Random Networks, Graphical Models and Exchangeability

Abstract

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.

Network cross-validation by edge sampling

“…Another direction we did not explore in this paper is cross-validation under alternatives to the inhomogeneous Erdös-Renyi model, such as Crane and Dempsey [2018] or Lauritzen et al [2018]. ECV may also be modified for the setting where additional node features are available [Li et al, 2016, Newman andClauset, 2016].…”

On Exchangeability in Network Models

“…et al (2018). In Lauritzen et al (2018), we have also investigated exchangeable network models as graphical models on binary data with symmetric restrictions. There we have shown that distributions in E n can only be compatible with few Markov properties, and we have identified all the possible conditional independence structures that such distributions may exhibit.…”

“…where for a (labeled or unlabeled) graph G, E(G) is the number of its edges, r U (G) is the number of graphs in L n containing G as a subgraph and belonging to the isomorphism class represented by U , and z(U ) is the common value of the coordinates of the Möbius parameters z corresponding to the graphs in the isomorphism class represented by U . See Examples 1 and 2 in Lauritzen et al (2018).…”

“…When n = 4, there are 11 isomorphism classes, shown below in Figure 1 as unlabeled graphs, along with their respective sizes. Lauritzen et al (2018)). Finally, direct calculations reveal that, with the exception of the point masses at the empty and complete graphs, none of the maximum likelihood estimates of the probability distributions extend to exchangeable distributions over larger networks.…”

“…In the companion paper Lauritzen et al (2018), we rely on tools from the theory of graphical models to study the Markov properties of finitely exchangeable network models. The results derived there complement the ones we obtain in the present paper.…”