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An Experimental Study of the Small World Problem Author(s): Jeffrey Travers and Stanley Milgram Source: Sociometry, Vol. 32, No. 4 (Dec., 1969), pp. 425-443 Published by: American Sociological Association Stable URL: http://www.jstor.org/stable/2786545 Accessed: 23/09/2010 13:05 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=asa. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected] American Sociological Association is collaborating with JSTOR to digitize, preserve and extend access to Sociometry. http://www.jstor.org

The Small Word Problem

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An Experimental Study of the Small World Problem Author(s): Jeffrey Travers and Stanley Milgram Source: Sociometry, Vol. 32, No. 4 (Dec., 1969), pp. 425-443 Published by: American Sociological Association Stable URL: http://www.jstor.org/stable/2786545 Accessed: 23/09/2010 13:05Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=asa. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]

American Sociological Association is collaborating with JSTOR to digitize, preserve and extend access to Sociometry.

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An ExperimentalStudy of the Small World Problem*JEFFREY TRAVERS Harvard UniversityAND

STANLEY MILGRAM The City University New York of individuals selected Arbitrarily (N-296) in Nebraskaand Bostonare asked chainsto a target to generate acquaintance personin Massachusetts, employmethod" (Milgram, 1967). Sixty-four chains reach ing "the small world thisgroupthe mean number intermediaries bethe target person.Within of the target chains reach tweenstarters and targetsis 5.2. Boston starting in withfewer than person intermediaries thosestarting Nebraska; subpopulaThe funneling tionsin the Nebraskagroupdo not differ amongthemselves. with48 per centof the chains of chainsthrough sociometric "stars"is noted, the three passingthrough personsbefore reaching target. Applications the of method studiesof large scale social structure discussed. to are the The simplest is way of formulating smallworldproblem "what is the from largepopulation, a thatany twopeople,selectedarbitrarily probability suchas thatof theUnitedStates,willknoweach other?"A moreinteresting takes accountof the fact that,whilepersonsa and z formulation, however, may not know each other directly, they may share one or more mutual acquaintances;that is, theremay exista set of individuals, (consisting B, of individuals b2 . . . bn) who knowboth a and z and thus link them bi, a More generally, and z may be connected by any single to one another. not common but acquaintance, by a seriesof suchintermediaries, a-b-c- . . -y-z; . i.e., a knowsb (and no one else in the chain); b knowsa and in addition knowsc, c in turnknowsd, etc. To elaboratethe problemsomewhat let further, us represent populathe* The study was carriedout while both authors were at Harvard University, and was financedby grants from the Milton Fund and from the Harvard Laboratory of Social Relations.Mr. JosephGerverprovidedinvaluable assistancein summarizing and criticizing work discussedin thispaper. the mathematical 425

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set tion of the United States by a partiallyconnected of points.Let each a that pointrepresent person,and let a line connecting pointssignify two the twoindividuals knoweach other.(Knowingis hereassumedto be symmetric:if a knowsb thenb knowsa. Substantively, "knowing" used to is denotea mutualrelationship; othersenses of the verb,e.g. knowing about a famousperson,are excluded.) The structure takes the formof a cluster of roughly200 millionpoints with a complexweb of connections among them.The acquaintancechains describedabove appear as pathwaysalong connected line segments. Unless some portionof the populationis totally isolated fromthe rest,such that no one in that subgroupknows anyone outsideit, there mustbe at least one chainconnecting twopeoplein the any population. generalthere In will be manysuch pathways, variouslengths, of betweenany two individuals. In view of such a structure, way of refining statement the one our of smallworld problem thefollowing: is giventwoindividuals selected randomly fromthe population, what is the probability that the minimum numberof mightask not about the minimum chains betweenpairs of people, but mean chain lengths, median chain lengths, etc.) the small worldproblemis to Perhapsthe most directway of attacking chainsin a largepopulation. This is the tracea number real acquaintance of in techniqueof the study reported this paper. The phrase "small world" are thatsocial networks in some sense tightly suggests woven,fullof unexfromone another individuals far pected strandslinking seemingly removed in physicalor social space. The principal questionof the presentinvestigacould be demonstrated tion was whethersuch interconnectedness experimentally. treatment with the The only exampleof mathematical dealing directly small world problemis the model providedby Ithiel Pool and Manfred Pool and Kochen assume a population Kochen (unpublished manuscript). each of whom knows,on the average,n othersin the of N individuals, that two persons population. They attemptto calculatePk, the probability from groupcan be linkedby a chainof k intermediaries. the chosenrandomly of Theirbasic modeltakesthe form a "tree"or geometric progression. Using of volumeprovided Gurevitch an estimate averageacquaintance by (1961), will to theydeducethattwo intermediaries be required link typicalpairs of in of individuals a population 200 million. Theirmodeldoes not take account of social structure. Instead of allowingacquaintance nets to definethe Pool and Kochenmust,forthepurof boundaries functioning social groups, into a number poses of theirmodel,conceiveof societyas beingpartitioned of hypothetical each withidentical groups, populations. They are thenableintermediariesrequired to link them is 0, 1, 2, . . . k? (Alternatively,one

