Grand canonical ensembles of sparse networks and Bayesian inference

TL;DR

This paper introduces a grand canonical Bayesian framework for modeling sparse exchangeable networks, enabling network size estimation and reconstruction, overcoming limitations of fixed-size models.

Contribution

It develops hierarchical, exchangeable network models in the sparse limit that allow for Bayesian inference of network size and structure, addressing previous fixed-node limitations.

Findings

Enables Bayesian estimation of network size from subgraphs

Handles networks with specified degree or latent variable distributions

Circumvents issues with infinite sparse exchangeable networks

Abstract

Maximum entropy network ensembles have been very successful in modelling sparse network topologies and in solving challenging inference problems. However the sparse maximum entropy network models proposed so far have fixed number of nodes and are typically not exchangeable. Here we consider hierarchical models for exchangeable networks in the sparse limit, i.e. with the total number of links scaling linearly with the total number of nodes. The approach is grand canonical, i.e. the number of nodes of the network is not fixed a priori: it is finite but can be arbitrarily large. In this way the grand canonical network ensembles circumvent the difficulties in treating infinite sparse exchangeable networks which according to the Aldous-Hoover theorem must vanish. The approach can treat networks with given degree distribution or networks with given distribution of latent variables. When only a subgraph induced by a subset of nodes is known, this model allows a Bayesian estimation of the network size and the degree sequence (or the sequence of latent variables) of the entire network which can be used for network reconstruction.