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Graphs and Social Network Analysis:

An Introduction

Michail TsikerdekisNovember 10, 2011

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Social Network Analysis

It is useful for investigations of kinship patterns,

community structure,interlockingdirectorships and soforth

It is mainly ananalysis for relationaldata

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Attribute data relate to the attitudes, opinionsand behaviour of agents, in so far as these areregarded as the properties, qualities or

characteristics that belong to them asindividuals or groups.

The methods appropriate to attribute data arethose of variable analysis, whereby attributes

are measured as values of particular variables(income, occupation, education etc.).

Variable Analysis

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Network Analysis

Relational data, on the other hand, are thecontacts, ties and connections, the groupattachments and meetings, which relate one

agent to another and so cannot be reduced tothe properties of the individual agentsthemselves.

The methods appropriate to relational data are

those of network analysis, whereby therelations are treated as expressing the linkageswhich run between agents.

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Traditional methods for producingdata for analysis

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What is a social network?

A set of concrete nodes(“actors”): individuals(e.g., persons),collectivities (e.g.,

organizations, countries) A set of concrete ties, all

of the same type, thatconnect them: each tie is

an element of a binarysocial relation such as “isa friend of” or “is teacher of”

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Kinds of Nodes

Individuals: persons, other animals

Collectivities: organizations, departments,teams, troops, countries, cities, species

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Relations Among People

Kinship: - mother of, wife of

Other role-based: boss of, teacher of, friend of

Cognitive/perceptual: knows, aware of whatthey know

Affective: likes, trusts

Interactions: give advice, talks to, fights with

Affiliations: belongs to same clubs, is physicallynear

Each relations yields a different structure & has

different effect!

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Why Social Network Analysis?

Popular culture: Games, plays, television,Forbes, Fortune, NY Times

Business Practitioners: New tools for

management consultants, new original forms;knowledge management

Academia: Multiple fields from linguistics toAIDS research to political science to sociology

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Examples of Networks

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Examples of Networks

Hi t S i t i A l i d

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History: Sociometric Analysis andGraph theory(1)

In the 1930s a group of German immigrantsinfluenced by Wolfgang Kohler's 'gestalt' theorywere working in the United States on cognitive

and social psychology. Using laboratorymethods or laboratory-like case studies, theylooked at group structure and at the flow of information and ideas through groups. (Kurt

Lewin, Jacob Moreno, and Fritz Heider)


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Moreno developedsociometry.

He believed that largescale social phenomena,such as the economyand state, weresustained andreproduced over time by

the small scaleconfigurations formed bypeoples patterns of friendship, dislike andother relations.


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Lewin studied groupbehavior

Lewin argued that the

structural propertiesof this social spacecould be investigatedmathematically using

vector theory andtopology.


History: Sociometric Analysis and

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Heider worked in thearea of social perceptionand attitudes.

He developed what isknown as balancetheory. The mind seeksbalance (an absence of tension) by trying to hold

ideas that are not inconflict with one another.This also applies toattitudes towards other people.

History: Sociometric Analysis and

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Cartwright and Hararyshowed mathematicallythat the outcome of thisprocess is necessarily a

group subdivided intocliques within which allties are positive andbetween which all tiesare negative. All groupsin which there is anyimbalance are in a stateof slow transitiontowards cliques.

History: Interpersonal

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History: InterpersonalConfigurations and Cliques (1)

The work of Warner,Mayo, Roethlisberger and Dickson on theHawthorne plant of theWestern ElectricCompany in chicago inthe 20s was a milestone

They discovered the"informal organization" of the organization -- thehidden social structurewhich seemed to haveas much effect on worker

productivity

History: Interpersonal

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History: InterpersonalConfigurations and Cliques (2)

Later on in the 50s, thedept of socialanthropology atManchester University in

England began lookingat conflict in groups.