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to devisea way to predict chainlengths within between and suchhypothesized groups. In an empirical studyrelatedto the small worldproblemRapoportand Horvath (1961) examinedsociometric nets in a junior high school of 861 students. The authorsasked studentsto name in order their eight best friends within school.They thentracedthe acquaintancechains created the in i.e. by the students'choices. Rapoportwas interested connectivity, the fraction the total populationthat would be contacted tracingfriendof by ship choices froman arbitrarystartingpopulation of nine individuals. Rapoportand his associates (Rapoport and Horvath, 1961; Foster et al., 1963; Rapoport, 1953; 1963) have developed a mathematical model to describethis tracingprocedure. The model takes as a point of departure random nets constructed the following in manner:a smallnumber points of is chosenfrom largerpopulationand a fixednumberof "axones" is exa tendedfrom each of thesepointsto a set of target pointschosenat random from population. the The same fixed number axonesis thenextended of from each of the targetpoints to a set of second generation targetpoints,and the process is repeatedindefinitely. targetpoint is said to be of the A if tthremove it is of the tthgeneration no lowergeneration. and Rapoport thensuggests formula calculating a for the fraction, of the population Pt, pointswhichare targetsof the tth remove.He is also able to extendthe formulato nonrandom nets, such as those created in the Rapoport and Horvath empiricalstudy,by introducing numberof "biases" into the a random model.Rapoportshowsthattwoparameters, net obtainablefrom the data, are sufficient producea closefitbetween predictions themodel to the of and the empirical outcome the traceprocedure.1 of Rapoport'smodel was designedto describea trace procedure quite difin it ferent fromthe one employed the presentstudy; however, has some If of relationto the small worldproblem. we set the number axones traced from givenindividualequal to the total numberof acquaintancesof an a average person, the Rapoport model predicts the total fractionof the populationpotentiallytraceable at each remove fromthe start, serving the precisely aims of the modelof Pool and Kochen. (It should,however, be noted that Rapoport'smodel deals with asymmetric nets, and it would the to be difficult modify modelto deal withgeneralsymmetric nets,which the characterize small worldphenomenon.) Despite the goodnessof fitbetweenRapoport'smodeland the data from1 Thereis additional evidence(Fararo and Sunshine, 1964) and theoretical empirical to are that 1967) fortheassumption twoparameters sufficient describe (Abelson, support on effects biaseshave minimal i.e. tracing procedure, thatmorecomplex the Rapoport nets. in connectivity friendship

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two large sociograms,there are unsolved problems in the model, as Rapoporthimself and others (Fararo and Sunshine,1964) have pointed difficult an empirifor out. The Pool-Kochenmodel involvesassumptions that society to cally oriented social scientist accept,such as the assumption and may be partitioned into a set of groupsalike in size and in internal to In externalconnectedness. the absence of empiricaldata, it is difficult On knowwhichsimplifying assumptions likelyto be fruitful. the other are hand, with regardto the empiricalstudy of Rapoport and Horvath,the and was small,well-defined, homofact that the total population employed geneousleaves open manyquestionsabout the natureof acquaintancenets in thelarger society.2 empirical An studyof American society a wholemay as well uncoverphenomena interest of both in theirown rightand as conon straints the natureof any correct model of the structure mathematical of large-scale acquaintanceship nets.PROCEDURE

