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Exponential Random Graph (p*) Models for Social Networks: The Global Outcomes of Local Model Specifications

Wednesday, November 19, 2003 - 11:00am - 11:50am
Keller 3-180
Garry Robins (University of Melbourne)
Exponential random graph models, when derived from a dependence graph using the Hammersley-Clifford theorem, are specified in terms of local network structures. But as these localized patterns agglomerate, the global outcomes are often not apparent. We review our recent work on simulating distributions of Markov random graphs, examining the resulting global structures by comparison with appropriate Bernoulli distributions of graphs. We provide examples of various stochastic global worlds that may result, including small worlds, long path worlds and dense non-clustered worlds with many four-cycles. Degeneracy in these models relates to the movement from structure to randomness, when parameter scaling results in a phase transition occurring at a certain temperature. Degenerate or frozen deterministic structures may be merely empty or full graphs, but also include more interesting highly clustered caveman graphs, bipartite structures, and global cyclic structures involving structurally equivalent groups.

But Markov random graphs are only one possible way to specify exponential random graphs. We present recent results from simulations for two other new model specifications. The first specification includes binary attribute measures on the nodes, with network ties and actor attributes mutually contingent, resulting in joint social influence/social selection models. The second specification includes aggregations of triangle and star counts, permitting an explicit model form for the degree distribution and a new transitivity concept, k-triangles, reflecting the distribution of triangles across the graph.