Dirichlet Meets Horvitz and Thompson: Estimating Homophily in Large Networks via Sampling

TL;DR

This paper introduces a sampling-based method using the Dirichlet energy and Horvitz-Thompson estimator to accurately estimate homophily in large networks from partial observations, addressing practical constraints.

Contribution

It proposes a novel unbiased estimation framework for network homophily using Dirichlet energy and HT estimator, applicable to various sampling designs.

Findings

Effective in estimating homophily from sampled data

Reliable in capturing heterophilic structures

Applicable to large, resource-constrained networks

Abstract

Assessing homophily in large-scale networks is central to understanding structural regularities in graphs, and thus inform the choice of models (such as graph neural networks) adopted to learn from network data. Evaluation of smoothness metrics requires access to the entire network topology and node features, which may be impractical in several large-scale, dynamic, resource-limited, or privacy-constrained settings. In this work, we propose a sampling-based framework to estimate homophily via the Dirichlet energy (Laplacian-based total variation) of graph signals, leveraging the Horvitz-Thompson (HT) estimator for unbiased inference from partial graph observations. The Dirichlet energy is a so-termed total (of squared nodal feature deviations) over graph edges; hence, estimable under general network sampling designs for which edge-inclusion probabilities can be analytically derived and used as weights in the proposed HT estimator. We establish that the Dirichlet energy can be consistently estimated from sampled graphs, and empirically study other heterophily measures as well. Experiments on several heterophilic benchmark datasets demonstrate the effectiveness of the proposed HT estimators in reliably capturing homophilic structure (or lack thereof) from sampled network measurements.