forWRO to 9 for thenewspaper. The density ofnetwork I is 0.54,considerably higher than

the density of M at 0.24.Expressing networkrelations in matrix formyields substantial benefitsover visual display. Perhapsforemost, it facilitates theanalysis of indirect relations.The indirect connectionsamong a set of N actors in anetwork can be uncoveredby raising an adjacencymatrix K to successivepowers, that is, bymultiplying the matrix of

binary ties by itself T times.The resulting elements inKT give the number of T-step connections leadingfrom actor i to actor j. Inmatrix multiplication, eachelement in the resulting(N,N) product matrix isfound by a pairwisemultiplication andsummation of thecorresponding row andcolumn vectors in theoriginal matrix. Forexample, in multiplying

matrix M by itself, the valueof the element in row 1,column 8 of M2 is found bymultiplying thecorresponding ten elementsin the first row and theeighth column of matrix M,then summing these tenproducts:(0) (1) + (0) (0) + (1) (1)+ (0) (1) + (1) (1) + (0)(0) + (0) (1) + (1) (0) +(1) (1) + (1) (0) = 3Thus, there are three two-

step connections betweenCOUN and UWAY, throughEDUC, MAYO, or WELF, afact that can easily beverified by inspection ofFigure 3, but which can betedious and time-consuming for a largematrix without resort tomatrix multiplication. Figure6 displays the completeresults of raising M to thesecond and to the sixthpowers, respectively. In the6-step matrix, we see that

ten distinct paths of lengthsix connect COUN to UWAY,although all are redundantpaths, such as the COUN-WELF-EDUC-UWAY-WELF-EDUC-UWAY sequence.Apart from showing howspecific pairs of actors areindirectly tied together, theset of T-step matricesreveals something aboutthe overall structure of thenetwork. For example,Figure 6 shows that the

majority of pairs areunreachable in the moneynetwork even throughlengthy chains. The reasonfor this pattern seems to bethat organizations specializeas sources or sinks of funds,with relatively fewtransmitters. In contrast,upon raising the informationexchange network, I, to the

M2C C E IO O D NU M U DN M C U

COUN 0 1 2 0COMM 0 0 0 0EDUC 0 0 0 0INDU 0 1 2 0MAYO 0 0 2 0WRO 0 0 0 0NEWS 0 0 1 0UWAY 0 0 1 0WELF 0 0 0 0

WELF 0 0 0 0WEST 0 0 0 0M6

C C E IO O D NU M U DN M C U

COUN 0 0 6 0COMM 0 0 1 0EDUC 0 0 1 0INDU 0 0 5 0MAYO 0 0 3 0WRO 0 0 0 0NEWS 0 0 1 0UWAY 0 0 1 0WELF 0 0 2 0

WELF 0 0 2 0WEST 0 0 0 0

Figure 6Two-and Six-Step Connections in Money Exchange Network

third power, every organization has one or more paths toevery other organization, indicating completereachability in this network in three or fewer steps.Two new matrices can be calculated to display thereachabilities and path distances among network actors.The first procedure sums a series of Kproduce a reachability matrix

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elements show whether ornot actor i can reach actor jin T or fewer steps:RT = K + K2 + K3 + + KT

(Matrix addition simplyinvolves adding togetherthe numerical values of thecorresponding i, j elementsof each matrix in the sum.)The elements rijk of RT arecounts of the total numberof connections betweenpairs of actors involving T or

fewer steps. A zero elementin RT means that that pairof actors is not reachable inT or fewer steps, althoughreachability may occur inchains of greater length.The second procedureresults in a path distancematrix, DT, whose elementsdijk reveal the shortestchains linking actors i and jin the kth network. It iscalculated as:DT = K + K2' + K3' ++ KT'

where KT' is KT in which allnonzero elements are setequal to T except for thoseelements that have beenreplaced by zeros becausethey were nonzero in matrixK raised to some power lessthan T. The diagonalelements always remainzero in the calculation. Thelongest path distance in anetwork will equal T whenKT+1 has no zero elementsin lower powers of thematrix. The DT procedure

excludes redundantconnections that occur inRT and thus may bepreferred as a relationaloperationalization. Figure 7illustrates path distancematrices for both M and I.Notice that although anyreachable pair requiresthree or fewer steps in eachmatrix and every actor canreach every other actor inthe information exchangenetwork, only about a thirdof the pairs are reachable in

the money exchangenetwork.At times, a normalization ofpath distances may beuseful, especially when thenetwork analyst isinterested in comparingsystems with differentdensities or differentnumbers of actors. Lincolnand Miller (1979) provideda simple equation forchanging the elements ofDT to proximities:

MC C E IO O D NU M U DN M C U

COUN 0 2 1 0COMM 0 0 1 0EDUC 0 0 0 0INDU 0 1 1 0MAYO 0 1 1 0WRO 0 0 0 0NEWS 0 1 2 0UWAY 0 0 2 0WELF 0 0 1 0

WELF 0 0 1 0WEST 0 0 0 0I

C C E IO O D NU M U DN M C U

COUN 0 1 2 2COMM 1 0 1 1EDUC 2 1 0 1INDU 1 1 2 0MAYO 1 1 1 1WRO 3 2 1 2NEWS 2 1 2 1UWAY 1 1 2 1WELF 2 1 2 2

WELF 2 1 2 2WEST 1 1 1 2

Figure 7Path Distance Matrices for Money and Information Exchange Networks

where dmax is the largest path distance observed in the knetwork between any reachable pair. When two actors aredirectly connected (i.e., their path distance is 1), theproximity value is 1. When two actors are unconnected (theirpath distance value is 0), pijkdistances then on values ranging between 0 and 1, with

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larger proximities occurringwhen fewer interveninglinks are required for i toreach j.Other methods proposed tonormalize the connectionsin a matrix include: (a)requiring each actor'schoices to sum to a total of1 (Hubbell, 1965); (b)subjecting path distances tosmallest space analysis tomeasure essential aspects

of relational intensity(Laumann and Pappi,1976); (c) using pathdistances in overlappingsocial circles (Alba andKadushin, 1976); and (d)basing normalization on anonlinear function ofdecreasing relationalintensity with increasingpath distance (Burt, 1982:28-29).The methods describedabove for calculating

reachability, path distance,and proximity in matrixrepresentations of networkdata focus on the relation ofone actor to another as adyad. They do not considerthe other N 2 actors exceptinsofar as those others arenecessary to complete thechain of steps from actor ito actor j. In the sectionsbelow we investigatedistance measures that takeinto account the relationsamong the other actors in

the network. The followingsection lays the groundworkby considering someindicators of actors'positions in networks.

Indices for Actors andNetworksMany different indices canbe computed from matricesto summarizecharacteristics for bothindividual actors and entirenetworks. The potential setof indices seems limited

only by analysts'imaginations, but spacelimitations allow coverage ofjust the more familiar andwidely used indices.An index of networkcohesion, G, divides thenumber of mutual choicesin a binary matrix ofdirected ties by themaximum possible numberof such choices:

where the term (zijk + zjik)takes the value of 1 if bothelements are 1s; otherwiseit takes the value of 0. Thecohesion index ranges from0 to 1.0, with larger valuesindicating that a greaterproportion of network

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relations are reciprocated.Note that the networkcohesion index resemblesthe density measuredescribed above, thedifference being thatasymmetric ties are countedin the latter but ignored inthe former.An index of networkmultiplicity, M, is based ontwo or more networks ofrelations for the same set of

N actors. A tie betweenactors i and j is said to bemultiplex if zijk = 1 in somespecified proportion of the k= 2 K networks underconsideration. For example,if actor i cites actor j in afriendship network actor ialso cites actor j in theadvice-giving network, themoney-lending network, thefavor-trading network, andso on. In its most generalform, network multiplexitymay be calculated as:

where zij(m) is 0 unlessmultiple zijk are nonzerofrom i to j, in which casezij(m) equals 1. Thenetwork multiplicity indexvaries from 0 to 1.0,reflecting the proportion ofall possible pairs of actorsthat have the specified levelof multiplex ties.At the individual actor level,

actor multiplexity can bemeasured as the proportionof an actor's ties with allother N 1 actors in thesystem that are multiplexacross K networks. Itsformula is:

again, where the zijk(m)value is 1 if actor i has therequisite number of linkswith a given actor j acrossthe K networks, and is 0 if

otherwise.A related index at the levelof the individual actor is anactor's ego network density,DE, which extends to theindividual the concept ofnetwork density consideredin the previous section.Actor i's ego networkconsists of that subsetamong the other N 1system actors with whichthe actor i has directconnections. This set is also

called the ''first star" or"primary star" (of course,actors who aredisconnected from anetwork have no egonetwork and their densitycan be considered

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either to be zero orundefined). If there are nEalters in actor i's egonetwork for the kth matrix,the density index iscalculated as:

which gives the proportionof potential linkages amongego's alters that actuallyoccur.

The various indices ofnetwork centrality tracetheir origins to Bavelas's(1950) and Leavitt's (1951)research on the effects ofsocial structure in humancommunication. Theyintroduced the idea that themore central an actor, thegreater his or her degree ofinvolvement in all thenetwork relations. Thesimplest centrality index foractor i is a ratio of theaggregate relations

involving i over all relationsin a network, i.e., theproportion of all networkrelations that involve i:

For example, in the moneyexchange network (M) theUnited Way's centralityscore is 0.36 and themayor's is 0.23, but in theinformation exchangenetwork (I) their centrality

scores are 0.16 and 0.33,respectively.Freeman's (1979) reviewidentified nine distinctcentrality indices based onthree structural propertiespossessed by the center ofa star network (a star is aconfiguration in which apoint is connected to the N1 other points, which arethemselves unconnected).The interested reader isreferred to the article for

further details, as we canpresent only two centralityindices based on"betweenness," whichFreeman (1979: 237)asserts provide "'finergrained' measures than theothers." Moreover, thecentrality indices below arerestricted to binary,symmetric data.

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The shortest path that linksa pair of points, i and j, in anetwork is called ageodesic. Any point orpoints that fall on ageodesic(s) linking a pair ofpoints is said to standbetween the two endpoints.For example, in Figure 4a,point 4 lies between 3 and6. For the endpoints 4 and7, two geodesics (of pathdistance two) exist, with

points 5 and 6 lyingbetween or on either one. IfFigure 4a represents anetwork of communicationchannels, then the betweenpoints have the potential tocontrol, disrupt, or distortthe flow of informationbetween the endpoints,either entirely as in the firstinstance or with somepartial control as in thelatter instance, wherealternative geodesics couldcarry the communication.

