Title: Random Networks, Graphical Models, and Exchangeability,
Abstract: We study conditional independence relationships for random networks and their
interplay with exchangeability. We show that, for finitely exchangeable network
models, the empirical subgraph densities are maximum likelihood estimates of
their theoretical counterparts. We then characterize all possible Markov
structures for finitely exchangeable random graphs, thereby identifying a new
class of Markov network models corresponding to bidirected Kneser graphs. In
particular, we demonstrate that the fundamental property of dissociatedness
corresponds to a Markov property for exchangeable networks described by
bidirected line graphs. Finally we study those exchangeable models that are
also summarized in the sense that the probability of a network only depends
onthe degree distribution, and identify a class of models that is dual to the
Markov graphs of Frank and Strauss (1986). Particular emphasis is placed on
studying consistency properties of network models under the process of forming
subnetworks and we show that the only consistent systems of Markov properties
correspond to the empty graph, the bidirected line graph of the complete graph,
and the complete graph.