Quantifying Multivariate Graph Dependencies: Theory and Estimation for Multiplex Graphs

TL;DR

This paper develops a comprehensive theoretical framework and estimators for measuring complex interdependencies and information sharing in multiplex graphs, advancing understanding of multivariate graph relationships.

Contribution

It introduces novel graphon-based information measures for multiplex graphs and provides consistent nonparametric estimators with convergence analysis.

Findings

Theoretical development of multivariate information measures for multiplex graphs.

Introduction of consistent estimators with proven convergence rates.

Empirical validation through simulations and real-world data applications.

Abstract

Multiplex graphs, characterised by their layered structure, exhibit informative interdependencies within layers that are crucial for understanding complex network dynamics. Quantifying the interaction and shared information among these layers is challenging due to the non-Euclidean structure of graphs. Our paper introduces a comprehensive theory of multivariate information measures for multiplex graphs. We introduce graphon mutual information for pairs of graphs and expand this to graphon interaction information for three or more graphs, including their conditional variants. We then define graphon total correlation and graphon dual total correlation, along with their conditional forms, and introduce graphon $O-$information. We discuss and quantify the concepts of synergy and redundancy in graphs for the first time, introduce consistent nonparametric estimators for these multivariate graphon information--theoretic measures, and provide their convergence rates. We also conduct a simulation study to illustrate our theoretical findings and demonstrate the relationship between the introduced measures, multiplex graph structure, and higher--order interdependecies. Real-world applications further show the utility of our estimators in revealing shared information and dependence structures in real-world multiplex graphs. This work not only answers fundamental questions about information sharing across multiple graphs but also sets the stage for advanced pattern analysis in complex networks.