An Introduction to Social Network Analysis James Moody Department of Sociology The Ohio State...

An Introduction to Social Network Analysis

James MoodyDepartment of SociologyThe Ohio State University

Introduction

The world we live in is connected:

Jim Moody

CraigCalhoun

Isaias Afworki

Introduction

These patterns of connection form a social space.

Social network analysis maps and analyzes this social space.

Adolescent Social Structure

Yet standard social science analysis methods do not take this space into account.

Moreover, the complexity of the relational world makes it impossible (in most cases) to understand this connectivity using only our intuitive understanding of a setting.

Introduction

Why networks matter:

• Intuitive: information travels through contacts between actors, which can reflect a power distribution or influence attitudes and behaviors. Our understanding of social life improves if we account for this social space.

• Less intuitive: patterns of inter-actor contact can have effects on the spread of “goods” or power dynamics that could not be seen focusing only on individual behavior.

Introduction

Social network analysis is:

•a set of relational methods for systematically understanding and identifying connections among actors

•a body of theory relating to types of observable social spaces and their relation to individual and group behavior.

Introduction

Network analysis assumes that:

• How actors behave depends in large part on how they are linked together

Example: Adolescents with peers that smoke are more likely to smoke themselves.

• The success or failure of organizations may depend on the pattern of relations within the organization

Example: The ability of companies to survive strikes depends on how product flows through factories and storehouses.

(continued…..)

Introduction

• Patterns of relations reflect the power structure of a given setting, and clustering may reflect coalitions within the group

Example: Overlapping voting patterns in a coalition government

Network analysis assumes that:

Introduction

An information network:

Email exchanges within the Reagan white house, early 1980s (source: Blanton, 1995)

Introduction

Power positions and potential influence

Introduction

Overview

Conclusions

Flows within Networks Structure of Social Space

Introduction

Basic Concepts

Tools, Models & Methods For Flows and Structures

• Actors are nodesIdeas, Papers, Events, Individuals,

Organizations, Nations

• Relations are lines between pairs of nodesSymmetric (shares a room with)Asymmetric (gives an order to)Valued (number of times seen together)

Basic Concepts

• Network data are familiar to you

• For example: - Personal, face-to-face contact - Telephone contact - Email contact - Contact through faxes or wires - Snail-mail contact - Membership in the same organization - Attendance at the same meetings - Graduates of the same university

Basic Concepts

For example, you might be tracking the activities of a number of people in related, but not identical cases, including meetings they attended. You may know little of the content of the event, or what they may have said to each other, only whether particular people were at the event.

Your data might look like:

Basic Concepts

Basic Concepts

11.19.2001. Meeting at Brussels. Attending:Smith, Johnson, Davis, James, Jackson

12.22.2001. Meeting at Paris. Attending:Johnson, James, Jones, Wilson

1.12.2001. Meeting in New York. Attending:Jones, Carter, Burns

2.14.2001. Meeting in Denver. Attending:Wilson, Burns, Wilf, Newman

(Red bold indicates people who are the focus of an investigation)

Basic Concepts

While perhaps not immediately apparent when looking at the list of names, a simple algorithm reveals connections among these actors.

Jackson

Davis

Smith Johnson Wilson Newman

Wilf

Burns

Carter

JonesJames

Basic concepts

Types of network data:

1) Ego-network - Have data on a respondent (ego) and the people they are connected to (alters)

- May include estimates of connections among alters

Basic concepts


2) Partial network- Ego networks plus some amount of tracing to reach contacts of contacts

- Something less than full account of connections among all pairs of actors in the relevant population

- Example: CDC Contact tracing data for STDs

Basic concepts


3) Complete- Data on all actors within a particular (relevant) boundary

- Never exactly complete, but boundaries are set

- Example: Coauthorship data among all writers in the social sciences

Contact’s contact

Trace RelationAlter Relation

Examples: linked levels of data

Actor

Key contact

Primary Relation


Consider the following (much simplified) scenario:

•Probability that actor i passes information to actor j (pij)is a constant over all relations = 0.6 •S & T are connected through the following structure:

S T

•The probability that S passes the information to T through either path would be: 0.09


Now consider the following (similar?) scenario:

S T

•Every actor but one has the exact same number of contacts•The category-to-category mixing is identical•The distance from S to T is the same (7 steps)•S and T have not changed their behavior•Their contacts’ contacts have the same behavior•But the probability of the information passing from S to T is:

= 0.148•Different outcomes & different potentials for intervention

Overview

Conclusions


Introduction

Basic Concepts


Network Flow

In addition to the simple probablity that one actor passes information on to another (pij), two factors affect flow through a network:

Topology-the shape, or form, of the network- Example: one actor cannot pass information to another unless they are either directly or indirectly connected

Time - the timing of contact matters- Example: an actor cannot pass information he has not receive yet

Topology

Two features of the network’s shape are known to be important: connectivity and centrality

Connectivity refers to how actors in one part of the network are connected to actors in another part of the network.