Assuming that twoendpoints i and j areindifferent as to whichgeodesic is used, theprobability of using any oneis 1/gij where gij is thenumber of geodesics linkingi and j. If actor m liesbetween the endpoints of ageodesic, the number ofsuch geodesics that involvem is gimj. Freeman (1977)showed that the maximumvalue on an index of partialbetweenness of a point is

attained only by the centralpoint in a star network.That value is (N2 3N +2)/2. Hence an appropriateindex of relative centralityfor point m is the ratio:

Values range between 0and 1, with higher scoresindicating greater actorcentrality relative to othernetwork members. Matrix

methods for locating andcounting the geodesics inlarge networks are detailedin Harary et al. (1965: 134-141) and require acomputer program toprocess the data. Therelative centrality index canbe calculated only on binarysymmetric matrices, butcan be applied todisconnected as well asconnected graphs. Forother actor centralityindices based on the degree

of a point (the number ofother points in directcontact with i) andcloseness of a point (pathdistances between points),consult Freeman (1979).The betweenness conceptof centrality can also beapplied to an entirenetwork. An index ofcentralization is based onthe difference

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between the centralityscore of the most centralactor and that of the N - 1other actors:

where CB(p*) is the relativecentrality score of the mostcentral point. This indextakes its greatest value,1.0, for a network star inwhich a single actor

dominates the connectionsamong all others, and ittakes its smallest value, 0,in circle or all-channel(complete) graphs in whichno actor is structurallydistinct from any other.The final indices consideredin this section measure theprestige (pi) of actorswithin the network. Anactor has greater prestigeto the degree that othersystem actors show

deference to him or her intheir relation. Practically,this definition means thatprestigious actors will moreoften tend to receive thanto initiate linkages, that is,to be chosen rather thanchoosers. As with centrality,several alternative operationalizations of prestige arepossible. The simplest,based on direct ties only, isan actor's indegree (seepreceding section) or,alternatively, the proportion

of all ties in the system thatare directed toward anactor. Better indices ofprestige take into accountindirect linkages as well asdirect ties.Lin (1976: 340-349) firstdefines an actor's influencedomain (Ii) as the totalnumber of actors thatdirectly or indirectly sendrelations to an individual.Next, the individual'scentrality (Ci) is defined as

the mean distance to allactors in the influencedomain, on the assumptionthat each linkage in a chainhas the same magnitude.Actor i's prestige is thencalculated as:

If centrality is zero, that is,if an actor is disconnected,the actor's prestige is alsozero. This index variesbetween zero and one, with

higher values indicatinggreater prestige and themaximum attained when allother actors directly sendrelations to actor i.

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A more sophisticatedprestige index corporatesthe prestige of the actorssending ties to actor i. Eachelement in the columnvector of actor i in matrix Kis multiplied by the prestigescores of the N 1 otheractors (subscripted j) andthe sum of these productsis actor i's prestige score:

Thus, actor i's prestige ishigher to the extent that itreceives many ties directedto it by many otherprestigious actors who arethemselves the recipients ofdirected ties from manyother actors. The scoresthereby take indirect as wellas direct linkages intoaccount. Note that, in amatrix of symmetricrelations, the prestige scoreis also a centrality measure(see Knoke and Burt,

1982). Because prestigescores appear on both sidesof the equation, asimultaneous solution to theN equations is required,using a computer algorithm.In matrix algebra notation,the system of equations is:O=P'(Z-I)where P' is a vector of Nprestige scores, Z is the(N,N) matrix of observeddirect relations, and I is the

identity matrix (a squarematrix with 1s on the maindiagonal and 0s elsewhere).If the relations in Z arenormalized so that thematrix is column stochastic(elements are nonnegativeand sum to 1.0 withincolumns), then theequation is the"characteristic equation" ofmatrix Z (van der Geer,1971: 64). The set ofprestige scores for all actorscan then be calculated (by

computer) as the firsteigenvector.A network analyst's choiceamong various indices forindividual actors or entirenetworks is not a simpledecision, but can berevealed only after carefulconsideration of theconceptual, substantive,and empirical features ofthe problem at hand. Forinstance, the decision toselect a measure of actor

centrality or prestige hingesin large part on whether therelations measured are trulyreciprocal (sent andreceived) or genuinelyasymmetric (received only),and on whether or not thequality of the other actors'network location should betaken into

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account (see Knoke andBurt, 1982, for a detaileddiscussion). Because thegrounds for index usageconstantly change acrosssituations, we can offer nouniversal rules for choice,but only counsel thenetwork analyst to proceedonly after thoroughinvestigation of theimplications of usingalternative measures.

Clique DetectionThe preceding sectionsconsidered network aspectsof individual actors and ofentire systems. In this andsubsequent sections,attention shifts to methodsfor partitioning networksinto subgroup components.Two predominant methodsfor determining the numberand composition of subsetsare the clique-detectionapproach, considered in thissection, and the structural

equivalence approach,presented in the nextsection.The clique is a centralconcept in much smallgroup research, and it hasbeen a mainstay oftheoretical and empiricalinvestigations for more thanthree decades. Manydefinitions have beenproffered over the years bydifferent analysts (Harary,1959; Lindzey and

Borgatta, 1954), but mostincorporate the idea that aclique is a highly cohesivesubset of actors within anetwork. In some versions,all or most clique memberspossess a specified relation(e.g., friendship,communication), while inother versions clique-matesare characterized as havingmore numerous or moreintense relations with eachother than with non-cliqueactors. In either case, the

implicit propositionmotivating clique analysis isthat actors who maintainespecially cohesive bondsamong themselves are morelikely to perform similarly(e.g., to share information,to develop similarpreferences, to act inconcert).Following Festinger (1949)and Luce and Perry (1949),the most stringent andrestrictive formal definition

of a clique is a maximalcomplete subgraph, a set ofcompletely linked points notcontained within a larger,completely linked set.Cartwright and Harary(1956) added the furtherstipulation that a cliqueinclude at least threemembers. For anundirected graph, theminimum configuration for aclique is three pointsconnected by three lines.For digraphs, the smallest

clique consists of threeactors each in mutualrelation (i.e., six directedlinkages among the threepoints); in other words, aclique is a stronglyconnected subset ofdigraph members. Everypair of actors in a clique isadjacent, while adding anyother network actor to theclique will make it less than3-connected. Moregenerally, if clique size is in,digraph

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clique will have (n2 n)directed relations, and anundirected graph clique willhave (n2 n)/2 lines.Although the digraph inFigure 3 contains no cliquesunder this definition, theundirected graph in Figure4a contains three suchcliques: (1)points 1, 2, 3,and 4; (2) points 4, 5, and6; and (3) points 5, 6, and7. As this illustration

implies, many networks donot partition neatly intounique, nonoverlappingcliques. The latteroccurrence suggests eithera disjointed nonsystem orfactions in overt conflict(Roistacher, 1974: 133).The basic methodologicalquestion is: Given a matrixK of N actors, how can aresearcher determine thenumber of cliques, theirconstituent actors, and

patterns of clique overlap?The stringent maximalcomplete subgraphdefinition and pre-computer-era difficulty inprocessing large networkssoon led networkresearchers to developmore ad hoc methods forclique detection (Hararyand Ross, 1957; McQuitty,1957; MacRae, 1960;Coleman and MacRae,1960; Hubbell, 1965;Doreian, 1969; Lankford,

1974). These procedureshad their owncomputational difficulties,especially with very largematrices, and theirtheoretical foundationswere often obscure (Albaand Moore, 1978). With therise of powerful computers,algorithms for locating allmaximal completesubgraphs were eventuallywritten (Auguston andMinker, 1970).

By that time, however, thecompleteness criterion hadcome to be seen as overlyrestrictive for manysubstantive applications. Asubset of actors might failto be identified as a cliquebecause only a few relationsamong its members werelacking. Realistically,measurement error couldresult in some truly strongand reciprocated linkagesgoing unmeasured in thedata collection phase.

Furthermore, as everyclique member has a pathdistance one to every othermember, no variation ininternal clique structureoccurs. Clearly, somerelaxation of the maximalcomplete subgraph criterionseemed desirable, thusallowing a stronglyconnected clique to beaugmented by moreperipheral actors that mightnot have reciprocal ties toall others in the clique.

Proposed alternative criteriafor clique detection havemostly been variants of themaximal strong componentconcept from graph theory(Harary et al., 1965: 53-55). A maximal strongcomponent is a networksubgroup in which eachactor can at least reachevery other actor directly orindirectly and no furtheractors can be addedwithout losing this mutualreachability. Actors i and j

can be joined together intoa clique if the smaller of zijand zji is at least greaterthan some criterion alpha(or greater than zero forbinary matrices). Hubbell(1965) and Doreian (1969)described clique-detectionprocedures using thisrelaxed

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criterion, and Laumann andPappi (1976: 102-105)applied it in investigatingnetworks among elites in asmall German city.Among the morenoteworthy uses of themaximal strong componentdefinition is to locate n-cliques (Luce, 1950). Actorsform an n-clique if therelations between all pairsare measured as the

reciprocal of their pathdistances and the smallervalue for any pair of actorsexceeds 1/(n + 1). Thusevery member of an n-clique can reach everyother in n or fewer binarylinks. An n-clique allows forindirect connectionsthrough intermediary actorsbut limits the maximumdistance across which suchindirect interactions canoccur. Luce's definitionpermits n-clique members

to be connected viaintermediaries that are notthemselves clique members.Alba (1973) suggestedfurther restricting a cliqueto be an n-clique in whichevery pair of actors isconnected by a geodesic (ashortest path, see above)composed solely of otherclique members.Another relaxation of thestringent clique definition isthe k-plex clique (Seidman

and Foster, 1978). A k-plexstructure is a graph with npoints in which each pointis connected by a path oflength 1 to at least n - k ofthe other points. Thus everyactor of a k-plex hasmaximum strong relationswith all except k cliquemembers. Becausecomplete graphs are 1-plexes, a true clique in themaximal complete subgraphdefinition is a special caseof the k-plex approach. But

in permitting more indirectlyconnected actors into the k-plex clique, the methodallows for internal variationin clique structure.The k-plex definition isconsistent with the conceptof a social circle, a set ofactors with shared interestshaving direct or minimallyindirect linkages with eachother (Kadushin, 1966,1968; also Alba andKadushin, 1976). As

operationalized by Alba andMoore (1978), detection ofsocial circles begins withfinding all maximalcomplete subgraph cliquesand then successivelyaggregating them whenthey overlap to a specifieddegree. For example, allsimple cliques of three ormore actors are firstidentified; then cliques thatdiffer by a single memberare joined. Next, if threequarters of the members of

a smaller group also belongto a larger circle or clique,the two are merged. (The75% overlap is arbitrary,and Alba and Mooreconsidered other levelsbefore deciding on thisvalue as most suitable forthe network they wereanalyzing.) Subsequentaggregations, using thesame or different criteria ofoverlapping strongcomponents, can be madeuntil the researcher decides

closure is warranted ontheoretical or substantivegrounds. The resultantsocial circles thus arecomposed of actors that,

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while not having maximumstrength relations with eachother nor even mutualreachability, are stillrequired to maintaincontacts with a largeproportion of the othercircle members.Most of the clique-detectionprocedures mentionedabove are sufficientlycomplex as to requirecomputer implementation

for application to empiricalresearch. Thus, spacelimitations prevent us fromproviding detailedexplanations of the stepsinvolved in moving frommatrix data on directrelations to theidentification of cliques, n-cliques, k-plexes, and socialcircles. Readers are urgedto consult the originalsources cited above.Many of the clique-

detection methods are notconcerned with inter-cliquerelations. Except for the n-clique technique, ties tononclique actors aregenerally ignored. Thisfailure to take into accountthe full set of relationsamong all network actors isa major criticism leveledagainst the clique approachto network partitioning byresearchers who prefer thestructural equivalenceapproach.

