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Random networks, graphical models and exchangeability
Author(s) -
Lauritzen Steffen,
Rinaldo Alessandro,
Sadeghi Kayvan
Publication year - 2018
Publication title -
journal of the royal statistical society: series b (statistical methodology)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 6.523
H-Index - 137
eISSN - 1467-9868
pISSN - 1369-7412
DOI - 10.1111/rssb.12266
Subject(s) - mathematics , markov chain , conditional independence , graphical model , random graph , exponential random graph models , combinatorics , discrete mathematics , markov property , line graph , variable order markov model , graph , consistency (knowledge bases) , markov model , statistics
Summary 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.