Harrison White in the60's and '70s further

developed themathematical side of social network analysis

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Development of the Field

1736 –Euler

1930s Sociometry: Moreno; Hawthorne studies

1940s Psychologists: Clique formally defined

1950s Anthropologists: Barnes, Bott & Manchester school 1960s Anthropologists: Kinship algebras; Mitchell

1970s Rise of Sociologists: Social Networks Journal &Assoc, Milgram's small-world, Granovetter’s weak ties

1980s Computation: IBM PC & network programs

1990s Adaptive Radiation: UCINET IV released, Spread of networks & dyadic thinking to

Source: Steve Borgatti, 2004

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Handling relational data

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Case-by-Variable Data matrix

Traditionally data wereheld in a data matrix aframework in which theraw or coded data can

be organized in a moreor less efficient way.

Today they can be foundin massive databases.

Each row contains acase (person) and eachcolumn a variable.

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Case-by-affiliation Matrix

The case-by-variable data matrix cannot beused for relational data. These data mustinstead, be seen in terms of a case-by-affiliationmatrix.

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Casuality

Traditional Variables → Network Variables →Traditional Variables

Personality → Centrality → Health

Gender → Friendship → Performance

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Graph Theory

Graph theoryconcerns set of elements and therelations amongthese the elementsbeing termedpoints(nodes) and

the relations lines.

M d D C ll i

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Modern Data Collection

Web data collection & automatic processing

– Easing burden on respondents and analysts

Passive electronic data collection

– Tacit Knowledge Systems Knowledge Mail

– Phone records

– Web cookies

– PDA beaming of business cards and me

C ti ith th d t !

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Caution with the data!

If you have all the data from a population youare allowed to make induction about thepopulation.

If you are sampling then the sample needs tobe representative of the population. Randomsampling is ideal here.

Do you have a sample of the population? What

guarantees it being representative.

Sociocentric(Complete) vs

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Sociocentric(Complete) vsEgocentric perspective

Sociocentric examines complete networks of actors and relations, studying the globalproperties of a network and characterizing theposition of any given actor by reference to allthe others

Egocentric network analysis takes theperspective of individual actors and the

‘‘personal networks’’ surrounding them,focusing on the local structure of each actor’sinterpersonal environment.

Ki d f N t k D t

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Kinds of Network Data

Complete Ego

1-mode Ties among “all”membersof a single class

of entities

Ties among theset of nodes(alters) directly

tied to aspecific individual(ego)

2-mode Ties between all

membersof two differentclasses of entities

Ties between two

sets of entities tied to aspecificindividual

1 d l t t k

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1-mode complete network

1 mode ego net ork

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1-mode ego network

2 mode ego network

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2-mode ego network

2 mode complete network

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2-mode complete network

Density and inclusiveness

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Density and inclusiveness

Inclusiveness of agraph is the totalnumber of nodesminus the number of

isolated nodes. A 20-point graph with

five isolated nodeswould have an

inclusiveness of 0.75 The more inclusive is

the graph, the more

dense will it be.

Density Undir.=

l

n (n−1)/2

Density Direct.=l

n(n−1)

Density and perspectives

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Density and perspectives

16 of the 91 nodesare connected

The density is 18%

from a sociocentricperspective

But from anegocentric

perspective thedensity for G is a lotdifferent than thedensity of H

Local Centrality

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Local Centrality

(Local)Degree centrality is defined as thenumber of links incident upon a node (i.e., thenumber of ties that a node has)

If the network is directed (meaning that tieshave direction), then we usually define twoseparate measures of degree centrality, namelyindegree and outdegree. (“in-centrality” and

“out-centrality” of a node) Measure of local centrality cannot be compared

with other graphs when the graphs differ significantly in size

Relative centrality

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Relative centrality

Relative centrality of local centrality in which theactual number of connections is related to themaximum number that it could sustain.

A degree of 25 in a graph of 100 nodesindicates a relative local centrality 0.25 while adegree of 25 in a graph of 30 points indicates arelative centrality of 0.86.

Can be used with other graphs regardless of size. But, even this cannot work with graphsthat were build upon different relations. 'Love'network versus 'awareness' network.