Structural EquivalenceThe second basic approachto partitioning networkactors into subgroupsinvolves the application of astructural equivalencecriterion to their relations.As briefly described inChapter 2, actors areaggregated into a jointlyoccupied position to theextent that they have acommon set of linkages toother system actors. More

formally, two objects a andb of a set C are structurallyequivalent if, for any givenrelation R and any object xof C, aRx if and only if bRx,and xRa if and only if xRb."In other words, a isstructurally equivalent to bif a relates to every object xof C in exactly the sameway as b does. From thepoint of view of the logic ofthe structure, then a and bare absolutely equivalent,they are substitutable"

(Lorrain and White, 1971:63).The structural equivalencecriterion requires a pair ofactors to have exactlyidentical patterns ofrelations with the N 2 otheractors in the network inorder to be placed togetherin the same networkposition. For most empiricalpurposes, this criterion istoo stringent andimpractical, just as the

maximal complete subgraphdefinition of a clique provedto be too restrictive. Asshown below, in practicenetwork researcherstypically relax the criterionof strong structuralequivalence, groupingactors in the same positionson the basis of theirsimilarity of relations toother actors. A structuralequivalence criterion forplacing a pair of actors inthe same position takes into

account not

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their relations to each other(as in the clique-detectionapproach), but only theirrelations with the othersystem actors. Indeed,structural equivalenceprocedures for partitioninga set of network actors donot require any members ofthe equivalent subset tomaintain relations with eachother. Such empiricalresults do occur and have

meaningful interpretations.Thus the most obviouscontrast between theclique-detection and thestructural equivalencemethods lies in theirdifferential emphasis onrelations within or betweensubgroup actors.Cliques have usually beenidentified for singlenetworks, while substantiveapplications of structuralequivalence to multiple

networks have implied thatthe latter technique ispreferable for analyses ofmultiple network systems.However, the structuralequivalence applicationssimply sum comparisonsacross networks,disregarding substantivegrounds for the summation.Comparable procedures inclique-detection would alsosimply add relations amongactors across severalnetworks to obtain an

aggregate relation matrixfor analysis. Hence bothapproaches are equally well(or ill) suited to multiplenetwork data.An appreciation of thestructural equivalenceapproach can be gained bya detailed examination, withexamples from the I and Mmatrices, of its two mostpopular operationalizations:as continuous and asdiscrete distance measures.

As the formerconceptualization subsumesthe latter (Burt, 1977a:124-127), we begin withthe more general method ofmeasuring structuralequivalence in continuousdistance terms.Continuous DistanceThe key assumption in thisoperationalization ofstructural equivalence isthat distance between apair of actors is measurable

in terms of dissimilarity intheir patterns of relationswith other system actors. Ifthey have exactly identicalrelations with the others,their distance is zero and,as noted above, theyoccupy an identical point inthe social space. As the pairof actors have increasinglydifferent patterns of tieswith the others, they areincreasingly more distantfrom each other in thesocial space.

Conceptualizing a socialspace in Euclidean terms,the distance betweenactors i and j (dij and dji,where distances are sym-

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metric) equals the squareroot of the sum of squareddifferences across all thirdactors q:

where (ziqzjq) is thedifference between the twoactors in the relations theyinitiate with a third actor(i.e., a pair of elements inrows i and j of matrix K),and (zqizqj) is the

discrepancy in relationsreceived from a third actor(i.e., a pair of elementsfound in columns i and j ofmatrix K).This distance metric readilygeneralizes to more thanone network of relations bysumming values across K³2matrices involving the sameset of N system actors:

In either case, regardless ofthe number of matrices ofrelations over which socialrole distance is calculated,the result is an N×Nsymmetric matrix ofdistances between everypair of actors. Maindiagonal elements are zero,to reflect no distance of anyactor from itself.To illustrate the use ofthese formulas, we willcalculate the distance

between the United Wayand the welfaredepartment, using both theinformation and moneyexchange networks shownin Figure 5 above. In effect,we first pull out the twopairs of row vectors fromboth matrices and placethem parallel to each other:

The underlined z's are the

direct relations between thetwo actors. Under somespecifications of thedistance, these relationsmay be

ignored in calculating distancebetween pairs of actors. If they areincluded, as they will be here, theyshould not be double-counted whentaking the sum of differences acrossthe pair of columns (below). Next,the pairwise differences in the rowvector elements are formed, squared,and summed for both matrices:

Turning to the columns for the twoactors, we pull out the pair of vectorsfrom both matrices:

UWAY WELF1011101**00100

100**0

The asterisks replace the redundantpairs of relations that have alreadybeen used in the row differences.Taking the sum of squareddifferences between column elementsyields 4. The square root of the sumof 7 and 4 is 3.317, the social roledistance between the United Way

and the welfare department for thetwo matrices.

DC C E IO O D NU M U DN M C U

COUN 0.00 3.87 3.87 3.00COMM 3.87 0.00 4.24 3.74EDUC 3.87 4.24 0.00 4.00INDU 3.00 3.74 4.00 0.00MAYO 3.46 3.00 4.24 3.32WRO 3.74 3.87 4.00 3.74NEWS 3.46 3.32 3.46 3.16UWAY 3.46 3.87 3.32 3.74WELF 3.16 3.87 3.00 3.32

WELF 3.16 3.87 3.00 3.32WEST 3.60 3.87 3.61 3.32

Figure 8Matrix of Social Role Distances Across Information and Money Exchange Networks

Omission of the redundant pairs need not be specified, and thedistance equations above include the difference between zThe examples in this monograph are based on so few cases that we feltthe adjustment should be made here.Figure 8 displays D, the distance matrix whose elements are the dbetween every pair of actors in both the information and moneynetworks, calculated as in the preceding example. The two actorsclosest to each other are the Women's Rights Organization and theWestend neighborhood organization (ddistant pairs of actors are the chamber of commerce and education

distant pairs of actors are the chamber of commerce and education(4.24) and the mayor and the United Way (4.24). Because none of theten actors has zero distance, imposing a strong structural equivalancecriterion would result in no actors being placed together in the samenetwork position. Measurement error, sampling variability, and otherfactors can conspire to render empirical network data less thanperfectly reliable even for pairs of actors that are essentially equivalentin their ties to other system participants. For these reasons, a weakstructural equivalence criterion is typically imposed to identify actorpositions. To be placed together in the same positions, actors are

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required only to have asocial distance equal to orless than some arbitrarilychosen nonzero value, a(alpha). That is,dij£a

Burt (1976, 1980), one ofthe most persistentproponents of thecontinuous distanceapproach to structuralequivalence, selectedvalues of alpha on the basis

of a hierarchical clusteringapplied to the matrix ofsocial role distances, D.Cluster analysis drawsboundaries around objectsin multidimensional spacethat result in maximalhomogeneity (minimumvariation) within eachcluster (Bailey, 1974: 61).A hierarchical clusteringmay proceed byagglomeration: startingwith N actors, the algorithmsuccessively combines and

recombines them into largerand larger clusters C, where2 £ C £ N1. Such clustersare mutually exclusive withregard to actor membershipand nested at successivelevels of aggregation. Themost similar actors arejoined in the early stepsand the least similar actorsjoin clusters only in the laststeps. The result is aresembling a tree: ''A treemay be regarded as ahierarchical grouping

structure, in which theobjects in B are groupedinto a set of clusters, theseclusters are grouped into aset of clusters, and so on"(Hartigan, 1975: 1140). Atree can be diagramed as adendogram to display thelevel of similarity at whichactors and clusters mergeinto more inclusive clusters.As applied to the matrix ofsocial role distances, D, ahierarchical clustering sets

the initial value of alpha atzero and combines into onecluster all actors with dij =dji = 0 (strong structuralequivalence). Insubsequent steps, the valueof alpha is increased tosuccessively higher levelsfound in the matrix D,combining the actors withthese values into subsetsthat have greater similarityto each other than to otherclusters. Successivelygreater values of alpha

permit the clustering ofclusters as structurallyequivalent under moregenerous weak criteriauntil, at the final step, allactors are merged into asingle group despite theirseparation by large socialdistances. At someintermediate point, thenetwork researcher mustselect an arbitrary butreasonable alpha value thatresults in a partitioning ofthe N actors into a set of C

clusters that will be theterminal set. No objectivestandard can be invoked asto the value of alpha atwhich clustering should behalted. Rather, theterminus must be selectedby the researcher within thecontext of a substantiveproblem, guided by theuses to which the resultsare to be put and aninspection of thedendogram pattern ofagglomeration.

Figure 9Dendogram Showing Hierarchical Clustering of

Social Role Distance Matrix in Figure 8The result of applying a hierarchicalclustering to the distance matrix inFigure 8 is displayed in the dendogram inFigure 9. This analysis was performedwith the STRUCTURE program (Project,1981), which uses Johnson's (1967)connectedness method of hierarchicalclustering. After a cluster is formed, it istreated as a new object and its distancefrom the other objects is determined,with agglomeration at the next stepbased on the smallest distance between

clusters (the alternative diameter methodproceeds by using the largest distance).Westend and WRO are the first set ofactors to be clustered (at alpha = 2.65).When alpha reaches 3.00, three newclusters emerge, each consisting of a pairof actors, while the newspaper and theUnited Way are the only isolates not yetjoined with at least one other actor. Ifthe network researcher decides toterminate the clustering at alpha = 3.16,three subsets would

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be identified (WRO,WEST),(COUN, INDU, NEWS,EDUC, WELF), and (COMM,MAYO), with a single isolate(UWAY). To proceed furtherwould be futile, since allactors merge into a singleheterogeneous cluster atthe next level of alpha.Substantively, the three-cluster partitioning seemsto make much sense. Bothnonestablishment voluntary

associations (WRO andWEST) are groupedtogether, based mainly ontheir absence of relationswith the other actors.Indeed, this pair does notform a clique: None of thefour possible exchanges ofmoney and informationoccurs between them.According to Table 1, thesetwo organizations are alsothe least influential actors,possibly as a consequenceof their sparse connections

in the two networks. Thefive-actor cluster consists ofthe city/county council andtwo of its bureaus, plusboth private sectororganizations (NEWS,INDU). This subset isinternally structured muchlike a social circle: about athird of the forty possibledyadic relations across bothnetworks actually occur.Finally, the two-actorcluster consisting of themayor and chamber of

commerce is almost a pureclique, as three of the fourdyadic exchanges of moneyand information occur. Also,with a mean influencereputation of 5.83, it issomewhat more powerfulthan is the five-actor cluster(mean influence = 5.40;see Table 1).Figure 10 rearranges therows and columns of themoney and informationexchange network matrices

to conform to the four-cluster partitioning obtainedwhen alpha = 3.16. Verticaland horizontal linesdemarcate the members ofthese clusters, as well asthe single unaggregatedactor, UWAY. In thecontinuous distanceapproach, a jointly occupiedposition can be defined as amaximal set of structurallyequivalent actors. Anetwork position jointlyoccupied by three or more

actors is called a status, S(Burt, 1980: 102). Thisminimum size requirementparallels the convention inclique detection. Under thisdefinition, only a singlestatus, S1, exists in Figure10. (In a system with alarger number of actors,presumably more statuseswould be identified.) Thefive actors COUN, INDU,NEWS, EDUC, and WELFjointly occupy this statusbecause of their

involvement in similarrelational patterns with theactors across bothnetworks.The role set that defines astatus is the mean oraverage of the relationsthat link the status'soccupants with each otherand with the occupants ofother statuses. In the kthnetwork, the typical relationbetween a pair of positions,S 1 and S 2, jointly occupied

by n1 and n2

Figure 10Rearrangement of Money and Information Exchange

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actors, respectively, is themean or density of the tiesfrom all occupants of S1 toall members of S2:

while the average relationwithin-position S1 is simplythe density of ties amongthe position's occupants(ignoring self-ties incalculating the proportion):

where the ith and jth actorsoccupy S1 and the qth actorbelongs to S2.Because only one statuswas found in the exampledata, we technically cannotcompute the role set for S1.However, for illustrativepurposes, we shall considerthe pair of two-actorclusters to constitute

statuses S2 and S3. Thenthe role set defining S1 isgiven by a ten-elementvector consisting of fivemoney and five informationrelations: (1) the meanrelations from theoccupants of S1 to each ofthe other two statuses; (2)the mean relations to S1from each of the other twostatuses; and (3) relationsamong the occupants of S1.The full vector is:

(z12M z13M z21M z31M z11Mz12I z13I z21I z31I z11I)

which is, using the densitiesexhibited in Figure 10:(0.2 0.1 0.3 0.0 0.3 1.0 0.21.0 0.6 0.4)Similar role-set vectors ofmean relations could becalculated for the networkstatuses S2 and S3. Theconcept of network statuswill be used below indiscussing hypothesis

testing.

