Mesoscale two-sample testing for networks

Networks arise naturally in many scientific fields as a representation of pairwise connections. Statistical network analysis has most often considered a single large network, but it is common in a number of applications to observe multiple networks on a shared node set. When these networks are grouped by case-control status or another categorical covariate, the classical statistical question of two-sample comparison arises. In this work, we address the problem of testing for statistically significant differences in a given arbitrary subset of connections. This general framework allows an analyst to focus on a single node, a specific region of interest, or compare whole networks. Our ability to conduct ``mesoscale’’ testing on a meaningful group of edges is particularly relevant for applications such as neuroimaging and distinguishes our approach from prior work, which tends to focus either on a single node or the whole network. In this mesoscale setting, we develop statistically sound projection-based tests for two-sample comparison in both weighted and binary edge networks. The key to our approach is to leverage network information from outside the set of interest to learn informative low-rank projections which leads to more powerful tests.

💡 Research Summary

The paper addresses the problem of two‑sample testing for multiple networks that share a common set of nodes, focusing on a user‑specified subset of edges rather than the entire graph (global testing) or a single edge (local testing). This intermediate scale, termed “mesoscale,” is motivated by applications such as neuroimaging where researchers are interested in comparing connectivity within or between anatomically defined brain regions.

The authors propose a flexible statistical framework based on independent‑edge exponential‑family models. Each edge, whether binary or weighted, follows a distribution with a known link function (logistic for binary, identity for Gaussian) and possibly a dispersion parameter. The mean of each edge is expressed as h(Θ), where Θ denotes the natural‑parameter matrix and h is the inverse link. The null hypothesis for a mesoscale set S ⊂ {1,…,n}² is simply Θ⁽¹⁾_S = Θ⁽²⁾_S, i.e., the expected adjacency entries are equal across the two samples for all edges in S.

A key methodological contribution is the use of low‑rank projections learned from the complement set –S. Because many modern network models (stochastic block model, random dot product graph, and other latent‑space models) imply that the full expectation matrix is approximately low‑rank, the authors estimate a low‑dimensional subspace from the edges outside S. They then project the observed edges in S onto this subspace, forming a linear statistic T = (1/√|S|)·1ᵀ P̂ y, where P̂ is the estimated projection matrix and y is the vector of edges in S. Under the null, T is asymptotically standard normal; under the alternative, its mean shifts proportionally to the true difference between Θ⁽¹⁾_S and Θ⁽²⁾_S.

The paper provides two main theoretical results. The first guarantees size control for any exponential‑family edge model with known dispersion: the test’s Type I error does not exceed a pre‑specified α in the large‑sample limit. The second analyzes power, showing that when the low‑rank structure is correctly captured, the mesoscale test dominates traditional local tests (which suffer from severe multiple‑testing penalties) and can approach the power of a global test while retaining specificity to the region of interest. Robustness to model misspecification is also discussed; because the projection is learned from data rather than imposed by a parametric model, the procedure tolerates errors in latent‑dimension choice or link‑function misspecification.

Empirical evaluations include extensive simulations under stochastic block models, random dot product graphs, and mixed‑membership models. The authors vary the size and shape of S (single edges, rows, columns, rectangular blocks) and the signal strength. Across all settings, the projection‑based mesoscale test achieves higher area‑under‑the‑ROC curve than competing global tests based on Frobenius or operator norms, and higher power than local multiple‑testing procedures, especially when the signal is weak and the sample size modest.

A real‑data application compares functional connectivity networks derived from resting‑state fMRI scans of Parkinson’s disease patients and healthy controls. Using a standard brain atlas, the authors define several rectangular blocks corresponding to known functional systems (e.g., fronto‑striatal, motor‑cerebellar connections). The mesoscale test identifies significant differences in these blocks, consistent with prior clinical findings, while global tests either reject the null indiscriminately or lack power to pinpoint specific systems.

Limitations noted include the need for a data‑driven method to select the latent dimension, extension to undirected or symmetric graphs (the current theory assumes directed edges), and handling of heterogeneous dispersion across edges. Future work may incorporate adaptive rank selection, non‑Gaussian edge models with unknown dispersion, and dynamic or multiplex network extensions.

In summary, the paper introduces a novel mesoscale two‑sample testing framework that leverages low‑rank structure via projection‑based statistics. It offers rigorous size control, favorable power properties, and practical applicability to scientific domains where researchers need to test hypotheses about meaningful sub‑networks rather than the whole graph.