The Spectral Barycentre of a Set of Graphs with Community Structure

TL;DR

This paper introduces a spectral method for computing a barycentre graph that summarizes a set of graphs with community structure, ensuring the preservation of topological features and demonstrating convergence properties.

Contribution

It proposes a spectral barycentre construction using eigenvalues and eigenvectors with structural constraints, applicable to graphs with community structure, and proves convergence in stochastic block models.

Findings

Eigenvalues of the barycentre can be derived from an optimization problem.

The eigenvectors are computed via Soules bases, preserving topological structure.

The method converges almost-surely to the population mean in large stochastic block models.

Abstract

The notion of barycentre graph is of crucial importance for machine learning algorithms that process graph-valued data. The barycentre graph is a "summary graph" that captures the mean topology and connectivity structure of a training dataset of graphs. The construction of a barycentre requires the definition of a metric to quantify distances between pairs of graphs. In this work, we use a multiscale spectral distance that is defined using the eigenvalues of the normalized graph Laplacian. The eigenvalues -- but not the eigenvectors -- of the normalized Laplacian of the barycentre graph can be determined from the optimization problem that defines the barycentre. In this work, we propose a structural constraint on the eigenvectors of the normalized graph Laplacian of the barycentre graph that guarantees that the barycentre inherits the topological structure of the graphs in the sample dataset. The eigenvectors can be computed using an algorithm that explores the large library of Soules bases. When the graphs are random realizations of a balanced stochastic block model, then our algorithm returns a barycentre that converges asymptotically (in the limit of large graph size) almost-surely to the population mean of the graphs. We perform Monte Carlo simulations to validate the theoretical properties of the estimator; we conduct experiments on real-life graphs that suggest that our approach works beyond the controlled environment of stochastic block models.