SNA.pdf

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I N F R A S T R U K T U R B E R K E L A N J U T A N

I S M U R I N I D W I A R I

SNA Density – Centrality – Degree of Participation

The Concept

Social structure is ‘an overall system, network or pattern’ of relations (Nadel,1957:12), which the analyst abstracts from the concretely observable actions of individuals (Scott,1991:30)

Network is ‘the interlocking of relationships whereby the interactions implicit in one determine those occurring in others’(Nadel,1957:16)

‘The whole of social life’ could be seen as ‘a set of points some of which are joined by lines’ to form a ‘total network’ of relations. The informal sphere of interpersonal relations was to be seen as one part, a ‘partial network’, of this total network (Barnes, 1954:43)

Durkheim and Tonnies (end of 18th C)

existence of social structure is as in the form of personal ties that either link people who share similar values and beliefs or impersonal, formal and instrumental social links.

S. Wasserman & K. Faust (1994)

“the social network perspective encompasses theories, models, & applications that are expressed in terms of relational concepts or processes. That is, relations defined by linkages among units are a fundamental component of network theories.”

The Definition

The four tenets of SNA (S. Wasserman & K. Faust,1994)

Actors & their actions are viewed as interdependent rather than independent, autonomous units.

Relational ties (linkages) between actors are channels for transfer of “flow” of resources (either material or non material).

Network models focusing on individuals view the network structural environment as providing opportunities for, or constraints on, individual action.

Network models conceptualize structure (social, economic, political, and so forth) as lasting patterns of relations among actors.

The Definition

Three Types of Data

Attribute data

relate to the attitudes, opinions and behavior of agents, in so far as these are regarded as the properties, qualities or characteristics that belong to them as individuals or group.

Relational data

the contacts, ties and connections, the group attachments and meetings, which relate one agent to another and so cannot be reduced to the properties of the individual agents themselves.

Ideational data

which describe the meanings, motives, definitions and typifications themselves.

Source: Scott, J., 2000, Social Network Analysis a Handbook, SAGE

Style of research Source of evidence Type of data

Type of Analysis

Survey research Questionnaires, interviews

Attribute Variable analysis

Ethnographic research

Observations Ideational Typological analysis

Documentary research

Texts Relational Network analysis

Three Types of Data


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The importance of studying affiliation networks is grounded in theoretical importance of individuals’ memberships in collectivities (Wasserman & Faust, 1994)

Affiliation Network: (Faust, 1997)

Such network of actors tied to each other through their participation in collectivities, and collectivities linked through multiple memberships of actors

Consist of two elements:

A set of actors and

A collection of subsets of actors (events)

Connections among members of one mode of the modes are

based on linkages established through the second mode (Wasserman & Faust, 1994)

Affiliation Network

Affiliation Network 8

The set of actors is labeled as

The set of events is labeled as

Affiliation Matrix

Hypothetical Example:

6 actors & 4 events

},...,,{ 21 gnnnN

},...,,{ 21 hmmmM

m1 m2 m3 m4

n1 1 0 1 1

n2 0 1 0 1

n3 0 1 1 0

n4 0 0 1 0

n5 1 1 0 1

n6 1 1 1 1

A ‘1’ in row actor , column event of A indicates that actor ni is affiliated with event mk

i k

gxhaA ik

Affiliation Network: one-mode & two-mode

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n1 n2 n3 n4 n5 n6

n1 0 1 1 1 2 3

n2 1 0 1 0 2 2

n3 1 1 0 1 1 2

n4 1 0 1 0 0 1

n5 2 2 1 0 0 3

n6 3 2 2 1 3 0

n1 n2 n3 n4 n5 n6 m1 m2 m3 m4

n1 0 0 0 0 0 0 1 0 1 1

n2 0 0 0 0 0 0 0 1 0 1

n3 0 0 0 0 0 0 0 1 1 0

n4 0 0 0 0 0 0 0 0 1 0

n5 0 0 0 0 0 0 1 1 0 1

n6 0 0 0 0 0 0 1 1 1 1

m1 1 0 0 0 1 1 0 0 0 0

m2 0 1 1 0 1 1 0 0 0 0

m3 1 0 1 1 0 1 0 0 0 0

m4 1 1 0 0 1 1 0 0 0 0

0'

0

A

AX NM

Two-mode Bipartite Matrix

'AAX N

One-mode matrix of actor co-memberships – indicates the number of memberships shared by each pair of actors

A B C D E

1 1 1 1 1 0

2 1 1 1 0 1

3 0 1 1 1 0

4 0 0 1 0 1

1 2 3 4

1 - 3 3 1

2 3 - 2 2

3 3 2 - 1

4 1 2 1 -

A B C D E

A - 2 2 1 1

B 2 - 3 2 1

C 2 3 - 2 2

D 1 2 2 - 0

E 1 1 2 0 -

Methodology

1. Incidence matrix typically rectangular matrix

represents the affiliations of agents in a network

2. Adjacency matrix always square matrix represents the ties among agents in a network

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Inclusiveness refers the number of connected points that are included within the various connected parts of the graph.

Inclusiveness = (n- n0 )/n

The degree of a point is a numerical measure of the size of its neighborhood.

Density of indirected graph = 2l/n(n-1)

Note: the directed graph has asymmetrical matrix wherein a directed line from A to B will not necessarily a reciprocated line directed from B to a.