Degree Centrality

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Degree Centrality

Index of exposure to what is flowing through thenetwork. i.e. Gossip network: central actor morelikely to hear a given bit of gossip

Interpreted as opportunity to influence & beinfluenced directly

Predicts variety of outcomes from virusresistance to power & leadership to job

satisfaction to knowledge

Global centrality(Closeness)

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Global centrality(Closeness)

Freeman (1979, 1990) has proposed a measure of global centrality based around what he terms the'closeness' of the points. It is expressed in terms of thedistances among the various points.

A path is a sequence of distinct lines connecting twonodes, and the length of a path is measured by thenumber of lines which make it up.

The length of the shortest path between two points is a

measure of the distance between them. A point is globally central if it lies at short distances

from many other points.

Closeness Centrality usefulness

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Closeness Centrality usefulness

Index of expected time until arrival for givennode of whatever is flowing through the network

I.e. Gossip network: central player hears things

first

Betweenness centrality

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Betweenness centrality

How often a node lies

along the shortest pathbetween two other nodes

The betweenness of apoint measures the

extent to which an agentcan play the part of a'broker' or 'gatekeeper'with a potential for

control over others. Interpreted as indicating

power and access todiversity of what flows;

potential for synthesizing

bk =∑i , j

g ikj

g ij

Degree centrality, ClosenessC t lit

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Centrality

Reach Centrality

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Reach Centrality

Number of nodes that each node can reach in kor less steps. For k = 1, this is equivalent todegree centrality

Two-step reach tells the analyst whatproportion of the total number of people in thenetwork can be reached by any particular person within one ‘link’ or step of the people

who comprise his/her immediate ties It is a good measure of the extent to which any

person could mobilize resources or conveyinformation by reaching out to others.

Centralization

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Centralization

Density describes the general level of cohesionin a graph: centralization describes the extentto which this cohesion is organized aroundparticular focal points.

Centralization definition

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Centralization definition

Difference betweeneach node’s centralityscore and that of themost central node

Centralization=∑

i

∣c MAX −ci∣

∑i∣ χ MAX − χ i∣

What is a star graph?

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g p

Transitivity

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y

The probability that two friends are friends withone another

Proportion of triples with 3 ties as a proportion

of triples with 2 or more ties: aka the clusteringcoefficient

Component

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p

A subgraph S of a graphG is a component if S ismaximal and connected.

Connected means that

every node can reachevery other by somepath (no matter howlong)

They can be discoveredwith random algorithmsor by algorithms thatbuild up “spanning trees”

A

B

Cutpoint

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p

A node which, if deleted, would increase thenumber of components

Components in Directed graphs

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p g p

S is a strong component if for all u,v in S, thereis a path from u to v

S is a weak component if it is a component of

the underlying (undirected) graph

Cyclic component

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y

A cycle is a path that returns to its own starting nodeand like a path, a cycle can be of any length.

A bridge is a line that does not itself lie in a cycle butthat may connect two or more cycles.

A cyclic component can be defined as a setintersecting cycles connected by those lines or nodesthat they have in common.

K-core (1)

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A k-core is a maximal subgraph in which eachpoint is adjacent to at least k other points.

S=G is 1-core & 2-core; S = {1..8} is 3-core

There is no 4-core or higher

K-core (2)

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Finds areas within which cohesive subgroupsmay be found

Identifies fault lines across which cohesive

subgroups do not span In large datasets, you can successively

examine the 1-cores, the 2-cores, etc.

