Role Analysis in Social Networks

1. Role Analysis in Social Networks Victor Lee

2. Outline • What are Social Networks? • Role and Position Analysis • Equivalence Models for Roles • Block Modelling

3. Social Networks • Not just Facebook and MySpace… • A social network is a collection of actors who are joined by pairwise ties. • Actors may be individual persons or organizations • Ties are any type of relationship between two actors, such as friendship, kinship, financial exchange, influence, or prestige.

4. Representing (Social) Networks • As a Graph • Actor  vertex or node • Tie  edge or link • Could be directed/undirected,weighted/unweighted • As an Adjacency Matrix • N x N matrix, N = number of vertices • aij = weight of edge from i  j 2 1 3 4

5. Social Position • Social Position = collection of actors similarly embedded in a network • Similar sets of ties to other actors • Not based on adjacency, proximity, or reachability • Example: Nurses at different hospitals occupy the position nurse, even though they don't work with each other, or even the same doctors or patients.

6. Social Role • Social Role = pattern of relationships that an actor has with other actors • May include both direct and compound relations • Example of Kinship Relations: combinations of relations marriage and descent. • Direct: sister, husband, son • Combination: sister-in-law, uncle, grandson • Position is a grouping;Role is what characterizes the group

7. Role & Positional Analysis • Positional analysis • Separate actors into subsets of positions  Partitioning • Each position is a mathematical equivalence class • Role Analysis • Discover and describe patterns of relationships  Pattern Mining or Motif Discovery • Global: look at the full set of edges • Local: look at the neighborhood of an individual

8. Analysis Approaches • Positional, then Role • Group actors into equivalence classes (positions) • Describe each position with an aggregate role description. • Role, then Positional • Describe the relationships of each individual • Group actors that have equivalent or similar patterns or relations

9. Equivalence Classes and Relations • An equivalence class C is a set of ordered pairs in which the following properties hold: • Reflexivity: (a,a) ∈ C • Symmetry: if (a,b) ∈ C, then (b,a) ∈ C • Transitivity: if (a,b) and (b,c) ∈ C, then (a,c) ∈ C • An equivalence relation for set A partitions A into equivalence classes {C1, …, Ck}

10. Equivalence Classes for Roles • Goal: define a rule-based equivalence relation that will partition a set of actors into positions and roles • Three common definitions: • Structural Equivalence • Automorphic Equivalence • Regular Equivalence

11. Structural Equivalence • Two actors are structurally equivalent if they have identical ties to and from the other actors (Lorrain and White 1971). • Example: • C1 = {1, 4} • C2 = {2,5} • C3 = {3} 1 4 2 5 3

12. Automorphic Equivalence • Two actors are automorphically equivalent if they have identical ties to equivalent actors. • Must have same number of ties • Recursive definition • If we color the vertices by position, • Vertices are equivalent if their neighborhoods consist of the same number of the same colors • Example: Two families with exactly the same number of children, parents, etc.

13. Isomorphism and Automorphism • A graph isomorphism of graphs G and H is a bijective mapping of vertices f(V(G))  V(H) • such that all the edges remain the same • That is, e(a,b) ∈ G if and only if e(f(a),f(b)) ∈ H • A graph automorphism is when the isomorphism is to G itself • That is, we rearrange the vertex labels of G • Example: Every possible labeling of a clique is automorphically equivalent • What about a ring? A binary tree?

14. Regular Eqivalence • Two actors are regularly equivalent if they have ties to equivalent positions. • Need not have the same number of ties • Recursive definition • If we color the vertices by position, • Vertices are equivalent if their neighborhoods consist of the same set of colors • But the quantity of each color does not matter • Example: Two families with different numbers of children, parents, etc.

15. 1 2 3 4 5 6 7 8 9 Comparing Equivalences • Structural: connect to exactly the same neighbors • {5,6}, {8,9}, singletons • Automorphic: connect to the same distribution of colors • {5,6,8,9}, {2,4}, singletons • Regular: connect to the same colors • {5,6,7,8,9}, {2,3,4}, {1}

16. Block Modeling • Consider the matrix representations • Assume we arrange rows and columns by equivalence class, creating blocks • What do you notice about each block? • What rule would each type of equivalence follow?

17. Image Matrices & Reduced Graphs • If we can make a simple statement about each block: • “Every relation within the block is (almost) always 1.” • “Every relation within the block is (almost) always 0.” • We can form an image matrix/reduced graph

18. Density Table • In automorphic equivalence, we expect to see the same number of 1’s in each row and each column within a block • “Every relation within block Bij has a probability pij of being present.” • In-relations can be different from out-relations, so Bij may be different from Bji

19. Thinking more about Block Models • How would you construct a block model based on regular equivalence? • What if we have weighted edges?

20. Near-Equivalence • Exact equivalence is often too strict • Also want to know how similar are two actors • We can then cluster together similar actors

21. Traditional Approaches for Similarity • Euclidean distance • Each vertex or group is a dimension in space • Distance between row vectors & column vectors of two actors: • Correlation (Pearson product-moment) • If two actors are equivalent, the correlation between their rows and columns will be +1 ‏

22. Tools for Partitioning Actors • CONCOR • Iteratively: compute row & column correlations,then split • PCA might be superior • Hierarchical Clustering

23. Other Topics • Relational Algebra • Given multiple types of edges (say, Friend and Trust) • Form an image matrix for each edge type (F and T) • Image multiplication? Example: R = F x Trxy means “x has a friend z who trusts y” • Optimal Partitioning? • Exact equivalence  clear idea of “best” partition • Using similarity  trade-off between tight variance with a block and having a small number of blocks • Other tools and methods to compute similarity and to partition?

24. References These slides are based onWasserman and Faust (2004), Social Network Analysis: Methods and Applications, Ch. 9 – 12. Additional Reading • Concise online book on social network analysis:www.faculty.ucr.edu/~hanneman/nettext/index.html • Review paper from the sociologist’s perspective:Borgatti, S. and Everett, M., Notions of Position in Social Network Analysis, Sociological Methodology (22), 1-35, 1992.