Exchangeable Random Measures for Sparse and Modular Graphs with Overlapping Communities

Abstract

Summary We propose a novel statistical model for sparse networks with overlapping community structure. The model is based on representing the graph as an exchangeable point process and naturally generalizes existing probabilistic models with overlapping block structure to the sparse regime. Our construction builds on vectors of completely random measures and has interpretable parameters, each node being assigned a vector representing its levels of affiliation to some latent communities. We develop methods for efficient simulation of this class of random graphs and for scalable posterior inference. We show that the approach proposed can recover interpretable structure of real world networks and can handle graphs with thousands of nodes and tens of thousands of edges.

Sparse networks with core-periphery structure

“…Having defined what we mean by core-periphery structure, we now give a construction of a core-periphery network. Building on the model of [41], we also allow the network to exhibit community structure. Following [6], we represent a graph by the point process on the plane…”

“…A finite graph G α of size α > 0 is obtained by considering the restriction of Z to [0, α] 2 , see [6]. As in [41], we consider that the probability of a connection between nodes i and j is given by the link function…”

“…and dense otherwise. [41] considered a specific class of models for ρ, based on compound completely random measures (CRMs) [17], where ρ is concentrated on (0, ∞) K ; that is, for all nodes i and communities k we have w ik > 0. As a consequence, the sparsity pattern is homogeneous across the graph.…”

“…Firstly, we provide a precise definition of what it means for a graph to have a core-periphery structure, based on the sparsity properties of the subgraphs of core and periphery nodes. Secondly, building on earlier work from [41], we present a class of sparse network models with such properties, and provide methods to simulate from this class, and to perform posterior inference. Finally, we demonstrate that our approach can detect meaningful core-periphery structure in two real-world airport and trade networks, while providing a good fit to global structural properties of the networks.…”

“…The statistical network modelling framework our work is set in, introduced by [6], is based on representing the network as an exchangeable random measure. Sparse graph models with (overlapping) block structure have already been proposed within this framework [19,41]. However, these models cannot be applied directly to model networks with a core-periphery structure, as they make the assumption that a single parameter tunes the overall sparsity properties of the graph, with the same structural sparsity properties across different blocks.…”