Victor Lee

 What are Social Networks?  Role and Position Analysis  Equivalence Models for Roles  Block Modelling

 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.

 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  a ij = weight of edge from i  j

 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.

 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

 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

 Positional, then Role a.Group actors into equivalence classes (positions) b.Describe each position with an aggregate role description.  Role, then Positional a.Describe the relationships of each individual b.Group actors that have equivalent or similar patterns or 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 {C 1, …, C k }

 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

 Two actors are structurally equivalent if they have identical ties to and from the other actors (Lorrain and White 1971).  Example:  C 1 = {1, 4}  C 2 = {2,5}  C 3 = {3}

 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.

 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?

 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.

 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}

 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?

 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

 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

 How would you construct a block model based on regular equivalence?  What if we have weighted edges?

 Exact equivalence is often too strict  Also want to know how similar are two actors  We can then cluster together similar actors

 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 ‏

 Hierarchical Clustering  CONCOR  Iteratively: compute row & column correlations, then split  PCA might be superior

 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 T r xy 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?

These slides are based on Wasserman and Faust (2004), Social Network Analysis: Methods and Applications, Ch. 9 – 12. Additional Reading  Concise online book on social network analysis:  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.