Edge Exchangeable Models for Interaction Networks

Abstract

Many modern network datasets arise from processes of interactions in a population, such as phone calls, email exchanges, co-authorships, and professional collaborations. In such interaction networks, the edges comprise the fundamental statistical units, making a framework for edge-labeled networks more appropriate for statistical analysis. In this context we initiate the study of edge exchangeable network models and explore its basic statistical properties. Several theoretical and practical features make edge exchangeable models better suited to many applications in network analysis than more common vertex-centric approaches. In particular, edge exchangeable models allow for sparse structure and power law degree distributions, both of which are widely observed empirical properties that cannot be handled naturally by more conventional approaches. Our discussion culminates in the Hollywood model, which we identify here as the canonical family of edge exchangeable distributions. The Hollywood model is computationally tractable, admits a clear interpretation, exhibits good theoretical properties, and performs reasonably well in estimation and prediction as we demonstrate on real network datasets. As a generalization of the Hollywood model, we further identify the vertex components model as a nonparametric subclass of models with a convenient stick breaking construction.

Phase Transition in the Recoverability of Network History

“…The tools introduced in Ref. [36], further work on information content, and the concept of exchangeability [73,74] all might offer insights into this issue. Fourth, we have shown that nonlinear preferential attachment can sometimes act as a useful effective model of a network's growth, even when the network is definitely not generated by this model.…”

Next waves in veridical network embedding*

“…These splits are created via sub-sampling the original data in a way which "respects" the way the observed data are sampled from a hypothetical population distribution. In the case of networks, various candidates for population models of graphs (including [45][46][47][48][49][50][51]) have been proposed, each with their own shortcomings. It is therefore unclear if in practice one should accept a single routine way in which training splits should be chosen, although some work [52] has explored the implications of when a subsampling procedure defined on finite graphs leads to a suitable population limit.…”

Random-Walk Models of Network Formation and Sequential Monte Carlo Methods for Graphs

“…Random-walk models generate networks one edge at a time; this specification of a graph may be viewed as a sequence of edges. Recent work on so-called edge exchangeable graphs (Crane and Dempsey, 2017;Williamson, 2016;Cai et al, 2016;Janson, 2017) define network models in terms of an exchangeable sequence of edges. Edges in random-walk models are not exchangeable.…”

“…All these models constrain dependence by rendering edges conditionally independent given some latent variable: in a graphon model, this latent variable is a local edge density; in the related model of Caron and Fox (2017), a scalar variable associated with each vertex; in recent so-called edge exchangeable models (Crane and Dempsey, 2017;Williamson, 2016;Cai et al, 2016), a random measure associated with the edges; in a configuration model, the degree sequence; in PA models, the vertex arrival times and a vector of vertex-specific variables (Bloem-Reddy and Orbanz, 2017). Some of these models have been noted to be misspecified for network analysis problems, including the following examples.…”