for This paper follows procedure tracing the acquaintancechains devised and firsttestedby Milgram (1967). The presentpaper introduces exan perimental variationin this procedure, varying"starting by populations"; it also constitutes first a technicalreporton the small worldmethod. The procedure an maybe summarized follows: arbitrary as "target person" and a groupof "starting and an attempt persons"wereselected, was made to generatean acquaintancechain fromeach starterto the target.Each witha document asked to beginmoving by mail and starter was provided it The document the towardthe target described study, named the target, and asked the recipient becomea participant sendingthe document to by on. It was stipulatedthat the documentcould be sent only to a first-name acquaintanceof the sender.The senderwas urged to choose the recipient in sucha way as to advancetheprogress the document of towardthe target; severalitemsof information about the targetwere providedto guide each new senderin his choice of recipient. Thus, each document made its way along an acquaintancechain of indefinite length, chain whichwould end a only whenit reachedthe targetor when someonealong the way declined to participate. Certainbasic information, such as age, sex and occupation, was collectedforeach participant.2 In addition thePool-Kochen to and Rapoport work, there numerous are other studies of socialnetwork phenomena related thesmall-world tangentially to problem. Two wellknown examples Bailey'sThe Mathematical are Theory Epidemics Coleman, of and Katz and Menzel'sMedicalInnovation. Bailey'sworkdeals withdiffusion froma structured thanwithconvergence a target source, rather on from set of scattered a as sources, in thepresent study. The Coleman, Katz and Menzelstudydeals withan important substantive correlate acquaintance of nets,namely information diffusion.

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We wereinterested discovering in some of the internal structural features of chains and in makingcomparisons across chains as well. Among the questions hoped to answer we werethe following: How manyof the starters -if any-would be able to establishcontact with the targetthrougha chain of acquaintances?How many intermediaries would be requiredto link the ends of the chains? What formwould the distribution chain of lengths take? What degreeof homogeneity age, sex,occupation, other in and characteristics participants of wouldbe observed within chains?How would complete chainsdiffer from incomplete theseand otherdimensions? on An additionalcomparison was set up by using three distinctstarting subpopulations. The targetpersonwas a Boston stockbroker; two of the starting populations were geographically removedfromhim, selectedfrom the state of Nebraska. A thirdpopulationwas selected fromthe Boston while area. One of the Nebraska groupsconsisted bluechipstockholders, of the secondNebraskagroupand the Bostongroupwere"randomly" selected and had no specialaccess to the investment business.By comparisons across thesegroupswe hoped to assess the relativeeffects geographical of distance and of contactwith the target'soccupationalgroup. Moreoverwe hoped to establisha strategy future for experimental extensions the procedure, of in whichthesociological characteristics thestarting target of and populations wouldbe systematically variedin orderto exposefeatures social structure. of The primary researchquestions,then,involveda test of the feasibility and fruitfulness the methodas well as an attempt discover of some eleto mentary features real social nets. Several experimental of extensions the of procedure alreadyunderway. moredetaileddescription the current are A of method givenin the following is sections. Starting Population.The starting populationforthe study PARTIcIPANTS. of was comprised 296 volunteers. these,196 were residents the state of Of of Nebraska,solicitedby mail. Withinthis group,100 were systematically chosen ownersof blue-chipstocks; these will be designated"Nebraska the stockholders" this throughout paper. The restwerechosenfrom populationat large; thesewill be termed "Nebraska random"group.In addithe tion to the two Nebraska groups,100 volunteers were solicitedthrough an in advertisement a Boston newspaper(the "Boston random"group). Each memberof the startingpopulationbecame the firstlink in a chain of directed the target acquaintances at person. in Intermediaries. The remaining participants the study,who numbered solicited otherparticipants; 453 in all, werein effect by theywereacquainas tancesselectedby previous participants people likelyto extendthe chain was voluntary. towardthe target.Participation werenot paid, Participants offered incentive completion chains. norwas money otherreward or as for of

