An Introduction to Exponential-Family Random Graph Models

Transcription

1 An Introduction to Exponential-Family Random Graph Models Luo Lu Feb.8, / 11

2 Types of complications in social network Single relationship data A single relationship observed on a set of nodes at a single point in time Time series data More than one point in time at which the relation on the set of nodes is observed Covariates With information about nodal attributes in addition to the relationship information Valued relationship Some types of relationships exist in varying degrees or strengths Multiple relationships More than one type of relationship studied 2 / 11

3 Basic Notation Assume G = (V, E) to be a directed graph with no self-loops or multiple edges. Define the adjacency matrix X: The adjacency matrix for is given below. { 1 if i relates to j X ij = 0 otherwise (a) A directed graph (b) The adjacency matrix 3 / 11

4 Reciprocation and differential attaction Let M denote the number of pairs i, j for which X ij = X ji = 1. M = i<j X ij X ji. The in-degree of node j is X +j = n i=1 X ij The out-degree of node j is X j+ = In the earliest sociometric studies, Moreno (1934) found that the directed graph s observed edges are not distributed randomly. n i=1 X ji 4 / 11

5 Reciprocation and differential attaction To test it, Moreno posited a null model for X in which all adjacency matrices with out-degrees agreeing with those in the data are equally likely. For any i j, E(X ij X ji {X i+ } n i=1) = P(X ij = 1 {X i+ } n i=1)p(x ji = 1 {X i+ } n i=1) = X i+x j+. (n 1) 2 Hence E(M {X i+ } n i=1) = n X 2 2(n 1) nvar(x i+) 2(n 1) 2 and E(Var(X +i ) {X i+ } n i=1) = X X 2 2(n 1) (n 2)Var(X i+) (n 1) 2 M and Var(X +i ) usually exceeded their chance expected values, which implies the existance of reciprocation. Similarly, from intuition, it should not be surprising that nodes are differentially attractive and that some are involved in more relational ties than are others. Two kinds of nodes: stars and isolates. This will result in large values of Var(X +i ). 5 / 11

6 The p 1 distribution Let G denote the set of all n by n adjacency matrices so that X may be thought of as a random matrix taking values in G. p 1(x) = P(X = x) = exp{ρm+θx +++ i α i x i+ + j β j x +j } K(ρ, θ, {α i }, {β j }) with constraints α + = β + = 0. Df = (n 1) + (n 1) = 2n. 6 / 11

7 Special Cases of p 1 p 1(x) = P(X = x) = exp{ρm+θx +++ i α i x i+ + j β j x +j } K(ρ, θ, {α i }, {β j }) D ij = (X ij, X ji ); m ij = P(D ij = (1, 1)); a ij = P(D ij = (1, 0)); n ij = P(D ij = (0, 0)); 7 / 11

8 Dependence Graphs D = (E, E ) is the conditional dependence graph of G: {(i, j), (k, l)} E iff X ij! X kl Xij,kl c From D to G: by Hammersley-Clifford Theorem - a probability distribution can be represented as a Markov network if and only if it is a Gibbs random field, that is, its density can be factorized over the cliques (or complete subgraphs) of the graph. where K D be the clique set of D. Pr(X = x θ) = 1 Z(θ) exp( S K D θ S (i,j) S X ij ) 8 / 11

9 Dependence Graphs Simple examples: Independent edges: X ij! X kl X c ij,kl iff (i, j) = (k, l) The only cliques are the nodes of D themselves, thus Pr(X = x θ) exp( i j θ ij X ij ) Independent dyads: X ij! X kl X c ij,kl iff (i, j) = (k, l) or (i, j) = (l, k) Each dyad of D contributes a clique, as does each edge, thus Pr(X = x θ) exp( i<j θ ij X ij X ji + i j θ ijx ij ) 9 / 11

10 More Complex example-markov Graphs X ij! X kl X c ij,kl iff {i, j} {(k, l} > 0. Intuitively, edge variables are conditionally dependent iff they share at least one endpoint. D now has a large number of cliques; these are the edges, stars and triangles of G 1 trivially, includes density and reciprocity k-stars equivalent to degree count statistics, hence includes degree distribution 2 through triads, includes local clustering as well as local cyclicity and transitivity in digraphs 10 / 11

11 Conclusion Many ways to define the sufficient statistics, more statistics more parameters to estimate Many ways to describe dependence among elements Once one leaves simple cases, not always clear where to begin 11 / 11