One advantage to thecontinuous distanceapproach to structuralequivalence is the ability torepresent actors' relationsvisually in a

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manner analogous to thesociogram of direct linkagesdescribed above. Thematrix of Euclideandistances between actors,D, can be entered in any ofa variety ofmultidimensional scalingprograms (Kruskal andWish, 1978), and ageometric display of socialspace will be produced in ddimensions (d < N), along

with a measure of fitbetween the data and theconfiguration. In Figure 11,we show the results of atwo-dimensional smallestspace analysis (Roskam andLingoes, 1970; McFarlandand Brown, 1973) based oninput of the distances inFigure 8. The stresscoefficient is 0.11, anacceptable value. We havedrawn closed curves aroundthe three clusters of actorsidentified in Figure 10. The

diagram clearly showsdispersion of actors andpositions across the socialspace underlying the twoexchange networks, with anoted tendency for eachactor's location to reflect itsinfluence reputation (thelinear correlations of theinfluence scores in Figure 3are .45 and .67 with SSAdimensions one and two,respectively).Discrete Distance

A second approach tostructural equivalencepartitioning of networkmatrices is the blockmodelmethod introduced byWhite and his colleagues(Breiger et al., 1975; Whiteet al., 1976; Boorman andWhite, 1976; Arabie et al.,1978; Arabie and Boorman,1981). Blockmodelinginvolves both a procedurefor grouping actors intostructurally equivalentposition (blocks) and a

technique for analyzing therole structure of multiplerelations among the blocks.This subsectionconcentrates on theblocking methodology anddefers until later aconsideration of the rolealgebra component ofblockmodeling.The most widely usedalgorithm for partitioningmultiple network matrices isCONCOR (CONvergence of

iterated CORrelations), ahierarchical clusteringprocedure that starts with aset of K observed matrices.The matrices, eachrepresenting a distinctnetwork of relations amongN actors, are "stacked" tocreate a single KN×N matrixfor input into a correlationprogram. The program firstcalculates the Pearsonproduct-moment correlationcoefficients (rij) betweenevery pair of columns (i.e.,

between every pair ofactors). The resulting N×Nmatrix of correlations, R,thus contains linearmeasures of similaritybetween every pair ofactors, based on their tiesfrom all other actors. Thelarger the positivecorrelation, the greater thestructural equivalence of apair.

Figure 11Smallest Space Analysis of Distances in Figure 6

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In the next step, CONCORsubmits the R matrix itselfto the correlation program,this time correlating all pairsof columns (each columnconsisting of N correlationcoefficients). The resultingmatrix of second-ordercorrelations is likewise fedback into the correlationprogram, and the process isrepeated until all entries inR are either+1.0 or 1.0.

The final result is a two-block partitioning of theoriginal set of networkactors, with all actorspositively correlatedbelonging to one block andall actors negativelycorrelated with members ofthat block belonging to thesecond block. The CONCORalgorithm may then berepeated separately oneach subset, producingadditional bipartitions. Thecycle can continue until

every actor is placed in itsown block (N-blockpartition). Thus, CONCOR isa hierarchical clusteringprogram based onsuccessively dividing theentire network into smallerand smaller blocks, indiametrical contrast to thecontinuous distanceapproach that agglomeratesindividual actors intosuccessively larger clusters.Both procedures require thenetwork researcher to make

a substantive decisionabout where to halt thelumping or splitting.CONCOR calculatescorrelations using thecolumns of the K stackedmatrices, that is, on the setof relations received from allsystem actors across all Knetworks. Alternatively,iterative correlations couldbe calculated using the rowvectors (i.e., the set of allties sent to other actors). A

third possibility is to useboth the ties sent and theties received, by includingthe K transposed matricesin the set of stacked inputdata. (A transposed matrixis one in which the zij andzij elements are exchanged.See Schwartz, 1977, for anargument that all diagonalelements should beexcluded in computingcorrelations in CONCOR.)Because our illustrationabove of the continuous

distance approachcalculated Euclideandistances on both rows andcolumns, the followingCONCOR example will alsobe based on both relationssent and received in themoney and informationexchange matrices.To return to the same dyaddiscussed before, thecorrelation between theUnited Way and the welfaredepartment using the forty

paired binary values is .26,indicating a modest level ofsimilarity in these twoorganizations' patterns ofrelations with other systemactors. Figure 12 displaysthe full matrix ofcorrelations among the tenorganizations produced inthe first CONCOR step. Acomparison with the valuesreported in the socialdistance matrix (Figure 8)indicates a high degree ofinverse covariation (high rij

with low dij, and viceversa), but not a simplelinear transformation ofvalues. The correlation ofthe 45 non-diagonaldistance values in the Dmatrix with thecorresponding correlationsin the R matrix is .84.

RC C E IO O D NU M U DN M C U

COUN 1.00 .14 .15 .45COMM .14 1.00 -.06 .14EDUC .15 -.06 1.00 .04INDU .45 .14 .04 1.00MAYO .28 .40 -.04 .38WRO .10 .35 -.10 .10NEWS .30 .30 .32 .30UWAY .26 .14 .38 .15

WELF .34 .14 .47 .34WEST .11 .21 .17 .36

Figure 12Matrix of Column and Row Correlations Across Information and Money Exchange Networks

The contrast between the continuous and discrete distance algorithms isfacilitated by comparing the ways that correlations and distances are computedon binary data. Consider the standard 2× 2 layout with cell and marginalfrequences indicated by letters:

Y0 1

x 1 a b0 c d

Total a+c b+d

The formula for the Pearson correlation is simply that for the non-parametricstatistic phi:

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The Euclidean distanceformula boils down to:

Thus distance andcorrelation are not simplylinear transformations ofone another. Distance isalways nonnegative, whilecorrelations vary on astandardized scale from 1to +1. Given thosedifferences in the basicsimilarity measures, a

substantive difference inthe identification ofpositions and theiroccupants isunderstandable.Proceeding with theCONCOR analysis,convergence to the ± 1.0matrix occurred on thetenth iteration. One blockconsisted of COMM, MAYO,WRO, and WEST, and theother block's members wereCOUN, INDU, NEWS, EDUC,

UWAY, and WELF.Subsequent partitions ofboth blocks resulted in thefollowing four-blockassignments:(1) COMM, MAYO(2) WRO, WEST(3) COUN, INDU, NEWS(4) EDUC, UWAY, WELFThis partition bears someresemblance to thecontinuous distance

clusterings above, exceptthat EDUC and WELF havebeen split off the third blockand placed in a positionwith UWAY. Thesedifferences underscore thefact that two methodsoperationalize structuralequivalence somewhatdifferently and approachhierarchical clustering in adifferent fashion.The main distinction,however, between the

discrete and continuousdistance methods lies intheir assumptions aboutspace. CONCOR produces aclassification of networkactors into discrete,mutually exclusive andexhaustive categories. Itobtains neither measures ofproximity between blocksnor even an intrinsicordered relation among theblocks. Thus, theirnumbering from 1 to 4 isarbitrary. In contrast, the

continuous distance methodpreserves Euclideandistances during itsaggregation of actors intojointly occupied positions.Retaining the metricpermits the application ofpowerful statistical methodstesting hypotheses, asdiscussed below. For furtherobservations on the basicspatial assumptions of thetwo approaches tostructural equivalence, seeBurt (1977a: 125-127).

CONCOR, like thecontinuous distanceapproach, is an algorithmthat searches forstructurally equivalent setsof actors, useful where

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no strong a priorihypothesis exists aboutnetwork structure. TheCONCOR algorithm is not avalidated procedure: Thereis no proof of the invariantconvergence to the ± 1.0matrix form, and it isunclear what objectivefunction (if any) isminimized or maximized bythe process (Schwartz,1977). Nonetheless,

CONCOR has been foundrepeatedly to be''empirically useful inproducing interpretableblock-models" (Arabie andBoorman, 1982), and forthat reason its popularityremains high.

Matrix Permutation andImagesAfter the network actorshave been partitioned intosubsets of structurallyequivalent positions, the

original data matrices canbe permuted to reveal theunderlying intra-andinterpositional relations.Permutation is arearrangement ofcorresponding rows andcolumns to bring togetherin adjacent portions of eachnetwork matrix those actorsthat jointly occupy thesame position or block.Vertical and horizontal linescan be drawn atappropriate places in the

permuted matrices to showthe distinct positionsidentified by thehierarchical clustering. Thisrearrangement is essentiallythe same manipulationdescribed above for theresults of the continuousdistance approach. Figure13a shows the moneyexchange matrix permutedaccording to both two-andfour-block partitioningresulting from the CONCORanalysis of the money,

information, and their twotransposed matrices. Thecomparable permutation ofthe information matrix isnot shown. Note again thatthe sequence of blocks andof actors within blocks isentirely arbitrary, asCONCOR imposes no orderamong positions.After permutation, theoriginal matrix may bereduced to its image. Animage matrix is obtained

from a blocked matrix byreplacing each of itssubmatrices by either a 0 ora 1 according to the densityof relations within eachsubmatrix. Typically, one oftwo density criteria is used:(1) submatrices with no tiesamong actors are coded 0s(zero-blocks) andsubmatrices with one ormore ties are coded as 1s(one-blocks), or (2) somecutoff density value, alpha,is chosen at the

researcher's discretion andall submatrices withdensities less than alphaare set to 0, while allsubmatrices with alphadensity and above are setto 1. (A frequently imposedcutoff density is the granddensity for the matrix as awhole.) The first criterion isextremely restrictive inrequiring zero-

Figure 13Permuted Money Exchange Matrix with Two-

BlockDensities and Image Matrix

blocks to be strictly filled with 0s,probably an unrealistic assumption forfallible social data. Using a highercutoff density to decide whichsubmatrices will be coded 0 and 1 inthe image matrix is usually morerealistic and substantively revealing.The more liberal criterion tolerates afew ties between dyads across whatare essentially unrelated positions.