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which weresent a document The 296 initialvolunteers THE DOCUMENT. contained: was thetool of the investigations The document principal becomea partica a. a description thestudy, requestthatthe recipient of ipant,and a set of rules forparticipation; him; information concerning personand selected b. thenameof the target his c. a roster, whicheach participant asked to affix name; was to about each businessreplycards asking information d. a stack of fifteen participant. specific Rules for Participation. The documentcontainedthe following instructions participants: to a. Add yourname to the roster thatthe nextpersonwho receivesthis so folder will knowwhomit came from. b. Detach one postcardfromthe bottomof this folder.Fill it out and No returnit to Harvard University. stamp is needed. The postcard of It is veryimportant. allows us to keep track of the progress the folder it movestowardthe target as person. c. If you know the targetperson on a personalbasis, mail this folder met the to directly him (her). Do this only if you have previously name basis. targetpersonand know each otheron a first d. If you do not know the targetperson on a personalbasis, do not Instead, mail this folderto a personal try to contacthim directly. acquaintancewho is morelikelythan you to know the targetperson. or but You may send thebookleton to a friend, relative, acquaintance, you knowpersonally. it mustbe someone who lives in Sharon, personwas a stockholder TargetPerson.The target a Massachusetts, suburbof Boston, and who worksin Boston proper.In and place of employment, particadditionto his name,address,occupation service his military ipants were told his college and year of graduation, One questionunderindates, and his wife'smaidenname and hometown. whichpeople would use in reaching was vestigation the typeof information the target. of was to prevent function the roster Roster.The primary "looping,"i.e., to to prevent sendingthe document someonewho had already people from of was to motiit received and sentit on. An additionalfunction the roster the vate people to continue chains.It was hoped thata list of priorpartica ipants,including personalacquaintancewho had sent the documentto1969: 110-11.

of document appearsin Milgram, A photographic reproduction this experimental

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431

therecipient, wouldcreatewillingness the part of thosewho received on the document send it on. to TracerCards. Each participant asked to return us a businessreply was to cardgiving certain information about himself about thepersonto whom and he sent the document. The name, address,age sex and occupationof the senderand sender'sspouse were requested, were the name, address,sex as and age of the recipient. addition, natureof the relationship In the between senderand recipient-whether they were friends, relatives, businessassociates, etc.-was asked. Finally, participants were asked why they had selected particular the recipient the folder. of The businessreplycardsenabledus to keep running trackof theprogress of each chain. Moreover, theyassuredus of getting information even from chainswhichwerenot completed, allowing to make comparisons us between completeand incomplete chains. RESULTSCOMPLETIONS.217 of the 296 starting personsactuallysent the document could reach the target on to friends. Anyone of the documents persononly if thefollowing weremet: 1) recipients weresufficiently motivated conditions on to send the document to thenextlink in the chain; 2) participants were for closerto the target able to adopt some strategy movingthe documents further allow themto (this condition requiredthat the given information in selectthe nextrecipient a mannerthat increasedthe probability conof shortpaths were in fact requiredto link tactingthe target); 3) relatively fewchainswould remainactive long enough and target(otherwise starters therewas serious doubt to reach completion).Given these contingencies, of the documents, whether in the mindof the investigators any particularly in from target could movethrough thosestarting an area remote the person, on and converge him. The actual outinterlocking acquaintancenetworks or comewas that64 of the folders, 29 per centof thosesentout by starting reachedthe target. persons, eventually DISTRIBUTIONOF CHAIN LENGTHS. Complete Chains. Figure 1 showsthe

of as is heredefined thenumber intermediaries requiredto link starters and The mean of the distribution 5.2 links. is target. whether apparentdrop in frequency the It was unclearon first inspection of or at the medianlength fivelinkswas a statistical the accident, whether was revealed that the distribution actually bimodal. Furtherinvestigation relationgraphedin Figure 1 concealedtwo underlying distribusummary chainsweredividedinto thosewhichapproached tions: whenthe completed and those which approachedhim via the target throughhis hometown

frequency distribution of lengths of the completed chains. "Chain length"

432 20 -

SOCIOMETRY

15 -

Z~~~: A

U

\

Nx64

0 10 a: w

5

0

0

I

2

3

7 8 9 5 6 4 OF NUMBER INTERMEDIARIES FIGURE 1Chains Lengths Completed of

10

I1

12

two distinguishable distributions Boston businesscontracts, emerged.The is meanof theSharondistribution 6.1 links,and thatof the Bostondistribuis at tion is 4.6. The difference significant a level betterthan .0005, as U Mann-Whitney test. (Note that more assessed by the distribution-free of betweenmeans statisticaltests of the significance differences powerful of be appliedto thesedata, sincethosetestsassumenormality undercannot The shape of the true or theoretical distributions. distribution of lying what we do not know.) of acquaintancechains is precisely lengths what seems to occur is this. Chains which convergeon Qualitatively, reach his hometown information the targetprincipally using geographic by but once thereoftencirculatebeforeenareas readily, or the surrounding circleof acquaintances. There is no available information the tering target's to narrowthe fieldof potentialcontactswhich an individualmighthave as within town.Such additionalinformation a list of local organizations the

