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.