The abstraction of an image from ablocked matrix is a homomorphismthe observed relations, that is, amapping from a blocked matrix to animage matrix that maps one-blocks to1s and zero-blocks to 0s (Arabie et al.,1978; this use of homomorphism is notto be con-

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Figure 14Two-and Four-

Block Images of Money and

nformation Exchange Networks

fused with the semigrouphomomorphism developedby Boorman and White,1976). The homomorphicreduction of the two-blockpartition of the moneyexchange network to itsimage is illustrated in Figure13. Figure 13b showsdensities for the foursubmatrices or positions(density calculationsexclude the main diagonals,

since self-ties wereprohibited in the data). Asmentioned above, the meandensity for the entire matrixis 0.24. Only one of the foursubmatrices exceeds thiscutoff value, and thus theother three submatrices areset to zero, as shown infigure 13c. The resultingtwo-block money exchangeimage matrix thus revealsthat money flows only fromthe second position to thesecond position. This

position is not a true clique,however, since its density isnot 1.00, which wouldindicate a maximalcomplete subgraph.In Figure 14, we display thetwo-block and four-blockimages for both the moneyand information exchangenetworks, again using themean density cutoffcriterion for each full matrix.At the simpler two-

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block level, informationtrading between the twopositions is apparent: Thehighest densities occurredin the submatrices involvinginformation exchangesbetween sets of actorsoccupying differentpositions. In the morerefined four-block partition,a more complex pair ofimages emerges. Severalgenuine zero-blocks are

encountered, eight in themoney and two in theinformation networks. Andall five one-blocks in theinformation networks thatinvolve the first positionhave densities of 1.00. Themost obvious consistencyacross both networks is theabsence of connectionbetween position 2,consisting of the two non-establishment associationsWRO and WEST, with anyother position, including

themselves. This position iscompletely isolated from asustained relation with anyother positions. In themoney exchange image,position 4 (EDUC, UWAY,WELF) is the main recipientof money transactions,receiving support from theother two positions as wellas from its own members.Position 3 (COUN, INDU,NEWS) does not receivefunds from any subset ofnetwork members, but it

does supply both position 4and position 1 (COMM,MAYO), which also providessome of its own financialsupport.In the informationexchange matrix positions 1and 3 are the mainrecipients, from all sources(including within their ownpositions) except for theisolated position 2. Position4 is differentiated inreceiving community affairs

information only from thefirst position, in strikingcontrast to its role as afinancial recipient in theother network.

Hypothesis TestingThe techniques foranalyzing relations amongmultiple networks of actorspresented above areprimarily descriptive. Theprocedures would be morevaluable if networkresearchers could formulate

and test hypotheses aboutthe social structuresobserved in the data.Unfortunately, statisticaltests are not available forclique models. Therestrictive definition of aclique as three or moreactors that form a maximalcomplete subgraph is acondition either met or metby an empirical network,and hence no significancetest is feasible. Definitionsof less stringent cliquelike

subgroups are just toonebulous to provide abaseline for testing thedegree to which subsets ofempirical actors

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conform to these cliquelikestructures. Only whencliques are defined asstructurally equivalentactors also tied to eachother by a strong criterion(alpha = 0) can cliques betreated to statistical tests asa special case of a jointlyoccupied position, asdescribed next.Hypotheses about the triadcombinations in networks

can be subjected tostatistical tests, using thetriad census, a vector ofsixteen types of triads thatcan occur in network data.Description of the teststatistic would require toomuch space, so theinterested reader is advisedto consult Holland andLeinhardt (1975 and 1978).An even moremathematically arcaneapproach is the biased netsmodel for assessing

deviations from randomnessin relations among actors(Rapoport and Horvath,1961).Burt (1976, 1977a, 1977b,1980) developed factor-analytic procedures fortesting a center-peripheryhypothesis about thedistances of structurallyequivalent actors from astatus position representedas a single point in space.Page limitations allow us

only to sketch the generaloutlines of the test.Using structural equivalencemethods described above tocompute the distancesamong the N actors in amultiple network system,subsets of actors arehierarchically clustered intodistinct statuses, S, eachcontaining three or moreactors. For the K actorsjointly occupying statusposition Sk, we observe only

the K2 K distances betweenpairs of actors (the dij).Under the hypothesis to betested, a single unobservedvector of true distancesfrom a single point in thestatus to each actor isresponsible for (i.e.,"causes") the observedvariation in these dij inter-actor distances. Thus, theempirical distances amongthe structurally equivalentactors jointly occupying astatus may be treated as

fallible indicators of the truevector.The situation is depicted inpath diagrammatic terms inFigure 15. At the top, theunobserved status Sk isrepresented as producingvariation in the distance ofeach actor (di) from thecenter point of the status.The path coefficients (lik)from Sk to the dis reflectthe different role distancesof each of the K actors,

with larger values of likindicating greater proximityto the center and smallervalues reflecting greaterdistance.Because the relations maybe measured with randomerror, an actor's distancefrom the center of thestatus is shown also to beaffected by a random errorcomponent (di), whoseeffect (Qi) indicates thedegree to

Figure 15Path Diagram of Covariance Structure for the Center-

Periphery Hypothesiswhich the relations in which actor i areinvolved is an accurate indicator of the role setdefining status Sk. Under this restrictedcovariance structure for the center-peripheryhypothesis, the observed covariance betweenany pair of occupants of a status is equal tothe product of their lik coefficients, indicatingthe extent to which their relational patternsreflect the status's role set. If the center-periphery hypothesis is correct, then thecovariance structure depicted in Figure 15 willaccurately describe the observed covariance

matrix among the distances of the K actors(the covariance matrix of the dThe K × K observed variance/covariance, S(not to be confused with Sk, the kestimated from the observed distances. This S

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matrix can be approximatedby the basic factor analyticequation, in matrix form:

where S is the estimatedvariance/covariance matrix,L is a 1× K vectorcomposed of covariancesbetween distance to actorsand true distance to thecenter of status Sk to eachactor (i.e., a vector of factor

loadings);F is the(standardized) variance ofthe vector of true socialdistance between the Kactors and status Sk; andis a K × K diagonal matrixcontaining variances of theerror scores in the vector.Parameters of the specifiedmodel are estimatedthrough principalcomponents or maximumlikelihood methods bysubmitting a correlationmatrix, computed on the

pairwise correlation of thecolumns of the observedinteractor distances, to aconfirmatory factor analysisprogram (Jöoreskog, 1969).A chi-square test statisticfor the goodness of fit of Sto S is produced and maybe evaluated against theappropriate degree offreedom in estimating themodel (Burt, 1980: 109,notes adjustments to bothdegrees of freedom and c2required by the

specification).The null hypothesis assertsthat actors in status Sk areequivalent under a strongcriterion (alpha = 0). If thenull hypothesis is rejected,then Sk defines a statusonly under a weakstructural equivalencecriterion. That is, the roledistances among the actorsjointly occupying the statusSk defines a status onlyunder a weak structural

equivalence criterion. Thatis, the role distances amongthe actors jointly occupyingthe status Sk aresignificantly different from0, but still small relative toother network actors not inthe status.Under a weak structuralequivalence criterion, ajointly occupied status isnot defined by a singlepoint but as a fieldsurrounding a point in

multidimensional space. Totest hypotheses aboutdistance as a point inspace, two or more of thestatus occupants must bespecified as indicators ofthe role distance to thestatus. In this more typicalsituation, the standardizedvalues of the li express thecorrelation between actors'role distances to the statusas defined in terms of itsindicator actors and the roledistance to the observed

position of each actor.Actors on the periphery of astatus (i.e., thoseincorporated into a positionunder larger alpha valuesduring the hierarchicalclustering) will have lowercovariation with distance tothe status than will the coreactors.Hypothesis testing forblockmodels based ondiscrete distance betweenpositions in multiple

networks has not been fullydeveloped

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to date. BLOCKER, acomputer program forfinding simultaneoushomomorphiccorrespondences betweenbinarized matrices andimages (Heil and White,1976), allows aninvestigator to specify asinput a set of hypothesizedimages on the basis ofsome theoreticalconsiderations. The

program then looks at allpossible permutations ofthe original matricesinvolving a "lean fit" (purezero-blocks) that could fitthe hypothesized blocking.The number of blocks and alower bound on block sizemay be specified by theuser. The program hasbeen useful in identifying"floaters," actors whoseblock membership isambiguous, and who thuscould be assigned to two or

more different blockswithout violating thehypothesis. In practice,BLOCKER has fallen intodisfavor because it requiresa network researcher tohave a priori knowledge ofthe data structure and ittolerates no impurities inzero-blocks. The inductiveCONCOR algorithm avoidsboth these rigorousrequirements and has beenmore favored in practice asthe mean to identify blocks

and their occupants. Still,BLOCKER serves theimportant function offorcing the network analystto think clearly about thespecific structural patternsthat are theoreticallyexpected. Properly used,this procedure could makeimportant contributions totightening the link betweentheory and data in networkresearch.Goodness-of-fit tests for

proposed blockmodels havebeen suggested by White(1977), Carrington et al.(1980), Carrington and Heil(1981), and Morgan andWolfarth (1980), althoughthey have yet to see wideacceptance. The underlyingidea is to compare aproposed blocked binarymatrix or matrices with thematrix that would beexpected under a nullhypothesis, using eithercorrelations or a chi-square

statistic as the measure ofassociation. Several of thesetechniques have severeconstraints on the types ofdata to which they can beapplied and have unknowndistributional propertiesthat do not permitlegitimate tests of statisticalsignificance. And in somecases, the hypothesizedcomparison model is trivial,e.g., a matrix of randomrelations, whose rejectionadds little to what is already

known about the datastructure. Clearly, statisticsfor testing blockmodelhypotheses is an area ofnetwork research greatly inneed of much development.Another topic needingfurther development is therelative importance ofvarious network relations inmultiple network analyses.When several matrices areused jointly in partitioning aset of actors into

subgroups, their differentialdensities can affect theclustering results. Atpresent, adequate criteriahave not been developedfor determining

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the degree of redundancyamong multiple networkrelations or for assessingthe unique contributionsamong a set of matrices inproducing a joint partition.Weighting effects instructural equivalenceprocedures is an importanttopic for future research.

Analysis of StructuralCorrelates

The preceding sections ofthis chapter discussed atlength some procedures forinvestigating twofundamental components ofnetwork analysis. This firstcomponent consists ofmeans for detectingnetwork structures, such ascliques and structurallyequivalent positions, bypartitioning a relationalmatrix into subgroups. Thesecond component involvescharacterizing actors,

subgroups, or completenetworks in variousstructural terms, such ascentrality, prestige,connectedness,multiplexity, density, andthe like. Upon completion ofthese preliminary tasks, anetwork researcher mayapply a third component ofnetwork analysis, whichmay be called the analysisof homophyly or, moresimply, of structuralcorrelates.

