This document discusses structural equivalence and positional analysis in networks. It defines structural equivalence as two actors having identical ties to and from all other actors. It describes methods for measuring approximate structural equivalence using metrics like Euclidean distance and correlation. It also outlines techniques for partitioning actors into positions based on their structural equivalence, including CONCOR and hierarchical clustering algorithms. The document emphasizes that positional analysis aims to simplify network data by grouping similarly positioned actors.

1. Chapter 9.
Structural Equivalence
박건우
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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
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3. 9.1.1 Social Roles and Positions
• Position
– a collection of individuals who are similarity embedded in networks of
relations
– 같은 position에 있는 actor들은 직접 연결될 필요 없음
• Role
– the patterns of relations which obtain between actors or between
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
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4. 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 similar with
others
– modeling social roles as systems of ties between actors or between
positions
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5. 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
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6. 9.2 Definition of Structural Equivalence
• 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
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7. 9.2.1 Definition
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8. 9.2.2 Example
• Both 1,2 and 3,4 are structurally equivalent per each.
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9. 9.2.3 Some Issues in Defining
Structural Equivalence
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10. 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
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11. 9.3.1 Simplification of
Multirelational Network
• It is really difficult to find structural equivalent position intuitively
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12. 9.3.1 Simplification of
Multirelational Network
• If we permute rows(and columns simultaneously), we can find
intuitive positions
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13. 9.3.1 Simplification of
Multirelational Network
• Simplify the sociomatrix by collapsing same position
and represent them as reduced graph
– image matrices along with a description of which actors are assigned to
which position is called blockmodels(chapter 10)
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14. 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 equivalence
definition. Structural equivalence is just one case.
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15. 9.4 Measuring Structural Equivalence
• It is nearly impossible that two actors will be exactly structurally
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 Equivalence
• 9.4.2 Correlation as a Measure of Structural Equivalence
• 9.4.3 Some Considerations in Measuring Structural Equivalence
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16. 9.4.1 Euclidean Distance
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17. 9.4.2 Pearson Correlation
• If two actors are structurally equivalent, the value will be equal to +1
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18. 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
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19. 9.5 Representation of Network Positions
• Goal : assigning actors to positions, and presenting the information
in a network data set in simplified form and provide 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
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20. 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 equivalence definition, and actors in
different subsets are less equivalent.
• 1) Partitioning Actors using CONCOR
• 2) Partitioning Actors using Hierarchical Clustering
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21. 9.5.1 Partitioning Actors
• 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
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22. 9.5.1 Partitioning Actors
• CONCOR : for CONvergence of iterated CORrelations
– shortages
• CONCOR’s procedure of always splitting a set into exactly two subsets
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
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23. 9.5.1 Partitioning Actors
• Hierarchical clustering : agglomerative or divisive
• drawback of both CONCOR and hierarchical clustering
– “grouping” or a split cannot be undone at a later stage
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24. 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)
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25. 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)
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26. 9.5.3 Ties Between and Within Positions
• 2. calculating density of each position
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27. 9.5.3 Ties Between and Within Positions
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28. 9.5.3 Ties Between and Within Positions
• 4. Reduced Graphs
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29. 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 compared.
– Chapter 12.
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