Progressively narrowing to core of network

Cohesive subgroups

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Initially conceived of as formalizations of fundamental sociological concepts: Primarygroups, Emergent groups

Now typically thought of in terms of a techniquefor identifying groups within networks

Canonical Hypothesis

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Members of group will have similar outcomes:Ideas, attitudes, illnesses, behaviors

Due to interpersonal transmission:transference, Influence / persuasion, Co-construction of beliefs & practices

As in communities of practice

So group membership is independent variableused to predict commonality of attitudes,beliefs, etc

Cliques (1)

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Maximal, complete subgraph Every possible pair of points is directly connected by a

line and the clique is not contained in any other clique

Maximum density (1.0)

Minimum distances (all 1)

Overlapping

Strict

Clique (2)

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N-Cliques (1)

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It still has to be complete under a certaindistance

Distance among members less than specifiedmaximum

When n = 1, we have a clique

N-Cliques (2)

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If n is greater than 2 it is a challenge to beinterpreted sociologically

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Issues with N-Cliques

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Overlapping: {a,b,c,f,e}and {b,c,d,f,e} are both2-cliques

Membership criterion

satisfiable throughnonmembers

Even 2-cliques can befairly non-cohesive: Red

nodes belong to same 2-clique but none areadjacent

Subgraphs

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Set of nodes: Is just aset of nodes

A subgraph: Is set of nodes together with ties

among them An induced subgraph:

Subgraph defined by aset of nodes; like pulling

the nodes and ties out of the original graph

N-clan

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Members of set withinn links of each other without usingoutsiders

Like an n-cliquewithout the use of outsiders

More cohesive thann-cliques

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N-Clan Issues

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Is {a,b,c,f} a 2-clan? List all 2-clans

few found in data

overlapping

K-Plex

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Is a set of nodes in which each node isadjacent to all except k of other nodes.

An 1-plex is equivalent to a 1-clique.

When k is equal to 2, all members in a 2-plexare connected to at least n-2 of the other members

Very numerous & overlapping

K-Plex Examples

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An example of Social Network Analysisstudy

Use of social network analysis to map the socialrelationships of staff and teachers at school

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In schools, it is unusual for researchers to measuresocial relationships specifically as part of theintervention.

Indirectly obtaining information about social

relationships “In this paper, we present a method for assessing the

social relationships directly in a school by focusing onmultidimensional social interaction structures or

networks among the teachers and staff.”

Source: Penelope Hawe and Laura Ghali Use of social network analysis to map the socialrelationships of staff and teachers at school Health Educ. Res. (2008) 23(1): 62-69 first

published online February 7, 2007 doi:10.1093/her/cyl162

Goals

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How network analysis could allow us to capturethe social structure of the high school staff andteachers at the start of a whole-school healthpromotion intervention?

Identify key players or gatekeepers who mightbe crucial to getting the intervention off theground

Mapping networks systematically at the start of an intervention, and analyzing themmathematically

Location and sample

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High school in Alberta,Canada with totalstudent population 556.

Mental health promotion

intervention modeled onthe experience of thesuccessful Gatehouseproject in Australia

Staff and teachers wereour focus for the firststage of the intervention.

Design method

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Map five relations across all teachers and staff in theschool based on initial consultation and pilot.

Relations:

– knew a person by name

– knew a person more personally – engaged in regularly occurring conversations with aperson – sought advice from a person in relation to a school

matter – socialized with that person outside of school hours

Self administered questionnaires sent to staff andteachers with questions focused on usual transactions

and routine relationships

UCINET

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Design expectations

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Research was designed to support directed graphs Not all relationships were expected to be reciprocal.

For example, Teacher 1 could know Teacher 2 byname but the reverse may not be the true.

Two relationships expected to be reciprocal were‘regular conversations’ and ‘socialize with’.Symmetrizing the data by the minimum value. In other words they counted the relationship as being present

only if both people said the tie was present Each relationship, such as advice-seeking or knowing-

by-name, constitutes a separate network.

Analysis Methods

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Network degree centralization score

Network betweenness centralization score: measureof strategic advantage and information control.

Two-step reach measure of the extent to which any

person could mobilize resources or convey informationby reaching out to others.

Others are measures about an individual person'sposition in the network.