-STUDY OF THE SMALL WORLD PROBLEM

433

of which the targetis a membermighthave provideda natural funnel, facilitating progress the document the of fromtown to targetperson.By contrast,those chains which approach the target throughoccupational channelscan take advantageof just such a funnel, zeroingin on him first through brokerage the business, thenthrough firm. his completion dropor Incomplete Chains. Chains terminate eitherthrough out: each dropout results an incomplete in chain.Figure2 showsthenumber of chainswhichdroppedout at each "remove"from starting population. the The "Oth remove"represents starting the populationitself: the "firstredirectly from move" designates set of peoplewho received document the the membersof the startingpopulation.The "second remove" received the documentfromthe startersvia one intermediary, thirdthroughtwo the as intermediaries, The lengthof an incomplete etc. chain may be defined the number removesfromthe startat whichdropoutoccurs,or, equivof alently, the numberof transmissions the folderwhichprecededropas of of distribution the out. By this definition, Figure 2 represents frequency a is lengthsof incomplete chains. The mean of the distribution 2.6 links. The proportion chains which drop out at each removedeclines as of 80

Z 60X _ 60U I F \

_J\

\

Nx232

o 40U.0CU

Z\

REMOVE FROM STARTAT WHICH TERMINATION OCCURRED FIGURE 2

Lengths Incomplete of Chains

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chainsgrowin length, thatproportion based on all chainsactiveat each if is remove(those destinedforcompletion well as incompletion). as About 27 per centof the 296 folders sent to the starting population not sent on. are 27 Similarly, per cent of the 217 chains actuallyinitiatedby the starters die at the firstremove.The percentage dropoutsthen appears to fall. of It also beginsto fluctuate, the totalnumber chainsin circulation as of grows the small,and an increasing proportion completions further of complicates picture. It was arguedearlierthat,in theory, any two people can be linkedby at least one acquaintancechain of finitelength,barringthe existenceof totallyisolatedcliques withinthe populationunderstudy.Yet, incomplete in becausea certain chainsare found ourempirical tracing procedure propordo tionof thosewho receivethe document not send it on. It is likelythat thisoccursforone of two major reasons: 1) individuals not motivated are in the study; 2) they do not know to whom to send the to participate in document orderto advance it towardthe target. the of of it For purposes gauging significance our numerical results, would be useful to know whetherthe dropoutsare randomor systematic, i.e., for or whether not theyare relatedto a chain'sprognosis rapid completion. are thatdropouts precisely thosepeople who It seemspossible,forexample, towardthe target.If are least likelyto be able to advance the document of of chainswould understate so, the distribution actual lengths completed and target an unknown between starters the truesocial distance amount. by (Even if dropoutsare random,the observeddistribution understates the calculable amount.) We can offer but true distribution, by a potentially is that this effect not powerful. some evidence, however, First,it shouldbe clear that,though people may drop out because they that any of theiracquaintances can advance the folder see littlepossibility to are theirsubjectiveestimates irrelevant the question towardthe target, estimates just raised. Such subjective may accountfor individualdecisions chainsthatdie in factwould not to participate; theydo not tellus whether had theygone to completion. thanothers have been longer People have poor the of intuitions concerning lengths acquaintancechains. Moreover, people ownacquaintances;it is hard to guessthe circles see can rarely beyondtheir in whichfriends friends-notto mention of people even moreremotely connectedto oneself-may move. More directevidencethat dropouts may be treatedas "random"can be gleanedfrom tracercards. It will be recalledthat each participant the was asked for information only about himself not but also about the person to whomhe sent the document. Thus some data were available even for dropouts, namelyage, sex, the nature of theirrelationship the people to

TABLE 1

Activity Chainsat Each Remove ofAll Chains Remove Chains Reaching Completions this Remove at thisRemove Dropouts at this Remove Per cent Dropouts Remove

