Modeling Network Populations via Graph Distances

Simón Lunagómez,Sofia C Olhede,Patrick J Wolfe

doi:10.1080/01621459.2020.1763803

Abstract

This article introduces a new class of models for multiple networks. The core idea is to parameterize a distribution on labeled graphs in terms of a Fréchet mean graph (which depends on a user-specified choice of metric or graph distance) and a parameter that controls the concentration of this distribution about its mean. Entropy is the natural parameter for such control, varying from a point mass concentrated on the Fréchet mean itself to a uniform distribution over all graphs on a given vertex set. We provide a hierarchical Bayesian approach for exploiting this construction, along with straightforward strategies for sampling from the resultant posterior distribution. We conclude by demonstrating the efficacy of our approach via simulation studies and two multiple-network data analysis examples: one drawn from systems biology and the other from neuroscience. This article has online supplementary materials.

Highlights

This article introduces a new class of models for data consisting of observations of multiple networks
Inferences must often be combined on the same gene interaction network, where different inferences correspond to different data sets or to different analysis procedures applied to the t same data (Bartlett et al, 2014)
Distinct from the literature discussed above, the methodology we propose here achieves different goals: (1) It enables the modeller to characterise the variability of a set of observed networks in terms of a Fréchet mean and a measure of how concentrated the distribution is around this mean, and to perform Bayesian inference, without resorting to asymptotics; (2) It enables the practitioner interested in network data to perform prior elicitation on graph space by using an observed network as starting point; and (3) It provides tools for incorporating different metrics on graph space into the modelling procedure, enabling the encoding of different assumptions the practitioner may have regarding similarity among graphs

Summary

Introduction

This article introduces a new class of models for data consisting of observations of multiple networks. We will introduce a first example of a random graph distribution based on distance and entropy; we call it the Centred Erdös–Rényi Model. Ted As a second example of a unimodal network distribution based on location and scale, we introduce a model motivated by the notion that the similarity with p respect to the centroid is made concrete by the choice of d (·,·) G (e.g. the metrics e proposed by Zelinka (1975), Hammond et al (2013), or the ones discussed in c Donnat and Holmes (2018)), and covered by our discussion in Section 2 earlier. A main difference with respect to their approach is that the Spherical Network Family is aimed to serve as the functional form for both the likelihood and the prior This model relates to the similarity measure proposed by Dahl et al (2017) for random partitions. It is straightforward to set up a Metropolis/ Hastings algorithm to sample (·)

The argument that maximises the log of the function:

Findings

Investigate how the probability