This lab explores three different approaches to identifying structural equivalence in social networks: Euclidean Distance, CONCOR, and Optimization methods. Structural equivalence examines how actors occupy similar positions in a network based on their connection patterns.
Datasets
We’ll use two datasets for this analysis:
Wasserman & Faust (1994) Network: A small network ideal for illustrating blockmodeling techniques
Anabaptist Leadership Network: A larger network of 67 religious leaders during the Protestant Reformation
Euclidean distance measures similarity by comparing distances between nodes in n-dimensional space. Unlike graph theoretical distance (path length), Euclidean distance represents the most direct route between actors.
Code
# Calculate structural equivalence using Euclidean distance#wfse.eq <- equiv.clust(wfse.net, method="euclidean", mode="digraph")wfse.eq <- sna::equiv.clust( wfse.net,equiv.fun = sna::sedist, # force sna's sedist, not blockmodeling'smethod ="euclidean",mode ="digraph")# Create blockmodel with 3 blockswfse.bl <-blockmodel(wfse.net, wfse.eq, k=3, mode="graph")# Display block membershipwfse.bl$plabels
The optimization approach uses random partitions and iteratively improves the fit by minimizing error scores.
Run Optimization Algorithm
Code
#dim(anabaptist.mat)# Run optimization with 8 blocksanabaptist.opt <-optRandomParC(M=anabaptist.mat, k=8, rep=500, approaches="bin", blocks=c("nul","com"))
Starting optimization of the partiton 50 of 500 partitions.
Starting optimization of the partiton 100 of 500 partitions.
Starting optimization of the partiton 150 of 500 partitions.
Starting optimization of the partiton 200 of 500 partitions.
Starting optimization of the partiton 250 of 500 partitions.
Starting optimization of the partiton 300 of 500 partitions.
Starting optimization of the partiton 350 of 500 partitions.
Starting optimization of the partiton 400 of 500 partitions.
Starting optimization of the partiton 450 of 500 partitions.
Starting optimization of the partiton 500 of 500 partitions.
Optimization of all partitions completed
2 solution(s) with minimal error = 254 found.
Code
# Display optimization resultsanabaptist.opt
Network size: 67
Approachs (paramter): bin
Blocks (paramter)
nul com
Sizes of clusters:
1 2 3 4 5 6 7 8
1 7 1 37 4 6 3 8
IM
1 2 3 4 5 6 7 8
1 nul com com nul com nul nul nul
2 com com nul nul nul nul nul nul
3 com nul nul nul com nul com com
4 nul nul nul nul nul nul nul nul
5 com nul com nul nul nul com nul
6 nul nul nul nul nul com nul nul
7 nul nul com nul com nul com nul
8 nul nul com nul nul nul nul nul
Error: 254
2 solutions with minimal error exits. Only results for the first one are shown above!
# Run optimization with 12 blocksanabaptist.opt2 <-optRandomParC(M=anabaptist.mat, k=12, rep=500,approaches="bin", blocks=c("nul","com"))
Starting optimization of the partiton 50 of 500 partitions.
Starting optimization of the partiton 100 of 500 partitions.
Starting optimization of the partiton 150 of 500 partitions.
Starting optimization of the partiton 200 of 500 partitions.
Starting optimization of the partiton 250 of 500 partitions.
Starting optimization of the partiton 300 of 500 partitions.
Starting optimization of the partiton 350 of 500 partitions.
Starting optimization of the partiton 400 of 500 partitions.
Starting optimization of the partiton 450 of 500 partitions.
Starting optimization of the partiton 500 of 500 partitions.
Optimization of all partitions completed
1 solution(s) with minimal error = 222 found.
Structural Equivalence occurs when two actors have identical patterns of relationships with all other actors in the network. Actors who are structurally equivalent are perfect substitutes for one another in terms of their network positions.
Methodological Approaches
Euclidean Distance: Measures direct dissimilarity between connection patterns
CONCOR: Uses iterative correlations to identify structurally similar positions
Optimization: Iteratively improves partition fit by minimizing error scores
Blockmodeling Interpretation
Complete Blocks (1-blocks): Dense connection patterns within position
Null Blocks (0-blocks): Sparse or absent connections between positions
Image Matrix: Simplified representation of connection patterns between positions
Practical Applications
Structural equivalence analysis helps identify:
Functional roles within organizations
Competitive positions in markets
Redundancy in communication networks
Potential for substitution or replacement
Conclusion
This lab demonstrated three complementary approaches to structural equivalence analysis:
Euclidean Distance provides a straightforward dissimilarity measure
CONCOR offers a robust correlation-based partitioning method
Each method has strengths and is appropriate for different analytical contexts. The choice among them depends on research questions, network characteristics, and theoretical frameworks.
References
Wasserman, S., & Faust, K. (1994). Social Network Analysis: Methods and Applications. Cambridge University Press.
Breiger, R. L., Boorman, S. A., & Arabie, P. (1975). An algorithm for clustering relational data with applications to social network analysis. Journal of Mathematical Psychology, 12, 328-383.
Ziberna, A. (2007). Generalized blockmodeling of valued networks. Social Networks, 29, 105-126.
Burt, R. S. (1976). Positions in networks. Social Forces, 55, 93-122.