Classified people as: teachers, support staff,administration

Also conducted analysis of gender

Results

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50 out 53 were present and return thequestionnaire (94% response rate)

28 women, 22 men

18 support staff, 30 teachers, plus principal andvice principal

Basic characteristics of the fivenetworks (n = 50)

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Relationship Density

score(%)

Degree

centralization score(%)

Betweenness

centralizationscore (%)

Socialize withoutside of school

5.9 19.4 14.4

Seek advice 15.2 54.0 23.4Engage inconversationregularly

25.5 39.3 4.8

Know personally 29.0 38.9 4.63

Recognize byname

65.9 27.4 1.47

Advice seeking network

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Principal and Vice principal

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The Principal has 37 direct ties and the Vice Principalhas 35 direct ties in the advice-seeking network.

Freeman's degree centrality measure: 76%, 71%

Betweenness centrality: The Principal has four times

the score of Vice Principal for betweenness centralitybecause the Principal is connected to some peoplewho otherwise seek advice from no one.

This increases his power and potentially makes him

more important or crucial.

Ego network size and two-stepreach

Ego network size Two-step reach (%)

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Ego network size Two step reach (%)

Socialize with outside

school Principal 2 27

Vice principal 10 47

Seeking advice

Principal 37 85


Regular conversations

Principal 29 75


Know personally

Principal 33 98


Recognize by name

Principal 43 100


Support staff results

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Support staff are more marginal than teachersfor all relationships we mapped.

Women who occupy the bulk of the supportstaff positions, were more likely to be on the

edges of the network.

Conclusions (1)

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Density was related to what might thought of asthe intensity of the relationship.

Network density was higher for more superficialrelationships, such as knowing a person by

name, and smaller for socializing.

The density for knowing-by-name was lower than we had expected at 65%.

That is, more people are in that awkwardposition of encountering other staff andteachers, but not being addressed by their name.

Conclusions (2)

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No isolates in the know-by-name network.Everyone was linked to someone, including all10 newcomers.

Seeking advice was centered around the

Principal and Vice Principal

Seven people were unconnected in the advice-seeking network, a phenomenon which could

be addressed, if perceived as a problem.

Applications

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As an example, low density in the socializing networkis acceptable, but that low density, and in particular the presence of isolated people, in the advice-seekingnetwork is not

Another common type of analysis is to search for cliques or closely connected subgroups. Diagnosticsdepend on the goals and purpose of the researcher.

Identifying people of strategic influence, so that

interventions can be tailored to them. Identify andrecruit natural leaders or helpers in communities.

Summary (1)

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Social network analysis useful for: investigations of kinship patterns, community structure, interlockingdirectorships and so forth

You need to have relational data

Nodes and relationships between them Different kinds of perspectives for social networks:

sociocentric(complete), egocentric (ego), 1-mode, 2-mode. Also different relationships produce different

unique networks Density is the amount of ties that are present as a

proportion of the total possible ties; if everyone knowseach other, the density score is 100%.

Summary (2)

Network centralization calculates the extent to which a

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network is centralized or dominated by a few people.

Degree centrality and betweenness helpful to identifyopportunity to influence & be influenced directly aswell as key players withing the network

Cliques, n-cliques and similar concepts helpful to

identify groups within networks and associatebehaviors with these groups

There is a variety of methods for the analysis of socialnetworks. The usage needs to be determined by the

researcher's goals

Software can assist in the automated collection of dataand it is necessary for large datasets. Examples of social network analysis software is UCINET & Pajek

Further Reading / References

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John P. Scott. Social Network Analysis: AHandbook. Sage Publications Ltd; 2nd edition(March 2000)

John P Scott (Editor), Peter Carrington (Editor).

The SAGE Handbook of Social NetworkAnalysis. Sage Publications Ltd (May 25, 2011)

Penelope Hawe and Laura Ghali Use of social

network analysis to map the social relationshipsof staff and teachers at school Health Educ.Res. (2008) 23(1): 62-69 first published onlineFebruary 7, 2007 doi:10.1093/her/cyl162

Further Reading / References

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Steve Borgatti. Introduction to Social Networks.http://www.analytictech.com/essex/Lectures/Intr oduction.pdf (Retrieved on November 10, 2011)

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