Density of directed graph = l/n(n-1)

Inclusiveness & Density

…


Inclusiveness & Density


No. of connected nodes 4 4 4 3 2 0

Inclusiveness 1.0 1.0 1.0 0.7 0.5 0

Sum of Degrees 12 8 6 4 2 0

No. of lines 6 4 3 2 1 0

Density 1.0 0.7 0.5 0.3 0.1 0

Inclusiveness = ratio of the number of connected points to the total number of points

An isolated point is incident with no lines and so can contribute nothing to the density of network. The more inclusive, the more dense will it be.

Centrality (Scott J, 1991)

is a measure of the importance of a node in providing connectivity (bridging) to other nodes in the network.

is a measure of the contribution of network position to the importance, influence, prominence of an actor in a network.

Local centrality (LC) of a node Absolute LC is simply the degree of a node

Relative LC is ratio between the degree of connection to the total number of nodes exclude the node of itself

Global centrality a node is defined as the sum of all distances of that node to the rest of

the nodes of the network

A node is globally central if it lies at short distances from the other nodes of the network.

The Example (Scott J, 1991)

E F J K N

A G B C O D M

H L Q P

A,C B G,M J,K,L All other

nodes

Local

Centrality

Absolute 5 5 2 1 1

Relative 0.33 0.33 0.13 0.07 0.07

Global Centrality 43 33 37 48 57

Centrality for Affiliation Networks

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The notion of centrality is to identify the most important actors in a social network that located in strategic locations within the network (Wasserman and Faust,1994).

The measure of the prominence or importance of the actors in a social network

A prominent actor is if the ties of the actor make the actor particularly visible to the other actors in the network.

Based on the relational pattern of the row and column entries

The use of centrality measures is to understand group structure which those actors with the most access will be the most central in the network.

One-mode Degree Centrality (Wasserman & Faust, 1994)

The simplest definition of actor centrality is that central actors must be the most active in the sense that they have the most ties to other actors in the networks.

This very active actor should thus have a maximal centrality index.

The most visible actors in the network.

Index of actor degree centrality

Normalized degree centrality

An ego density for a nondirectional relation is simply the ratio of degree of an

actor to the maximum number of ties that could occur.

j

jij

ijiiiD xxxndnC )()(

1

)()('

g

ndnC i

iD

Two-Mode Degree Centrality

Affiliation network as two-mode networks, consists of subsets of actors wherein connections among members of one of the modes are based on linkages established through the second mode

(Wasserman & Faust, 1994)

Two-mode Degree Centrality

The actors are important because of their level of activity or the number of contact that they have (Faust, 1997)

The raw degree centrality for an actor is defined as the number of events that is participated by the actor )( iD nC

Actor ni normalized degree centrality (Everett & Borgatti, 2005)

h

nC iD )(

One-Mode Closeness Centrality (Wasserman & Faust, 1994)

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Closeness measures how close an actor is to all the other actors in the set of actors as a function of geodesic distance.

Actors are central if they can contact others through efficient (short) paths

Index of actor closeness centrality: Sabidussi’s (1966) in Wasserman &

Faust (1994)

Normalized closeness centrality: Beauchamp (1965) in Wasserman &

Faust (1994)

g

jji

nC nndiC

1

)( ),(

1

)()1()('1

),(

1

iCnnd

g

iC nCgnCg

j ji

),( ji nnd : the length of any shortest path between the actors for all j ≠ i

),( ki nnd : the geodesic distance of the isolate actor nk = 0 for all i ≠ k

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is that an actor is central if it lies between other actors on their geodesics, implying that to have a large “betweenness” centrality, the actor must be between many of the actors via their geodesic.

Index of actor betwenness for is the sum of the estimated probabilities over all pairs of actors not including the ith actor for i distinct from j and k.

Normalized betwenness centrality

The value is between 0 and 1

One-Mode Betwenness Centrality (Wasserman & Faust, 1994)

jki

kj

jkiB gngnC /)()(

]2/)2)(1/[()()(' ggnCnC iBiB

in

in

jkg : the number of geodesic linking the two actors

)( ijk ng : the number of geodesic linking the two actors that contain actor i

: the number of pairs of actors not including 2/)2)(1( gg

Rate of Participation

Describe people’s rates of participation in social activities (McPherson, 1982).

The mean rate of affiliation for actors in the bipartite matrix that might one to compare people’s rate of participation in voluntary organization between communities (Wasserman and Faust, 1994).

Source: Wasserman and Faust (1994)

The row totals of an affiliation matrix, A or

The entries on the main diagonal of actor co-memberships matrix, in a valued relation matrix.

The number of events with which actors is affiliated is denoted by .

= the number or the size of events, , to which actor belongs.

(i) Rate of Participation

NX

N

ii

h

j iji xaa 1

h M

g

x

g

a

g

aa

g

i

N

ii

g

i

h

j ij

i

11 1


The average number of actors in each event, or mean degree of events in the bipartite matrix.

The size of each event refers to the number of actors affiliated with event j is given by the column total of affiliation matrix, A, or the entries on the main diagonal of the event overlap matrix, , is denoted by

(ii) Size of Events


MX

M

jj

g

i ijj xaa 1

h

x

h

a

h

aa

h

j

M

jj

g

i

h

j ij

j

11 1

Pustaka

Faust, K., 1997, Centrality in Affiliation Networks, Social Networks 19:157-191

Sabidussi, G., 1966, The Centrality Index of a Graph, Psychometrika Vol. 31 No. 4.

Scott, J., 1991, Social Network Analysis: A Handbook, 2nd Edition, SAGE.

Wasserman, S. and Faust, K., 1994, Social Network Analysis: Methods and Applications, Cambridge University Press.

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