Dependencies in Multiplex Networks: A Motif Count Approach

Multiplex networks are a powerful framework for representing systems with multiple types of interactions among a common set of entities. Understanding their structure requires statistical tools capturing higher-order cross-layer correlations. We develop a comprehensive framework for estimating and testing dependence in exchangeable multiplex networks through motif counts. We first propose a moment-based estimation methodology that extends the multi-layer stochastic block model network histogram to arbitrary motif counts. This allows us to estimate the $2^d-1$ graphons defining a $d$-layer multiplex network. We then derive the joint asymptotic distribution of cross-layer motif counts, that is aligned motifs shared across layers. Extending existing results from the unilayer setting, we show that the limiting distribution in the motif-regular case exhibits a covariance structure involving minimum-based distances between graphons. Finally, we construct hypothesis tests to detect inter-layer similarity and dependence. This work provides a rigorous extension of motif-count asymptotics and inference procedures to the multiplex setting, providing new tools to study high-order dependencies in complex networks.

💡 Research Summary

This paper develops a comprehensive statistical framework for analyzing dependence across layers in dense, exchangeable multiplex networks by leveraging motif (subgraph) counts. The authors begin by formalizing a multiplex network as a collection of d simple graphs sharing a common vertex set, and model the edge indicators for each unordered vertex pair as a d‑dimensional Bernoulli vector following a Multivariate Bernoulli distribution with 2^d − 1 parameters. These parameters capture both within‑layer edge probabilities and all possible cross‑layer joint occurrences, and they constitute a multivariate graphon when the network is exchangeable.

To estimate this multivariate graphon, the paper extends the stochastic block model to a Multiplex Stochastic Block Model (MSBM). In the MSBM each vertex belongs to one of K blocks, and for each block pair (a,b) a 2^d − 1 dimensional parameter vector θ_{ab} governs the joint edge probabilities across layers. The authors propose a moment‑based estimation scheme that uses arbitrary cross‑layer motif counts—subgraphs that are simultaneously present in a specified subset of layers. By writing the expected counts of these motifs as linear functions of the θ‑parameters, they obtain a system of equations that can be solved by generalized least squares, yielding consistent estimates of the underlying graphon.

The core theoretical contribution is the derivation of the joint asymptotic distribution of the vector of cross‑layer motif counts. First, for the multiplex Erdős–Rényi model (constant graphon), they show that the appropriately normalized counts converge to a multivariate Gaussian distribution whose covariance matrix is expressed in terms of minimum‑based distances between the underlying probability parameters. Next, they extend this result to general exchangeable multiplex networks under an F‑regularity condition (the analogue of regularity for a single‑layer graphon). When a graphon is F‑regular, each motif count decomposes into a Gaussian component plus an independent non‑Gaussian term that is an infinite weighted sum of centered chi‑squared variables. If the graphon is F‑irregular, only the Gaussian component remains. Consequently, the limiting distribution may be a mixture of Gaussian and non‑Gaussian parts, with the Gaussian covariance again governed by minimum‑based distances between the multivariate graphon components.

Building on these asymptotics, the authors construct two hypothesis‑testing procedures. The first test assesses inter‑layer structural similarity by comparing the expected densities of a collection of motifs across layers; a Wald statistic based on the estimated covariance matrix follows a chi‑square distribution under the null of equality. The second test evaluates block‑by‑block edge‑wise independence between layers in an MSBM, essentially testing whether the off‑diagonal elements of the θ‑vectors factorize across layers. Both tests are shown to have asymptotically correct size and good power in simulations.

Empirical evaluations on synthetic data explore various numbers of layers, blocks, and network sizes, confirming that motif‑based MSBM estimation dramatically reduces mean‑squared error relative to edge‑count‑only methods and that the proposed tests reliably detect both similarity and dependence. A real‑world case study on multimodal brain connectivity (e.g., fMRI and DTI) demonstrates that the framework can uncover meaningful cross‑modal structural relationships consistent with neuroscience literature.

In summary, the paper makes four major contributions: (1) a rigorous definition of multivariate graphons for multiplex networks; (2) a moment‑based estimator using arbitrary cross‑layer motif counts; (3) a detailed joint asymptotic theory for these counts, including regular and irregular regimes; and (4) practical hypothesis‑testing tools for inter‑layer similarity and block‑wise independence. These results provide a powerful new toolkit for researchers studying high‑order dependencies in complex systems ranging from social networks to biological connectomes.