Bayesian Testing for Exogenous Partition Structures in Stochastic Block Models

TL;DR

This paper introduces a Bayesian testing method to evaluate whether an external node partition aligns with the endogenous clustering structure in stochastic block models, using Bayes factors and MCMC techniques.

Contribution

It provides a formal Bayesian framework and computational approach to test the consistency of exogenous partitions with network clustering patterns.

Findings

Effective in simulations for detecting alignment between external partitions and endogenous clusters.

Applied to brain networks of Alzheimer's patients, revealing insights into exogenous equivalence structures.

The method quantifies uncertainty in endogenous groupings through Bayes factors.

Abstract

Network data often exhibit block structures characterized by clusters of nodes with similar patterns of edge formation. When such relational data are complemented by additional information on exogenous node partitions, these sources of knowledge are typically included in the model to supervise the cluster assignment mechanism or to improve inference on edge probabilities. Although these solutions are routinely implemented, there is a lack of formal approaches to test if a given external node partition is in line with the endogenous clustering structure encoding stochastic equivalence patterns among the nodes in the network. To fill this gap, we develop a formal Bayesian testing procedure which relies on the calculation of the Bayes factor between a stochastic block model with known grouping structure defined by the exogenous node partition and an infinite relational model that allows the endogenous clustering configurations to be unknown, random and fully revealed by the block-connectivity patterns in the network. A simple Markov chain Monte Carlo method for computing the Bayes factor and quantifying uncertainty in the endogenous groups is proposed. This routine is evaluated in simulations and in an application to study exogenous equivalence structures in brain networks of Alzheimer's patients.