An Optimal Transport Approach for Network Regression

TL;DR

This paper introduces a novel network regression method leveraging Wasserstein metrics and Fréchet means to better model how network topology varies with covariates, showing improved accuracy in synthetic and real data.

Contribution

It proposes a new Wasserstein-based network regression approach using Fréchet means for graph representation, with efficient computation and demonstrated superior performance.

Findings

Improves prediction accuracy over existing methods.

Effectively accounts for graph size, topology, and sparsity.

Achieves higher R² and lower MSPE in experiments.

Abstract

We study the problem of network regression, where one is interested in how the topology of a network changes as a function of Euclidean covariates. We build upon recent developments in generalized regression models on metric spaces based on Fr\'echet means and propose a network regression method using the Wasserstein metric. We show that when representing graphs as multivariate Gaussian distributions, the network regression problem requires the computation of a Riemannian center of mass (i.e., Fr\'echet means). Fr\'echet means with non-negative weights translates into a barycenter problem and can be efficiently computed using fixed point iterations. Although the convergence guarantees of fixed-point iterations for the computation of Wasserstein affine averages remain an open problem, we provide evidence of convergence in a large number of synthetic and real-data scenarios. Extensive numerical results show that the proposed approach improves existing procedures by accurately accounting for graph size, topology, and sparsity in synthetic experiments. Additionally, real-world experiments using the proposed approach result in higher Coefficient of Determination ($R^{2}$) values and lower mean squared prediction error (MSPE), cementing improved prediction capabilities in practice.