I

Chains R thisR

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

296 217 158 122 95 76 50 38 17 10 6 4 1 1 1

0 0 2 3 8 14 8 16 6 2 2 3 0 0 0

79 59 34 24 11 12 4 5 1 2 0 0 0 0 1

27 27 22 20 12 16 8 13 6 20 0 0 0 0 100

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

2 1

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preceding themin the chain,and the reason the dropouthad been selected These fourvariableswere tabulatedfor dropouts to receivethe document. tables achievedthe versusnon-dropouts. None of the resulting contingency .05 level of statistical significance chi-squaretest; we are therefore by led of betweenthe two groups,at to accept the null hypothesis no difference answerto the least on this limitedset of variables.Of course,a definitive are reallyrandommustwait until the deterquestionof whether dropouts or minantsof chain lengthare understood, until a way is foundto force all chains to completion.4 A research SUBPOPULATION COMPARISONS. possible paradigmfor future usingthe tracing procedure described here involvessystematic variationof One such study, therelationship between starting target the and populations. and targetgroups,has alreadybeen comusingNegro and White starting pletedby Korteand Milgram(in press). In thepresent study, whichinvolved onlya singletarget person,threestarting populations wereused (Nebraska random, Nebraska stockholders, Boston random.) The relevant and experimentalquestions werewhether proportion completed the of chainsor mean chain lengths wouldvaryas a function starting of population. Chain Length.Lettersfromthe Nebraska subpopulations had to covera of geographic distance about 1300 milesin orderto reachthe target, whereas in lettersoriginating the Boston group almost all startedwithin25 miles of his home and/orplace of work.Since social proximity dependsin part one on geographic proximity, mightreadilypredictthat completechains in in originating the Boston area would be shorter than those originating Nebraska.This presumption confirmed the data. As Table 2 shows, was by chains originating with the Boston randomgroup showed a mean length of 4.4 intermediaries betweenstarters and target,as opposed to a mean lengthof 5.7 intermediaries the Nebraska randomgroup. (p?.001 by for4 Professor Harrison White Harvard of for University developed technique adjusthas a ing raw chainlength data to takeaccount the dropout of problem. method His assumes are in that dropouts "random," the following sense.An intermediary knowsthe who sends him the folder, an target completing chain,with probability Otherwise, the 1. throws intermediary away the folderwith fixedprobability I-a, or sendsit on with is probability If senton,there a probability (which a. Qi depends number removes on of from origin) the thatthenext intermediary knowsthetarget. data is consistent The with a valuefora of approximately independent remove and of 0.75, from origin, hence the witha "random" rateof 25 per cent.The limited data further thatQ, dropout suggest grows a "staircase" in pattern from zero (at zeroremoves from starting the population) to approximately one-third six removes, at remaining constant thereafter. Based on these the withno dropouts resembles observed values,the hypothetical curveof completions of curveshifted upward;themedian length completed chainsrisesfrom to 7, but no 5 in drawnfrom raw data. the substantial alteration required conclusions is

TABLE 2 Lengths of Completed ChainsFrequency Distribution Number of Intermediaries Population 0 1 2 3 4 5 6 7 8 9 10 11 Total

