Random Networks, Graphical Models, and Exchangeability

TY - JOUR

T1 - Random Networks, Graphical Models, and Exchangeability

AU - Lauritzen, Steffen L.

AU - Rinaldo, Alessandro

AU - Sadeghi, Kayvan

PY - 2018/6

Y1 - 2018/6

N2 - 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 depends only on the degree distribution, and we identify a class of models that is dual to the Markov graphs of Frank and Strauss. 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.

AB - 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 depends only on the degree distribution, and we identify a class of models that is dual to the Markov graphs of Frank and Strauss. 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.

U2 - 10.1111/rssb.12266

DO - 10.1111/rssb.12266

M3 - Journal article

SN - 1369-7412

VL - 80

SP - 481

EP - 508

JO - Journal of The Royal Statistical Society Series B-statistical Methodology

JF - Journal of The Royal Statistical Society Series B-statistical Methodology

IS - 3

ER -