The basis questionaddressed by theseprocedures is: What are thecauses, concomitants, orconsequences of socialstructure for actors,subgroups, or completenetworks? For example,after dividing a network ofcommunity elites intosubgroups based on thepatterns of communityaffairs discussions, apolitical sociologist maywish to determine the

extent to which thesestructurally equivalentpositions are ideologicallyhomogeneous. Given somemeasure of individuals'political ideologies, anappropriate test might bean analysis of varianceusing the groups as thetreatment variable. Asignificant F ratio wouldindicate that the positionsvary in their mean politicalviews, the correlation ratiowould reveal how strongly

structural position predictsthe elite individual'sideologies, and the effectcoefficients would showwhich groups are aboveand below the mean on theideology scale.Attribute data could becombined with relationalmeasures in a general linearmodel. Thus known orhypothesized variables suchas elites' ages, incomes,and party affiliations could

be entered into a regressionequation along with a set ofdummy variables reflectingposition membership andthe net effects of eachpredictor variable on thepolitical ideology scale couldbe estimated by ordinaryleast squares

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procedures. For examples ofsuch combined attribute-relation analyses seeBreiger (1976) and Mullinset al. (1977) on naturalscientists, Galaskiewicz(1979) and Knoke(forthcoming) oncommunity organizations,and Snyder and Kick (1979)on the world system ofnationstates.Laumann et al. (1974)

demonstrated anotherapproach to combiningrelational and attributedata. They developed acausal model in which theunits of analysis were notindividual actors but the(N2N)/2 interpointdistances between pairs ofelites in a small Germantown's communitydiscussion network. Thesedistances were derived froma smallest space solution toa matrix of path distances.

Antecedent variables intheir path model weremeasures of thesimilarity/dissimilarity of theactor pairs on such factorsas religion, party affiliation,and sociopolitical values.The substantive conclusionnetwork was most stronglyaffected by social tiesbetween the actor pairs(see Laumann et al., 1977,for a comparison of findingsfor two American cities).

Other variations on thestructural correlates themeinclude ego-alter strategies(e.g., Duncan et al. 's 1968analysis of occuptionalaspirations of pairs of highschool students; see alsoColeman et al.'s 1966classic study of medicalinnovation diffusion) andgeographic space analysis(Doreian, 1981). Theanalysis of structuralcorrelates is still in thepreliminary stage and the

years ahead will likely see arapid proliferation of newapproaches and formalanalyses of theirmethodological properties.

Other IssuesWe have not begun toexhaust the wide variety ofmethodological topics innetwork analysis.Unfortunately, spacelimitations permit only acursory overview of otherissues, some of which are

still in the developmentalstage.Clusteringwith exceptionally largedata setsfor example, morethan 500 actors or morethan 20 distinctnetworksconventionalpartitioning methods suchas CONCOR may beinefficient and costly.Adaptation of alternativeclustering algorithms, suchas the K-means

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approach (Macqueen,1967; Hartigan and Wong,1979) or a combinatorialapproach (see Arabie andBoorman, 1982), mayprovide a better solution forclustering actors intosubsets.The discrete andcontinuous distancemethods for clusteringstructurally equivalentactors both result in

hierarchical partitionings ofthe networks. That is,mutually exclusiveequivalence classes ofactors are identifiedsimultaneously across all Knetwroks, as shown by adendogram or tree diagram.However, such restrictionsare often conceptuallyunrealistic, as actors maybelong to more than onesocial group (e.g., mayparticipate in several socialcircles, formal

organizations, familialsegments, and so on). Thusnonhierarchical proceduresthat allow overlappingclusters to emerge canprovide an alternativeapproach (Arabie, 1982).The recently developedMAPCLUS (MathematicalProgramming Clustering)algorithm (Arabie andCarroll, 1980) allows theuser to specify the numberof subsets in the finalsolution and does not

restrict subsets to bemaximal completesubgraphs.Structural RelatednessThe structural equivalenceconcept identifies actorsjointly occupying a socialposition on the basis ofsimilar relations to the sameother network actors. Analternativeconceptualization of socialroles argues that actorsshould occupy the same

role to the extent that theymaintain relations toequivalent other networkactors. For example, a setof foremen supervisesdifferent workers, a set offathers disciplines differentchildren, and a set ofprofessors teaches differentstudents. Thus two actorsin a network are structurallyrelated if they areconnected in the same orsimilar ways to structurallyrelated actors (Sailer,

1978). Thisreconceptualization of socialrole requires a differentalgorithm than theblockmodel or continuousdistance procedures tolocate structurally relatedactors (see Mandel andWinship, 1981). White andReitz (1982) provided auseful overview of theseissues.Role AlgebraAs developed by White and

his colleagues (Lorrain andWhite, 1971; Boorman andWhite, 1976),blockmodeling involvesmore than the clusteringtechnique described above.The structural regularitiesamong positions, asrepresented in the binaryimage matrices, could beuncovered by the set ofindirect or compoundrelations formed from all thedirect relations for the Knetworks. The technical

details of this role algebraare too complex to presentin

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this monograph, so theinterested reader is referredto the original articles, aswell as to modificationsproposed by Bonacich(1980; also Bonacich andMcConaghy, 1979; Breigerand Pattison, 1978;McConaghy, 1981).Change Over TimeThe methods and examplesdiscussed in thismonograph all refer to data

collected on networksobserved at a single point intime. Dynamic methods formodeling changing relationsare just coming to theattention of socialresearchers (Wasserman,1979, 1980; Runger andWasserman, 1980;Galaskiewicz andWasserman, 1981). Thesemodels are based oncontinuous-time or discrete-time Markov chains thatenable a researcher to

estimate parametersdescribing the rates offormation or disappearanceof linkages in the networks.As with many aspects ofnetwork research, themethodologicalsophistication in this areaseems to have run ahead ofthe theoretical andsubstantive developmentsthat could take advantageof the methods.Microstructure

Another complex networkmethodology whoseexposition cannot beencompassed within thepresent space is theanalysis of local structures.In a network of N actors,there are (N33N + 2N)/6possible triads of actors, butonly sixteen unique triadstructures involving mutualstrong, asymmetric, ormutual null ties betweenpairs. Various theoreticalmodelsbalance, cluster,

ranked cluster, andtransitivitymake predictionsabout the permissibility ofvarious triads (see Leik andMeeker, 1975: 53-74). Theresearch program of Davis,Holland, and Leinhardt(Davis, 1979) madesubstantial progress inspecifying and testingvarious aspects of suchmodels on triad data (seeHolland and Leinhardt1970, 1975, 1978; Davisand Leinhardt, 1972, for

some basic results).Microstructural analysesdescribe relations amongindividual actors. Stillunresolved is the questionof how such local structurescontribute to structuresobserved at the totalnetwork level of analysis.Computer ProgramsFinally, construction andmaintenance of a data filefor relational and attributedata can be a complex and

time-consuming task.Although every networkresearcher will discoverunique elements to beincorporated for specificobjectives, some features

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will be common to mostresearch projects, especiallythe ability to rapidlyassemble matrices, collapseor reduce the rows andcolumns, symmetrizeentries, and multiply andadd matrix elements.Roistacher (1979) andSonquist (1980; see alsoMulherin et al., 1981) offeruseful suggestions fororganizing network data

files. Sonquist (1980) alsoprovides a census of 28computer programs forvarious types of networkanalyses, includinginformation on themachines and computerlanguages in which theywere originally written, aswell as references forlocating further details.Many designers make theirprograms and manualsavailable at the nominalcost of materials and

postage. Readers areadvised to consult recentissues of the journals SocialNetworks and Connectionsfor updated listings of newprograms.

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ReferencesALBA, R.D. (1973) ''Agraph-theoretic definition ofa sociometric clique."Journal of MathematicalSociology 3: 113-126.and C. KADUSHIN (1976)"The intersection of socialcircles: a new measure ofsocial proximity innetworks." Sociological

Methods & Research 5: 77-102.ALBA, R.D. and G. MOORE(1978) "Elite social circles."Sociological Methods &Research 7: 167-188.ALLEN, M.P. (1974) "Thestructure ofinterorganizational elitecooptation: interlockingcorporate directors."American SociologicalReview 39: 393-406.

ANDERSON, B. (1979) "Cognitive balance theoryand social network analysis:remarks on somefundamental theoreticalmatters," pp. 453-469 inPaul W. Holland andSamuel Leinhardt (eds.)Perspectives on SocialNetwork Research. NewYork: Academic.ARABIE, P. (1982)"Conceptions of overlap insocial structure," in L.

Freeman, A.K. Romney, andD.R. White (eds.) Methodsof Social Network Analysis.and S. BOORMAN (1982)"Blockmodels:developments andprospects," in H. Hudson(ed.) Classifying SocialData: New Applications ofAnalytic Methods for SocialScience Research. SanFrancisco: Jossey-Bass.and P.R. LEVITT (1978)"Constructing blockmodels:

how and why." Journal ofMathematical Psychology17: 21-63.ARABIE, P. and J.D.CARROLL (1980) MAPPLUS:a mathematicalprogramming approach tofitting the ADCLUS model."Psychometrika 45: 211-235.ATKIN, R.H. (1977)Combinatorial Connectivitiesin Social Systems: AnApplication of Simplicial

Complex Structures to theStudy of LargeOrganization. Basel:Birkhauser Verlag.(1974) MathematicalStructure in Human Affairs.London: Heinemann.AUGUSTSON, J.G. and J.MINKER (1970) "Ananalysis of some graphtheoretical clustertechniques." Journal of theACM 17: 571-588.

BAILEY, K. (1974) "Clusteranalysis," pp.59-128 in D.R.Heise (ed.) SociologicalMethodology 1975. SanFrancisco: Jossey-Bass.BARNES, J.A. (1979)"Network analysis: orientingnotion, rigorous techniqueor substantive field ofstudy?" pp.403-423 in PaulHolland and SamuelLeinhardt (eds.)Perspectives on SocialNetwork Research. New

York: Academic.BAVELAS, A. (1950)"Communication patterns intask oriented groups."Journal of the AcousticalSociety of America 22: 271-282.BEHZAD, M. and G.CHARTRAND (1971)Introduction to the Theoryof Graphs. Boston: Allyn &Bacon.BENIGER, J.R. (1976)

"Sampling social networks:the subgroup approach."Proceedings of the Businessand Economic StatisticsSection, AmericanStatistical Association: 226-231.BERNARD, H: R. and P.D.KILLWORTH (1978) "Areview of the small-worldliterature." Connections 2:15-24.(1977) "Informant accuracyin social network data II."

Human CommunicationResearch 4: 3-18.

Page 88

and L. SAILER (1980)"Informant accuracy insocial network data IV: acomparison of clique-levelstructure in behavioral andcognitive network data."Social Networks 2: 191-218.BOISSEVAIN, J. (1974)Friends of Friends:Networks, Manipulators,and Coalitions. New York:St. Martin's.

BONACICH, P. (1980) "The'common structure,' areplacement for theBoorman and White 'jointreduction'." AmericanJournal of Sociology 86:159-166.and M.J. (1979) "Thealgebra of blockmodeling,"pp.489-532 in K.F.Schuessler (ed.)Sociological Methodology1980. San Francisco:Jossey-Bass.

