social network analysis:methods and applications chapter 9

04/08/2023 1

Chapter 9. Structural Equivalence

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Wasserman and Faust (1994)

04/08/2023 Wasserman and Faust (1994) 2

9.1 Background

• 9.1.1 Social Roles and Positions• 9.1.2 An Overview of Positional and Role Analysis• 9.1.3 A Brief History


9.1.1 Social Roles and Positions

• Position– a collection of individuals who are similarity embedded in

networks of relations– 같은 position 에 있는 actor 들은 직접 연결될 필요 없음

• Role– the patterns of relations which obtain between actors or be-

tween positions– network role refers to associations among relations that

link social positions– collections of relations and the associations among relations– roles can be modeled at three different levels

: actors, subset of actors, and the network as a whole


9.1.2 An Overview of Positional and Role Analysis

• Key aspects to the positional and role analysis– identifying social positions as collections of actors who are sim-

ilar with others– modeling social roles as systems of ties between actors or be-

tween positions


9.1.2 An Overview of Positional and Role Analysis

• Here we focus on positional analysis based on the similarity of actors in this chapter


9.2 Definition of Structural Equiva-lence

• Two actors are structurally equivalent if they have identical ties to and from all other actors in the network

• 9.2.1 Definition• 9.2.2 An Example• 9.2.3 Some Issues in Defining Structural Equivalence


9.2.1 Definition

• Actors i and j are structurally equivalent if, for all actors, k =1,2,…, g and all relations r = 1,2,…,R, actor i has a tie to k, iff j also has a tie to k, and I has a tie from k iff j also has a tie from k

• i j if i and j are structurally equivalent.

• Extended term : position– collection of equivalent ( or approximately equivalent ) actors


9.2.2 Example

• Both 1,2 and 3,4 are structurally equivalent per each.


9.2.3 Some Issues in Defining Structural Equivalence

• Multiple Relations– 모든 relations 에 대해서 structurally equivalent 할 경우 two actors

are structurally equivalent

• Valued Relations– tie 의 pattern 뿐 아니라 value 까지 같아야 structurally equivalent

이다 .– pattern 만 같으면 approximately structurally equivalent

• Nondirectional Relations• Self-ties and Graph Equivalence

– self-tie(reflexive tie) 가 허용되는 graph 의 경우 , 일 경우 graph equivalent 이다 .

– in other words, if actor j has a tie to actor i, actor i must have reflexive tie.


9.3 Positional Analysis

• major objective : simplify the information in a network data set

• 9.3.1 Simplification of Multirelational Network• 9.3.2 Tasks in a Positional Analysis


9.3.1 Simplification of Multirelational Network

• It is really difficult to find structural equivalent position intu-itively



• If we permute rows(and columns simultaneously), we can find intuitive positions



• Simplify the sociomatrix by collapsing same positionand represent them as reduced graph

– image matrices along with a description of which actors are as-signed to which position is called blockmodels(chapter 10)


9.3.2 Tasks in a Positional Analysis

• Steps of positional analysis– A formal definition of equivalence– A measure of the degree to which subsets of actors approach

that definition in a give set of network data– A representation of the equivalences– An assessment of the adequacy of the presentation

• Positional analysis can be done by using a variety of equiva-lence definition. Structural equivalence is just one case.


9.4 Measuring Structural Equiva-lence

• It is nearly impossible that two actors will be exactly struc-turally equivalent in actual networks

• we should identify subset of actors who are approximately structurally equivalent for positional analysis

• 9.4.1 Euclidean Distance as a Measure of Structural Equiva-lence

• 9.4.2 Correlation as a Measure of Structural Equivalence• 9.4.3 Some Considerations in Measuring Structural Equiva-

lence


9.4.1 Euclidean Distance

• 0 if two actors i and j are structurally equivalent• if i and j are totally structurally different


9.4.2 Pearson Correlation

• If two actors are structurally equivalent, the value will be equal to +1


9.4.3 Some Considerations in Measuring Structural Equivalence

• Comparison of Measures of structural equivalence

– correlation and Euclidian distances are not totally same– solution : standardize value of row i and row j


9.5 Representation of Network Po-sitions

• Goal : assigning actors to positions, and presenting the in-formation in a network data set in simplified form and pro-vide an interpretation for the results

• 9.5.1 Partitioning Actors• 9.5.2 Spatial Representations of Actor Equivalence• 9.5.3 Ties Between and Within Positions


9.5.1 Partitioning Actors

• If there are perfectly structurally equivalent positions, we can analyze positions by using the way introduced in 9.3.

• Since it is difficult to find them, we seek a partition of the actors into subsets(positions) so that actors within each subset are more nearly equivalent, according to the equiva-lence definition, and actors in different subsets are less equivalent.

• 1) Partitioning Actors using CONCOR• 2) Partitioning Actors using Hierarchical Clustering



• CONCOR : for CONvergence of iterated CORrelations– procedures

• starts with a sociomatrix• computes correlations among the rows(or columns)• construct a correlation matrix using the value• repeat same procedures until it converges(remain +1 or -1)• permutes rows(or columns) and represent matrix as below form:

• it can be repeated to find more positions– it can be thought of as a (divisive) hierarchical clustering

method• divisive vs agglomerative



• CONCOR : for CONvergence of iterated CORrelations– shortages

• CONCOR’s procedure of always splitting a set into exactly two sub-sets imposes a particular form on the resulting positional structure in the network

• the resulting partition is often different with social positions being understood intuitively

• formal properties of the procedures are not well understood



• Hierarchical clustering : agglomerative or divisive

• drawback of both CONCOR and hierarchical clustering– “grouping” or a split cannot be undone at a later stage


9.5.2 Spatial Representations of Actor Equivalence

• Multidimensional scaling: input is a one-mode symmetric matrix consisting of pairwise measures of similarity(Euclidian distance or correlation in this case)


9.5.3 Ties Between and Within Positions

• Describe how the positions relate to each other• 1. permute matrix based on the results(of any methods)



• 2. calculating density of each position



• 3. transform density matrix to image matrix(using density rule)

– density rule : set 1 if density of the position is greater than the whole density



• 4. Reduced Graphs


9.6 Summary

• Structural equivalence requires that equivalent actors have identical ties to and from identical others

• Therefore, it is difficult for different networks to be com-pared.– Chapter 12.

social network analysis:methods and applications chapter 9

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