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Dependencies in Multiplex Networks: A Motif Count Approach

Karl Sawaya, Sofia Olhede

Abstract

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.

Dependencies in Multiplex Networks: A Motif Count Approach

Abstract

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 graphons defining a -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.

Paper Structure

This paper contains 32 sections, 18 theorems, 272 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Basic Notation
  4. Multivariate Graphon Framework
  5. Graph Limit Theory
  6. Subgraph Counts in a Multiplex Network
  7. Vector-Motif Counts in a Multiplex
  8. Cross-layer Motif Counts in a Multiplex
  9. Multivariate Graphon Estimation
  10. Case $d =1$
  11. Case $d=2$
  12. Case $d>2$
  13. Asymptotic Joint Fluctuation of Multiplex Moments
  14. Asymptotic Distribution of Multiplex Erdős–Rényi Moments
  15. Asymptotic Distribution of Exchangeable Multiplex Moments
  16. ...and 17 more sections

Key Result

Theorem 2.1

Let $A = (A_{ij})$ be a jointly exchangeable random array. Then there exists an i.i.d. sequence $\xi = (\xi_1, \dots, \xi_n)$ with $\xi_i \sim \mathbf{U}[0,1]$, a random variable $\gamma \sim \mathbf{U}[0,1]$ independent of $\xi$, and a measurable function $W: [0,1]^3 \to [0,1]$ such that If $A$ is the adjacency matrix of a random graph $G_n$ and $G_n$ is disassociated, then the dependence on $\g

Figures (3)

  • Figure 1: A $2$-layer multiplex network with $11$ vertices. Edges of layer 1 are colored in red, edges of layer 2 colored in blue and edges present in both layers are colored in green.
  • Figure 2: Examples of bichromatic alternate motifs.
  • Figure :

Theorems & Definitions (62)

  • Definition 2.1: Multiplex Network Ganguly2025
  • Remark 2.1: Multiplex network as a decorated graph
  • Definition 2.2: Multiplex Erdős--Rényi network
  • Definition 2.3: Joint exchangeability DiaconisJanson2008
  • Theorem 2.1: Aldous-Hoover Aldous1981
  • Definition 2.4: Exchangeable graph SkejaOlhede2024
  • Remark 2.2
  • Theorem 2.2: Multivariate Aldous-Hoover SkejaOlhede2024
  • Remark 2.3
  • Definition 2.5: Multiplex Stochastic Block Model
  • ...and 52 more