BOORMAN, S.A. and H.C.WHITE (1976) "Socialstructure from multiplenetworks. II. rolestructures." AmericanJournal of Sociology 81:1384-1446.BOTT,E. (1955) "Urbanfamilies: conjugal roles andsocial networks." HumanRelations 8: 345-383.BREIGER, R.L. (1976)"Career attributes andnetwork structure: a

blockmodel study of abiomedical researchspecialty." AmericanSociological Review 41:117-135.(1974) "The duality ofpersons and groups." SocialForces 53: 181-190.S.A. BOORMAN, and P.ARABIE (1975) "Analgorithm for clusteringrelational data, withapplications to socialnetwork analysis and

comparison withmultidimensional scaling."Journal of MathematicalPsychology 12: 328-383.BREIGER, R.L. and P.E.PATTISON (1978) "Thejoint role structure of twocommunities' elites."Sociological Methods &Research 7: 213-226.BURT, R.S. (1982) Towarda Structural Theory ofAction: Network Models ofSocial Structure,

Perceptions and Action.New York: Academic.(1981) "Studyingstatus/role-sets as ersatznetwork positions in masssurveys." SociologicalMethods & Research 9:313-337.(1980) "Models of networkstructure." Annual Reviewof Sociology 6: 79-141.(1979) "A structural theoryof interlocking corporate

directorates." SocialNetworks 1: 415-435.(1978) "Cohesion versusstructural equivalence as abasis for networksubgroups." SociologicalMethods & Research 7:189-212.(1977a) "Positions inmultiple network systems.Part one: a generalconception of stratificationand prestige in a system ofactors cast as a social

typology." Social Forces 57:106-131.(1977b) "Positions inmultiple network systems.Part two: stratification andprestige among elitedecision-makers in thecommunity of Altneustadt."Social Forces 56: 551-575.(1976) "Positions innetworks." Social Forces55: 93-122.(1975) "Corporate society:

a time series analysis ofnetwork structure." SocialScience Research 4: 271-328.and W. M. BITTNER (1981)"A note on inferencesregarding network sub-groups." Social Networks 3:71-83.BURT, R.S., K.P.CHRISTMAN, and H.C.KILBURN, Jr. (1980)"Testing a structural theoryof corporate cooptation:

interorganizationaldirectorate ties as astrategy for avoiding marketconstraints on profits."American SociologicalReview 45: 821-841.BURT, R.S. and N. LIN(1977) "Network time seriesfrom archival records," pp.224-254 in D.R. Heise (ed.)Sociological Methodology1977. San Francisco:Jossey-Bass.

Page 89

CAPOBIANCO, M.F. (1970)"Statistical inference infinite populations havingstructure." Transactions ofthe New York Academy ofSciences 32: 401-413.CARRINGTON, P.J. andG.H. HEIL (1981)"COBLOC: a hierarchicalmethod for blockingnetwork data." Journal ofMathematical Sociology 8:103-131.

and S.D. BERKOWITZ(1980) "A goodness-of-fitindex for blockmodels."Social Networks 2: 219-234.CARTWRIGHT, D. and F.HARARY (1956) "Structuralbalance: a generalization ofHeider's theory."Psychological Review 63:277-292.COLEMAN, J.S., E. KATZ,and H. MENZEL (1966)Medical Innovation: A

Diffusion Study.Indianapolis: Bobbs-Merrill.COLEMAN, J.S. and D.MacRAE, Jr. (1960)"Electronic processing ofsociometric data for groupsup to 1000 in size."American SociologicalReview 25: 722-727.COOK, K.S. and R.M.EMERSON (1978) "Power,equity and commitment inexchange networks."American Sociological

Review 43: 721-739.CRANE, D. (1969) "Socialstructure in a group ofscientists: a test of the'invisible college'hypothesis." AmericanSociological Review 34:335-352.DAVIS, J.A. (1979) "TheDavis/Holland/Leinhardtstudies: an overview," pp.51-62 in P.W. Holland andS. Leinhardt (eds.)Perspectives on Social

Network Research. NewYork: Academic.and S. LEINHARDT (1972)"The structure of positiveinterpersonal relations insmall groups," pp. 218-251in J. Berger (ed.)Sociological Theories inProgress, Vol. 2. Boston:Houghton-Mifflin.DOREIAN, P. (1981) "Onthe estimation of linearmodels with spatiallydistributed data," pp. 359-

388 in Samuel Leinhardt(ed.) SociologicalMethodology 1981. SanFrancisco: Jossey-Bass.(1980) "On the evolution ofgroup network structure."Social Networks 2: 235-254.(1969) "A note on thedetection of cliques invalued graphs." Sociometry32: 237-242.DUFFIN, R.J., E.L.

PETERSON, and C. ZENER(1967) GeometricProgramming. New York:John Wiley.DUNCAN, O.D., A.O.HALLER, and A. PORTES(1968) "Peer influences onaspirations: areinterpretation." AmericanJournal of Sociology 74:119-137.ERICKSON, B.H., T.A.NOSANCHUK, and E. LEE(1981) "Network sampling

in practice: some secondsteps." Social Networks 3:127-136.EULAU, H. (1980) "TheColumbia studies ofpersonal influence." SocialScience History 4: 207-228.FESTINGER, L. (1949) "Theanalysis of sociograms usingmatrix algebra." HumanRelations 2: 153-158.FISCHER, C.S. (1982) ToDwell Among Friends:

Personal Networks in Townand City. Chicago:University of Chicago Press.FLAMENT, C. (1963)Applications of GraphTheory to Group Structure.Englewood Cliffs, NJ:Prentice-Hall.FORSYTH, E. and L. KATZ(1946) "A matrix approachto the analysis ofsociometric data:preliminary report."Sociometry 9: 340-347.

Page 90

FRANK, O. (1979)"Estimation of populationtotals by use of snowballsamples," pp. 319-347 inPaul W. Holland andSamuel Leinhardt (eds.)Perspectives on SocialNetworks Research. NewYork: Academic.(1978) "Sampling andestimation in large socialnetworks." Social Networks1: 91-101.

(1971) Statistical Inferencein Graphs. Stockholm,Sweden: Swedish ResearchInstitute of NationalDefense.FREEMAN, L.C. (1979)"Centrality in socialnetworks. I. Conceptualclarification." SocialNetworks 1: 215-239.(1977) "A set of measuresof centrality based onbetweenness." Sociometry40: 35-41.

GALASKIEWICZ, J. (1979)Systems of CommunityInterorganizationalExchange. Beverly Hills, CA:Sage.and S. WASSERMAN (1981)"A dynamic study of changein a regional corporatenetwork." AmericanSociological Review 46:475-484.GOODMAN, L.A. (1961)"Snowball sampling."Annals of Mathematical

Statistics 32: 148-170.GOULD, P. and A. GATRELL(1980) "A structuralanalysis of a game: theLiverpool v Manchesterunited cup final of 1977."Social Networks 2: 253-274.GRANOVETTER, M. (1977)"Reply to Morgan andRytina." American Journalof Sociology 83: 727-729.(1976) "Network sampling:

some first steps." AmericanJournal of Sociology 81:1287-1303.(1974) Getting a Job: AStudy of Contacts andCareers. Cambridge, MA:Harvard University Press.(1973) "The strength ofweak ties." AmericanJournal of Sociology 78:1360-1380.HALLINAN, M.T. (1978)"The process of friendship

formation." Social Networks24: 193-210.(1974) The Structure ofPositive Sentiment.Amsterdam: Elsevier.HANSEN, K.L. (1981)"Black' exchange and itssystem of social control,"pp. 70-83 in David Willerand Bo Anderson (eds.)Networks, Exchange, andCoercion. New York:Elsevier-North Holland.

HARARY, F. (1969) GraphTheory. Reading, MA:Addison-Wesley.(1959) "On themeasurement of structuralbalance." BehavioralScience 4: 316-323.R. NORMAN, and D.CARTWRIGHT (1965)Structural Models. NewYork: John Wiley.HARARY, F. and I.C. ROSS(1957) "A procedure for

clique detection using thegroup matrix." Sociometry20: 205-215.HARTIGAN, J.A. (1975)Clustering Algorithms. NewYork: John Wiley.and M.A. WONG (1979) "K-means clusteringalgorithm." Journal of theRoyal Statistical Society C(Applied Statistics) 28:100-108.HEIDER, F. (1979) "On

balance and attribution,"pp. 11-23 in Paul W.Holland and SamuelLeinhardt (eds.)Perspectives on SocialNetwork Research. NewYork: Academic.(1946) "Attitudes andcognitive organizations."Journal of Psychology 21:107-112.

Page 91

HEIL, G. and H.C. WHITE(1976) "An algorithm forfinding simultaneoushomomorphiccorrespondences betweengraphs and their imagegraphs." Behavioral Science21: 26-45.HOLLAND, P.W. and S.LEINHARDT (1978) "Anomnibus test for socialstructure using triads."Sociological Methods &

Research 7: 227-256.(1975) "Local structure insocial networks," pp. 1-45in D.R. Heise (ed.)Sociological Methodology1976. San Francisco:Jossey-Bass.(1970) "A method fordetecting structure insociometric data.""American Journal ofSociology 76: 492-513.HOMANS, G. (1950) The

Human Group. New York:Harcourt, Brace,Jovanovich.HUBBELL, C.H. (1965) "Aninput-output approach toclique identification."Sociometry 28: 377-399.HUNTER, J.E. (1979)"Toward a generalframework for dynamictheories of sentiment insmall groups derived fromtheories of attitudechange," pp. 223-238 in

Paul W. Holland andSamuel Leinhardt (eds.)Perspectives on SocialNetwork Research. NewYork: Academic.JOHNSON, S. C. (1967)"Hierarchical clusteringschemes." Psychometrika32: 241-254.JÖRESKOG, K.G. (1969) "Ageneral approach toconfirmatory maximumlikelihood factor analysis."Psychometrika 34: 183-

202.KADUSHIN, C. (1968)"Power, influence, andsocial circles: A newmethodology for studyingopinion makers." AmericanSociological Review 33:685-699.(1966) "The friends andsupporters ofpsychotherapy: on socialcircles in urban life."American SociologicalReview 31: 786-802.

KANDEL, D.B. (1978)"Homophily, selection, andsocialization in adolescentfriendships." AmericanJournal of Sociology 84:427-436.KAPFERER, B. (1969)"Norms and themanipulation ofrelationships in a workcontext," pp. 181-244 in J.Clyde Mitchell (ed.) SocialNetworks in UrbanSituations. Manchester,

England: ManchesterUniversity Press.KILLWORTH, P.D. and H.R.BERNARD (1979)"Informant accuracy insocial network data III. Acomparison of triadicstructure in behavioral andcognitive data." SocialNetworks 2: 19-46.(1976) "Informant accuracyin social network data."Human Organization 35:269-286.

KLOVDAHL, A.S. (1981) "Anote on images ofnetworks." Social Networks3: 197-214.KNOKE, D. (forthcoming)"Organization sponsorshipand influence reputation ofsocial influenceassociations." Social Forces.(1981) "Commitment anddetachment in voluntaryassociations." AmericanSociological Review 46:141-158.

and R.S. BURT (1982)"Prominence," in Ronald S.Burt and Michael J. Minor(eds.) Applied NetworkAnalysis: StructuralMethodology for EmpiricalSocial Research. BeverlyHills, CA: Sage.KNOKE, D. andE.O.LAUMANN (1982) "Thesocial organization ofnational policy domains," inNan Lin and Peter V.Marsden (eds.) Social

Structure and NetworkAnalysis. Beverly Hills, CA:Sage.KNOKE, D and J.R. WOOD(1981) Organized forAction: Commitment inVoluntary Associations. NewBrunswick, NJ: RutgersUniversity Press.

Page 92

KRUSKAL, J.B. and M.WISH (1978)Multidimensional Scaling.Beverly Hills, CA: Sage.LANKFORD, P.M. (1974)"Comparative analysis ofclique identificationmethods." Sociometry 37:287-305.LAUMANN, E.O. (1973)Bonds of Pluralism. NewYork: John Wiley.