• Reachability: Is it possible for actor i to reach actor j? This can only be true if there is a chain of contact from one actor to another.

• Distance: Given they can be reached, how many steps are they from each other?

• Number of paths: How many different paths connect each pair?

Network topology: reachability

Without full network data, you can’t distinguish actors with limited information from those more deeply embedded in a setting.

a

b

c

Network topology: distance & number of paths

Given that ego can reach alter, distance determines the likelihood of information passing from one end of the chain to another.

• Because information spread is never certain, the probability of transfer decreases over distance.

• However, the probability of transfer increases with each alternative path connecting pairs of people in the network.

Network topology: distance & number of paths

a

Distance is measured by the (weighted) number of relations separating a pair:

Actor “a” is: 1 step from 4 2 steps from 5 3 steps from 4 4 steps from 3 5 steps from 1

Network topology: distance & number of pathsPaths are the different routes one can take. Node-independent paths are particularly important.

a

b

There are 2 independent paths connecting a and b.

There are many non-independent paths

0

0.2

0.4

0.6

0.8

1

1.2

2 3 4 5 6Path distance

pro

ba

bil

ity

Probability of information transferby distance and number of paths, assume a constant p ij of 0.6

10 paths

5 paths

2 paths

1 path

Reachability in Colorado Springs(Sexual contact only)

(Node size = log of degree)

•High-risk actors over 4 years•695 people represented•Longest path is 17 steps•Average distance is about 5 steps•Average person is within 3 steps of 75 other people•137 people connected through 2 independent paths, core of 30 people connected through 4 independent paths

Centrality refers to (one dimension of) location, identifying where an actor resides in a network.

• For example, we can compare actors at the edge of the network to actors at the center.

• In general, this is a way to formalize intuitive notions about the distinction between insiders and outsiders.

Network topology: centrality

Centrality example:

At the local level, we expect people like NSJMP and NSOLN to have greater access to information than others in the network.

Network analysis gives us a set of tools to quantify this difference.

(Node size proportional to betweenness centrality )

Centrality example:

Actors that appear very different when seen individually, are comparable in the global network.

Information flows

Two factors that affect network flows:

Topology- the shape, or form, of the network- simple example: one actor cannot pass information to another unless they are either directly or indirectly connected

Time - the timing of contacts matters- simple example: an actor cannot pass information he has not receive yet

Timing in networks

A focus on contact structure often slights the importance of network dynamics

Time affects networks in two important ways:1) The structure itself goes through phases that are correlated with information spread

2) The timing of contact constrains information flow

Sexual Relations among A syphilis outbreak

Jan - June, 1995

Rothenberg et al map the pattern of sexual contact among youth involved in a Syphilis outbreak in Atlanta over a one year period.

(Syphilis cases in red)

Changes in Network Structure

Sexual Relations among A syphilis outbreak

July-Dec, 1995

Data on drug users in Colorado Springs, over 5 years

Drug Relations, Colorado Springs, Year 1


Drug Relations, Colorado Springs, Year 2Current year in red, past relations in gray

What impact does timing have on flow through the network?

In addition to changes in the shape over time, contact timing constrains how information can flow through the network.

Consider the following example:

B

C E

D F

A2 - 5

3 - 7

0 - 1

8 - 9

3 - 5

A hypothetical contact network

Numbers above lines indicate contact periods

B

C E

D F

A

The path graph for the hypothetical contact network

Direct contact network of 8 people in a ring

(adjacency matrix: cell = number of paths from row to column)

Implied contact network of 8 people in a ringAll contacts concurrent

Implied contact network of 8 people in a ringMixed Concurrent

2

2

1

1

2

2

3

3

Density = 0.57

Implied contact network of 8 people in a ringSerial (1)

1

2

3

7

6

5

8

4

Density = 0.73


1

2

3

7

6

1

8

4

Density = 0.51


1

2

1

1

2

1

2

2

Density = 0.43

Information flows

Summary:

Topology:- Information requires connected communication chains- Real-world networks are too complex to map these without specialized tools.

Time: - Network topology changes over time. This has implications for information flow.- Because small changes in relationship timing can have dramatic effects on information flow, it is impossible to know this intuitively.

Overview

Conclusions


Introduction

Basic Concepts


Structure of Social Space

Information flows are only one use of networks

It is also possible to characterize the key topological features of any social network. These features include things such as the extent of hierarchy and clustering.

1) Identify core groups & patterns of relations among groups

a. embeddedness in groups constrains actionb. group structure affects stability & resource distribution

2) Locate tensions or inconsistencies in a relational structure that might indicate sources of social change.


