A Spike-and-Slab Prior for Dimension Selection in Generalized Linear Network Eigenmodels

TL;DR

This paper introduces a Bayesian spike-and-slab prior for selecting the latent dimension in generalized linear network eigenmodels, improving interpretability and consistency across various network types.

Contribution

It formalizes GLNEMs for diverse edge types and proposes a novel Bayesian dimension selection method with theoretical guarantees.

Findings

Posterior concentrates on low-dimensional models near the true dimension.

Consistent dimension selection demonstrated on simulated networks.

Application to real-world biological, ecological, and economic networks.

Abstract

Latent space models (LSMs) are frequently used to model network data by embedding a network's nodes into a low-dimensional latent space; however, choosing the dimension of this space remains a challenge. To this end, we begin by formalizing a class of LSMs we call generalized linear network eigenmodels (GLNEMs) that can model various edge types (binary, ordinal, non-negative continuous) found in scientific applications. This model class subsumes the traditional eigenmodel by embedding it in a generalized linear model with an exponential dispersion family random component and fixes identifiability issues that hindered interpretability. Next, we propose a Bayesian approach to dimension selection for GLNEMs based on an ordered spike-and-slab prior that provides improved dimension estimation and satisfies several appealing theoretical properties. In particular, we show that the model's posterior concentrates on low-dimensional models near the truth. We demonstrate our approach's consistent dimension selection on simulated networks. Lastly, we use GLNEMs to study the effect of covariates on the formation of networks from biology, ecology, and economics and the existence of residual latent structure.