(1966) Prestige andAssociation in an UrbanCommunity. Indianapolis:Bobbs-Merrill.J. GALASKIEWICZ, and P.V.MARSDEN (1978)"Community structure asinterorganizationallinkages," pp. 455-484 in A.Inkeles et al. (eds.) AnnualReview of Sociology, Vol. 4.Palo Alto, CA:AnnualReviews.LAUMANN, E.O., P.V.

MARSDEN, and J.GALASKIEWICZ (1977)"Community influencestructures: Replication andextension of a networkapproach." AmericanJournal of Sociology 85:594-631.LAUMANN, E.O., P V.MARSDEN, and D.PRENSKY (1982) "Theboundary specificationproblem in networkanalysis," in R. S. Burt and

M. J. Minor (eds.) AppliedNetwork Analysis: StructuralMethodology for EmpiricalSocial Research. BeverlyHills, CA:Sage.LAUMANN, E.O. and F.U.PAPPI (1976) Networks ofCollective Action: APerspective on CommunityInfluence Systems. NewYork: Academic.LAUMANN, E.O., L.M.VERBRUGGE, and F.U.PAPPI (1974) "A causal

modelling approach to thestudy of a community elite'sinfluence structure," inEdward O. Laumann andFranz U. Pappi, Networks ofCollective Action. New York:Academic.LAZARSFELD, P.F., B.R.BERELSON, and H. GAUDET(1948) The People'sChoice: How the VoterMakes Up His Mind in aPresidential Campaign. NewYork: Columbia University

Press.LEAVITT, H.J. (1951)"Some effects ofcommunication patterns ongroup performance."Journal of AbnormalPsychology 46: 38-50.LEIK, R.K. and B.F.MEEKER (1975)Mathematical Sociology.Englewood Cliffs, NJ:Prentice-Hall.LEIK, R.K. and R.

NAGASAWA (1970) "Asociometric basis formeasuring social status andsocial structure."Sociometry 33: 55-78.LEINHARDT, S. (1972)"Developmental change inthe sentiment and structureof children's groups."American SociologicalReview 37: 202-212.LEVINE, J. (1972) "Thesphere of influence."American Sociological

Review 37: 14-27.LIN, N. (1976) Foundationsof Social Research. NewYork: McGraw-Hills.(1975) "Analysis ofcommunication relations,"in Gerhard J. Hannemanand William J. McElwen(eds.) Communication andBehavior. Reading, MA:Addison-Wesley.P.W. DAYTON, and P.GREENWALD (1978)

"Analyzing the instrumentaluse of relations in thecontext of social structure."Sociological Methods &Research 7: 149-166.(1977) "The urbancommunication networkand social stratification: a'small world' experiment,"pp. 107-119 in D.B. Ruben(ed.) CommunicationYearbook, Vol. 1. NewBrunswick, NJ: TransactionBooks.

LINCOLN, J.R. and J.MILLER (1979) "Work andfriendship ties inorganizations: Acomparative analysis ofrelational networks."Administrative ScienceQuarterly 24: 181-199.

Page 93

LINDZEY, G. and E. F.BORGATTA (1954)"Sociometricmeasurement," pp. 405-448 in Gardner Lindzey(ed.) Handbook of SocialPsychology, Vol. 1.Cambridge, MA: Addison-Wesley.LINTON, R. (1936) TheStudy of Man. New York:Appleton-Century-Crofts.LORRAIN, F. and H. C.

WHITE (1971) "Structuralequivalence of individuals insocial networks." Journal ofMathematical Sociology 1:49-80.LUCE, R. D. (1950)"Connectivity andgeneralized cliques insociometric groupstructure." Psychometrika15: 169-190.and A. PERRY (1949) "Amethod of matrix analysis ofgroup structure."

Psychometrika 14: 94-116.McCALLISTER, L. and C. S.FISCHER (1978) "Aprocedure for surveyingpersonal networks."Sociological Methods &Research 7: 131-148.McCONAGHY, M. J. (1981)"The common rolestructure: improvedblockmodeling methodsapplied to two communities'elites." Sociological Methods& Research 9: 267-285.

McFARLAND, D. and D.BROWN (1973) "Socialdistance as a metric: asystematic introduction tosmallest space analysis,"pp. 213-253 in Edward O.Laumann, Bonds ofPluralism: The Form andSubstance of Urban SocialNetworks. New York: JohnWiley.MacQUEEN, J. (1967)"Some methods forclassification and analysis of

multivariate observations,"pp. 281-297 in L. M. LeCanand J. Neyman (eds.)Proceedings of the FifthBerkeley Symposium onMathematical Statistics andProbability, Vol. 1. Berkeley:University of CaliforniaPress.McQUITTY, L. L. (1957)"Elementary linkageanalysis for isolatingorthogonal and obliquetypes and typal

relevancies." Educationaland PsychologicalMeasurement 17: 207-229.MacRAE, D., Jr. (1960)"Direct factor analysis ofsociometric data."Sociometry 23: 360-371.MANDEL, M. and C.WINSHIP (1981) "Rolesand positions: an extensionof the blockmodelapproach." NorthwesternUniversity, Evanston,Illinois. (unpublished)

MARIOLIS, P. (1975)"Interlocking directoratesand control ofcorporations." SocialScience Quarterly 56: 425-439.MARSHALL, J. F. (1971)"Topics and networks inintra-villagecommunication," pp. 160-166 in Steven Polgar (ed.)Culture and Population: ACollection of CurrentStudies. Monograph 9.

Chapel Hill, NC: CarolinaPopulation Center.MILGRAM, S. (1967) "Thesmall world problem."Psychology Today 1: 61-67.MILLER, J., J. R. LINCOLN,and J.OLSON (1981)"Rationality and equity inprofessional networks:gender and race as factorsin the stratification ofinterorganizationalsystems." American Journalof Sociology 87: 308-335.

MITCHELL, J. C. (1969)"The concept and use ofsocial networks," pp. 1-50in J. Clyde Mitchell (ed.)Social Networks in UrbanSituations. Manchester,England: ManchesterUniversity Press.MORENO, J. L. (1934) WhoShall Survive? Foundationof Sociometry, GroupPsychoterapy, andSociodrama. Washington,DC: Nervous and Mental

Disease Monograph 58.MORGAN, D. L. and S.RYTINA (1977) "Commenton 'Network Sampling:some first steps' by MarkGranovetter." AmericanJournal of Sociology 83:722-727.

Page 94

MORGAN, D. L. and L. S.WOLFARTH (1980) "Testingthe goodness-of-fit of block-models." Indiana University.(mimeo)MULHERIN, J. P., H. M.KAWABATA, and J. A.SONQUIST (1981) "Relateddata-bases for combinednetwork and attribute datafiles: an SASimplementation."Connections 4: 22-31.

MULLINS, N., L.HARGENS,P. HECHT, and E. KICK(1977) "The groupstructure of cocitationclusters: a comparativestudy." AmericanSociological Review 42:552-562.NADEL, S. F. (1957) TheTheory of Social Structure.London: Cohen & West.PERRUCCI, R. and M.PILISUK (1970) "Leadersand ruling elites: the

interorganizational bases ofcommunity power."American SociologicalReview 35: 1040-1056.Project in StructuralAnalysis (1981)STRUCTURE: a computerprogram providing basicdata for the analysis ofempirical positions in asystem of actors. ComputerProgram 1. Berkeley:University of California,Survey Research Center.

PROCTOR, C. H. (1979)"Graph sampling comparedto conventional sampling,"pp. 301-318 in Paul W.Holland and SamuelLeinhardt (eds.)Perspectives on SocialNetwork Research. NewYork: Academic.RAPOPORT, A. and W. J.HOVARTH (1961) "A studyof a large sociogram."Behavioral Science 6: 279-291.

ROGERS, E. M. and D. L.KINCAID (1981)Communication Networks:Toward a New Paradigm forResearch. New York:Macmillan.ROISTACHER, R. C. (1979)"Acquisition andmanagement of socialnetwork data," pp. 471-487in P. W. Holland and S.Leinhardt (eds.)Perspectives on SocialNetwork Research. New

York: Academic.(1974) "A review ofmathematical models insociometry." SociologicalMethods & Research 3:123-171.ROSKAM, E. and J. C.LINGOES (1970)"MINESSA-1: a FORTRANprogram for the smallestspace analysis of squaresymmetric matrices."Behavioral Science 15: 204-220.

RUNGER, G. andS.WASSERMAN (1980)"Longitudinal analysis offriendship networks." SocialNetworks 2: 143-154.SAILER, L. D. (1978)"Structural equivalence:meaning and definition,computation andapplication." SocialNetworks 1: 73-90.SAMPSON, S. F. (1969)"Crisis in a cloister." Ph.D.dissertation, Cornell

University, Ithaca, NewYork.SCHWARTZ, J. E. (1977)"An examination ofCONCOR and relatedmethods for blockingsociometric data," pp. 255-282 in D. R. Heise (ed.)Sociological Methodology1977. San Francisco:Jossey-Bass.SEIDMAN, S. B. and B. L.FOSTER (1978) "A graph-theoretic generalization of

the clique concept." Journalof Mathematical Sociology6: 139-154.SHEINGOLD, C. A. (1973)"Social networks andvoting: the resurrection of aresearch agenda."American SociologicalReview 39: 712-720.SNYDER, D. and E. L.KICK(1979) "Structural positionin the world system andeconomic growth, 1955-1970: a multiple network

analysis of transnationalinteractions." AmericanJournal of Sociology 84:1096-1126.SONQUIST, J. (1980)"Concepts and tactics inanalyzing social networkdata." Connections 3: 33-56.

Page 95

and T. KOENIG (1975)"Interlocking directorates inthe top U.S. corporations: agraph theory approach."Insurgent Sociologist 5:196-229.TRAVERS, J. and S.MILGRAM (1969) "Anexperimental study of thesmall world problem."Sociometry 32: 425-443.Van der GEER, J. P. (1971)Introduction to Multivariate

Analysis for the SocialSciences. San Francisco:Freeman.WASSERMAN, S. (1980)"Analyzing social networksas stochastic processes."Journal of the AmericanStatistical Association 75:280-294.(1979) "A stochastic modelfor directed graphs withtransition rates determinedby reciprocity," pp. 392-412in K. Schuessler (ed.)

Sociological Methodology1980. San Francisco:Jossey-Bass.WHITE, D. R. and K. P.REITZ (1982) "Rethinkingthe role concept: socialnetworks, homorphisms andequivalence of positions," inL. C. Freeman, A. K.Romney, and D. R. White(eds.) Research Methodsand Social NetworkAnalysis. Irvine: Universityof California, School of

Social Sciences.(unpublished)WHITE, H. C. (1977)"Probabilities ofhomomorphic mappingsfrom multiple graphs."Journal of MathematicalPsychology 16: 121-134.(1963) An Anatomy ofKinship. Englewood Cliffs,NJ: Prentice-Hall.(1961) "Managementconflict and sociometric

structure." AmericanJournal of Sociology 67:185-187.A. BOORMAN, and R. L.BREIGER (1976) "Socialstructure from multiplenetworks 1. Blockmodels ofroles and positions."American Journal ofSociology 81: 730-780.WILLIAMSON, O. E. (1970)Corporate Control andBusiness Behavior.Englewood Cliffs, NJ:

Prentice-Hall.WOOD, J. R. (1981)Leadership in VoluntaryOrganizations: TheControversy Over SocialAction in ProtestantChurches. New Brunswick,NJ: Rutgers UniversityPress.