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

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.

💡 Research Summary

The paper tackles the problem of measuring homophily (or smoothness) in large graphs when the full topology and node attributes are unavailable—a common situation in massive, dynamic, privacy‑sensitive, or resource‑constrained environments. The authors propose a sampling‑based framework that estimates the Dirichlet energy (also known as Laplacian‑based total variation) of a graph signal, which is a widely used quantitative proxy for homophily. The Dirichlet energy is defined as
 TV_G(X) = trace(XᵀLX) = Σ_{(i,j)∈E} A_{ij}‖x_i – x_j‖²,
where X contains node features, L is the combinatorial Laplacian, and A is the adjacency matrix.

Theoretical Foundations
The authors first show that the Dirichlet energy is a “testable” graph parameter in the sense of graph limit theory. By mapping a finite graph–signal pair (G, X) to a step‑graphon W_G and a step‑signal X_G, they define a continuous functional on the space of graphons:
 Φ(W, X) = ∬_{