Social network analysis is:

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

Introduction

• Actors (nodes, points, vertices): - Individuals, Organizations, Events …

• Relations (lines, arcs, edges, ties): between pairs of actors.- Undirected (symmetric) / Directed (asymmetric)- Binary / Valued

Basic conceptsNetwork Components

1) Egocentered Networks• Data on a respondent (ego) and the people they are connected to.

Measures:SizeTypes of relations

Basic concepts

Types of network data:

2) Complete Networks

• Connections among all members of a population.

• Data on all actors within a particular (relevant) boundary.

• Never exactly complete (due to missing data), but boundaries are set

• Ex: Friendships among workers in a company.

Measures:Graph properties DensitySub-groupsPositions

Background

Types of network data:

The unit of interest in a network are the combined sets of actors and their relations.

We represent actors with points and relations with lines.

Example:

Social Network data

a

b

c e

d

In general, a relation can be:Undirected / DirectedBinary / Valued

a

b

c e

d

Undirected, binary Directed, binary

a

b

c e

d

a

b

c e

d

Undirected, Valued Directed, Valued

a

b

c e

d1 3

4

21

Social Network data

From pictures to matrices

Undirected, binary Directed, binary

a b c d ea

b

c

d

e

1

1

1 1 1

1 1

a b c d ea

b

c

d

e

1

1 1

1 1 1

1 11 1

Basic Data Structures

Social Network data

a

b

c e

d

a

b

c e

d

d e

c

Indirect connections are what make networks systems. One actor can reach another if there is a path in the graph connecting them.

a

b

c e

d

f

b f

a

Connectivity

Measuring Networks

Distance is measured by the (weighted) number of relations separating a pair, Using the shortest path.

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

Distance & number of paths

Measuring Networks

a

An information network:

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

Measuring Networks

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

Centrality

Measuring Networks

Centrality is fairly straight forward: we want to identify which nodes are in the ‘center’ of the network. In the sense that they have many and important connections.

Three standard centrality measures capture a wide range of “importance” in a network:

DegreeClosenessBetweenness

The most intuitive notion of centrality focuses on degree. Degree is the number of lines, and the actor with the most lines is the most important:

Centrality

Measuring Networks

Centrality

Measuring Networks

Relative measure of Degree Centrality:

1

),()(' 1

n

ppaPC

ki

n

ikD

Degree Centrality:

),()(1

ki

n

ikD ppaPC

A second measure is closeness centrality. An actor is considered important if he/she is relatively close to all other actors.

Closeness is based on the inverse of the distance of each actor to every other actor in the network.

Closeness Centrality:

Relative Closeness Centrality

Centrality

Measuring Networks

1

1)],([)(

ki

n

ikC ppdPC

),(

1

1

),()('

1

1

1

ki

n

i

ki

n

ikC

ppd

n

n

ppdPC

Closeness Centrality

Centrality

Measuring Networks

Betweenness Centrality:Model based on communication flow: A person who lies on

communication paths can control communication flow, and is thus important. Betweenness centrality counts the number of shortest paths between i and k that actor j resides on.

b

a

C d e f g h

Centrality

Measuring Networks

Centrality

Measuring Networks

Betweenness centrality can be defined in terms of probability (1/gij),

CB(pk) = iij(pk) = =

gij = number of geodesics that bond actors pi and pj.gij(pk)= number of geodesics which bond pi and pj and content pk.iij(pk) = probability that actor pk is in a geodesic randomly chosen among the ones which join pi and pj.

Betweenness centrality is the sum of these probabilities (Freeman, 1979).

)(*g

1

ijkij pg

ij

kij

g

)(pg

Normalizad: C’B(pk) = CB(pk) / [(n-1)(n-2)/2]

Betweenness Centrality:

Centrality

Measuring Networks

If we want to measure the degree to which the graph as a whole is centralized, we look at the dispersion of centrality:

Freeman’s general formula for centralization (which ranges from 0 to 1):

)]2)(1[(

)()(1

*

nn

pCpCC

n

i iDDD

Centralization

Measuring Networks

Degree Centralization Scores

Freeman: 1.0 Freeman: .02 Freeman: 0.0

Centralization

Measuring Networks

Density

Measuring Networks

The more actors are connected to one another, the more dense the network will be. Undirected network: n(n-1)/2 = 2n-1 possible pairs of actors.

Δ =

Directed network: n(n-1)*2/2 = 2n-2possible lines.

ΔD =

2/)1( nn

L

)1( nn

L

Freeman: .25 Freeman: .23 Freeman: 0.25

Density

Measuring Networks

UCINET•The Standard network analysis program, runs in Windows•Good for computing measures of network topography for single nets•Input-Output of data is a special 2-file format, but is now able to read PAJEK files directly. •Not optimal for large networks•Available from:

Analytic Technologies

Social Network Software

PAJEK •Program for analyzing and plotting very large networks•Intuitive windows interface•Started mainly a graphics program, but has expanded to a wide range of analytic capabilities•Can link to the R statistical package•Free•Available from: http://vlado.fmf.uni-lj.si/pub/networks/pajek/

Social Network Software

NetDraw•Also very new, but by one of the best known names in network analysis software. •Free

Social Network Software