Start

Nebraska Random Nebraska Stock AllBostonRandom

0 00

0 0 22

0 0 33

1 34

4 6 144

3 4 81

6 6 164

2 22

0 1 21

1 10

1 1 31

0 00

18 24 6422

Nebra Nebra

0

8

6

2

0

Bosto All

All Neb

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SOCIOMETRY

U to a one-tailed Mann-Whitney test.) Chain lengththus provedsensitive and target. of one demographic variable-place of residence starters group was presumedto have easy access to The Nebraska stockholder business.Because the targetpersonwas a stockcontactsin the brokerage in broker, chains originating this groupwere expectedto reach the target moreefficiently chains fromthe Nebraska randomgroup.The chainthan meansforthe two groups,5.7 intermediaries the randomsample for length but the differed the expecteddirection, in and 5.4 for the stockholders, test. The significant the Mann-Whitney by difference not statistically was used the brokerage channelmore businessas a communication stockholders in oftenthan did the randomgroup; 60.7 per cent of all the participants with connected occupations chainsoriginating thestockholder groupreported in in while 31.8 per cent of participants chains originating with finance, the Nebraska randomgroupwere so classified. As of Proportion Completions. indicatedin Table 3, the proportions of and chains completedfor the Nebraska random,Nebraska stockholder, Boston subpopulations were 24 per cent, 31 per cent and 35 per cent, the differences not statistically significant, there are respectively. Although for is a weak tendency highercompletion rates to occur in groupswhere mean length of completedchains is shorter.This result deserves brief discussion. rate is constant each removefrom at the Let us assumethat the dropout start.If, forexample,the dropoutrate were 25 per cent then any chain of one wouldhave a 75 per centprobability reaching link,(.75)2 of reaching a etc. twolinks, Thus,thelonger chainneededto be in orderto reachcomplelong enoughto run its full tion,the less likelythatthe chainwouldsurvive differences amongthe three groups chain-length course.In thiscase,however, in were not sufficiently differences completion large to producesignificant if rate declinesas chainsgrowlong,such a derate. Moreover, the dropout and weakentheobserved inverse creasewouldoff-set effect discussed the just of relation betweenchain lengthand proportion completions.TABLE 3 Populations for Proportion Completions ThreeStarting ofStarting Population Stock. Nebraska Random Nebraska Boston Total

Complete Incomplete

18 58 76

(24%) (76%) (100%o)

24 54 78

(31%) (69%) (100%)

22 41 63

(35%) (65%) (100o)

64 153 217

(29%) (71%) (100%)

df.=2,P>.3, N.S. X3=2.17,

STUDY OF THE SMALL WORLD PROBLEM

439

COMMONCHANNELS. chainsconverge the target, As on common channels appear-that is, some intermediaries appear in morethan one chain. Figure 3 showsthepattern convergence. 64 letters The whichreachedthe target of were sent by a total of 26 people. Sixteen,fully25 per cent, reachedthe 10 targetthrough singleneighbor. a Another made contactthrough single a businessassociate,and 5 through second businessassociate. These three a links" together accountedfor 48 per cent of the total com"penultimate an divisionof labor appears. Mr. G, pletions. Amongthe three, interesting64 INDIVIDUALS 55 INDIVIDUALS 5 CHAINS 26 INDIVIDUALS 16 CHAINS

CHAINS 2 CHAINS 3

FIGURE 3

Common Paths Appearas ChainsConverge the Target on

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SOCIOMETRY

in merchant the target's is who accountedfor 16 completions, a clothing towardthe targetthosechainswhich hometown Sharon; Mr. G funnelled of Twenty-four place of residence. wereadvancingon the basis of the target's Mr. G accountedfor 2/3 of chainsreachedthe targetfromhis hometown; residents whichreachedMr. G came from thosecompletions. the letters All Mr. D and Mr. P, who accountedfor 10 and 5 of Sharon.By contrast, were contactedby people scatteredaround the respectively, completions, Boston area, and in severalcases, by people livingin othercitiesentirely. from Sharonresidents the Mr. G received folder On theotherhand,whereas D in a wide varietyof occupations, and P receivedit almostalways from A of stockbrokers. scattering names appear two or three times on the links; seventeennames appear once each. list of penultimate link. Going one step appeared even beforethe penultimate Convergence we further back, to people two removesfromthe target, findthat the 64 55 One man, Mr. B, appeared 5 times, chainspassed through individuals. appeared to and on all occasionssentthedocument Mr. G. Otherindividuals or three times each. two per OF ADDITIONAL CHARACTERISTICSCHAINS. Eighty-six centof the parand acquaintas to sent the folder personstheydescribed friends ticipants had been ances; 14 per cent sent it to relatives.The same percentages observedin an earlierpilot study. Data on patternsof age, sex and occupationsupportthe plausible hysimilar a from pool of individuals selectrecipients thatparticipants pothesis The data on age supportthe hypothesis unequivocally;the to themselves. of by data on sex and occupationare complicated the characteristics the of contactwithhim. targetand the special requirement establishing categoriesand the ages of those who Age was bracketedinto ten-year tabled against the ages of those to whomtheysent it. sent the document to the table showed a strongtendency clusteraround the On inspection test showedthe associationto be significant at diagonal,and a chi-square betterthan the .001 level. the sex of each sender was tabled against the sex of the Similarly, Men were ten timesmorelikelyto send the docurecipient. corresponding ment to othermen than to women,while women were equally likely to wereaffected to send thefolder malesas to females(p