Two features of interest related to network structure:

1) Cohesive groups: Sets of people who interact frequently with each other. These are often groups that work together. Groups are often organized into positions within a network that indicate particular roles or access resources

2) Hierarchy: Relational structure can identify the leadership positions within a network, though either direction of ties or periphery status


Structure of cohesive groups

A cohesive group is a set of actors with more interaction inside the group than outside the group, mutually connected through multiple paths.

Cohesive Group Structure

“Immaculate Preparatory High School”

Cohesive Group Structure: 3 types of positions


Cohesive Group Structure: Group member


Cohesive Group Structure: Group Member


Cohesive Group Structure: Bridge between groups


Cohesive Group Structure: Outsider


Cohesive Groups: Relevance

• Identify people who bridge important constituencies - people who are between groups have a unique ability to control information •Such actors are said to bridge structural holes, the number of “holes” an actor bridges gives insight into an actor’s power position in the network.

Hierarchy and network position

Many cohesive groups are embedded within a hierarchy, which one can map using relational tools.

Changes in the hierarchical position indicate changes in the power structure.

Examples of Hierarchical Systems

Linear Hierarchy(all triads transitive)

Simple Hierarchy

Branched Hierarchy

Mixed Hierarchy

Hierarchy and network position

If you don’t know the hierarchy of the network, asymmetry optimization techniques allow one to identify levels in a hierarchy

Start with some basic ideas of what a role is: An exchange of something (support, ideas, commands, etc) between actors. Thus, we might represent a family as:

H W

C

C

C

Provides food for

Romantic Love

Bickers with(and there are, of course, many other relations inside the family)

Group structure through multiple relations

The key idea, is that we can express a role through a relation (or set of relations) and thus a social system by the inventory of roles. If roles equate to positions in an exchange system, then we need only identify particular aspects of a position. But what aspect?

Structural Equivalence

Two actors are structurally equivalent if they have the same types of ties to the same people.

Group structure through multiple relations


A single relation


Graph reduced to positions

Alternative notions of equivalence

Instead of exact same ties to exact same alters, you look for nodes with similar ties to similar types of alters

Overview

Conclusions


Introduction

Basic Concepts


Tools, Methods & ModelsData Representations

1 2

3

5 4

Graph Adjacency Matrix

Arc ListNode List

Send11234444555

Recv23421235134

Tools, Methods & ModelsGraphical Display

Benefits:•Intuitive way to display networks.•Helps people see the social space – it is a map.•A concise presentation of a great deal of data.

Costs:•Lack of standards for how to display can create misleading images.•Displays of large networks tend to reveal only the roughest properties of the network

Tools, Methods & ModelsGraphical Display: Software

PAJEK •Program for analyzing and plotting very large networks•Intuitive windows interface•Used for most of the real data plots in this presentation•Mainly a graphics program, but is expanding the analytic capabilities•Free•Available from:


Cyram Netminer for Windows•Very new: largely untested•Price range depends on application•Limited to smaller networks O(100)


NetDraw•Also very new, but by one of the best known names in network analysis software. •Free•Limited to smaller networks O(100)

Tools, Methods & ModelsAnalysis Methods: Descriptive / Measurement

Wasserman, Stanley and Katherine Faust. 1994. Social Network Analysis. Cambridge: Cambridge University Press.

The key text for methods and measurement is:

The basic network measures use graph theory to formalize aspects of the network, and always work from either an adjacency matrix (slow for large graphs) or an edge/node list.

Tools, Methods & ModelsAnalysis Methods: Descriptive / Measurement

Properties of interest include:

Individual Level:Degree: Number of contacts for each person - Sum over the row/column of the adjacency matrix.Closeness Centrality: Inverse of the distance to every other node in the network. Count path distances from ego to alters.

Sub-group Level:Group Membership: Which groups are there? Various search algorithms for identifying groups.Group Position: Where does a given group fit in the overall flow of relations? Various Equivalence algorithms.

Graph Level:Density: Number of ties present as a percentage of all possible ties.Centralization: To what degree are edges focused through a small number of nodes. Various formulas for different centrality indices.

Tools, Methods & ModelsAnalysis Methods: Descriptive / Measurement: Software

1) UCI-NET•General Network analysis program, runs in Windows•Good for computing measures of network topography for single nets•Input-Output of data is a little clunky, but workable.•Not optimal for large networks•Available from:

Analytic [email protected]

2) STRUCTURE •“A General Purpose Network Analysis Program providing Sociometric Indices, Cliques, Structural and Role Equivalence, Density Tables, Contagion, Autonomy, Power and Equilibria In Multiple Network Systems.”•DOS Interface w. somewhat awkward syntax•Great for role and structural equivalence models•Manual is a very nice, substantive, introduction to network methods•Available from a link at the INSNA web site:

http://www.heinz.cmu.edu/project/INSNA/soft_inf.html

Tools, Methods & ModelsAnalysis Methods: Descriptive / Measurement: Software

3) NEGOPY•Program designed to identify cohesive sub-groups in a network, based on the relative density of ties.•DOS based program, need to have data in arc-list format•Moving the results back into an analysis program is difficult.•Available from:

William D. Richardshttp://www.sfu.ca/~richards/Pages/negopy.htm

4) SPAN - Sas Programs for Analyzing Networks (Moody, ongoing)•is a collection of IML and Macro programs that allow one to:

a) create network data structures from nomination datab) import/export data to/from the other network programsc) calculate measures of network pattern and compositiond) analyze network models

•Allows one to work with multiple, large networks•Easy to move from creating measures to analyzing data•All of the Add Health data are already in SAS•Available by sending an email to:

[email protected]

http://www.sfu.ca/~richards/Pages/negopy.htm

Tools, Methods & ModelsAnalysis Methods: Statistical Models

There are two general classes of statistical models for networks:

1) Models of the network itselfThe statistical question is how an observed network fits into the class of all possible random graphs with a given set of topological characteristics. The whole network is the substantive unit of analysis, though technically one works with the dyads from the network.

Examples: p* models (Wasserman and Pattison), MCMC random graph models (Tom Snijders, Mark Handcock)

2) Models of individual behavior that incorporate network characteristicsThe statistical question is whether or not network properties affect individual behaviors.

Examples: Network regressive-autoregressive models (Doriean), Peer influence models (Friedkin)


Exponential Random Graph Models

)(

))(exp()(

xz

xXp

Where:z is a collection of r explanatory variables, calculated on x is a collection of r parameters to be estimatedk is a normalizing constant that ensures the probability sums to 1.

As it turns out, k is incredibly difficult to identify, introducing a number of complexities to the model.



To estimate the model, we work with the conditional probabilities (Xij|Xcij)

instead of the full graph. This transforms the exponential model to a logit model on the dyads:

)}(exp{

)}(exp{

)|0(

)|1(

ij

ij

cijij

cijij

xz

xz

XXp

XXp

)]()([)|0(

)|1(log

)]}()([exp{

ijijcijij

cijij

ij

ijij

xzxzXxp

Xxpw

xzxz

Analysis Methods: Statistical Models


Software for analyzing these models is available from:Logit Pseudo-Likelihood estimation:

http://kentucky.psych.uiuc.edu/pstar/index.html (SPSS programs)http://www.sfu.ca/~richards/Pages/pspar.html (Program for Large graphs)

Empirically, these models are tricky to estimate, as the potential result space can easily become degenerate, particularly as z starts to include a more complicated rage of dependencies.

MCMC Estimation:Ongoing work by Mark Handcock, Tom Snijders and Co.

http://kentucky.psych.uiuc.edu/pstar/index.html

http://www.sfu.ca/~richards/Pages/pspar.html


Network Effect Models

Question is whether or not being connected to a particular set of people affects an individual’s behavior. The key statistical point is that we have abandoned the assumption that our cases are independent.

These models originated in spatial statistics – looking at the effect of an adjacent geographic area on outcomes for any given area.

Basic Peer Influence ModelFormal Model

XBY )1( (1)

)1()1()( )1( YWYY αα Tt (2)

Y(1) = an N x M matrix of initial opinions on M issues for N actors

X = an N x K matrix of K exogenous variable that affect YB = a K x M matrix of coefficients relating X to Y = a weight of the strength of endogenous interpersonal

influencesW = an N x N matrix of interpersonal influences

Basic Peer Influence ModelFormal Model

XBY )1( (1)

This is the basic general linear model.

It says that a dependent variable (Y) is some function (B) of a set of independent variables (X). At the individual level, the model says that:

k

kiki BXY

Usually, one of the covariates is , the model error term.

Basic Peer Influence Model

)1()1()( )1( YWYY αα Tt (2)

This part of the model taps social influence. It says that each person’s final opinion is a weighted average of their own initial opinions

)1()1( Yα

And the opinions of those they communicate with (which can include their own current opinions)

)1( TαWY


The key to the peer influence part of the model is W, a matrix of interpersonal weights. W is a function of the communication structure of the network, and is usually a transformation of the adjacency matrix. In general:

jij

ij

w

w

1

10

Various specifications of the model change the value of wii, the extent to which one weighs their own current opinion and the relative weight of alters.


Formal Properties of the model

The model is directly related to spatial econometric models:

If we allow the model to run over t, we can describe the model as:

XBWYY )1()()( αα

XWYY~)()( α

Where the two coefficients ( and ) are estimated directly (See Doreian, 1982, SMR)

Overview

Conclusions


Introduction

Basic Concepts


An Introduction to Social Network Analysis James Moody Department of Sociology The Ohio State...

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