Quantifying Multivariate Graph Dependencies: Theory and Estimation for Multiplex Graphs

TL;DR

This work develops a comprehensive theory for multivariate information in multiplex graphs using graph limit (graphon) representations. It defines and analyzes graphon mutual information for pairs, multivariate measures such as graphon interaction information, graphon total correlation, dual total correlation, and O-information, including their conditional variants, within a unified d-variate graph limit framework. The authors provide consistent nonparametric estimators for these measures, establish convergence rates, and validate them through simulations that illustrate redundancy and synergy in multiplex structures. Real-data experiments on temporal networks demonstrate the practical utility of the estimators in uncovering shared information and higher-order dependencies, highlighting the method's potential for complex multiplex pattern analysis.

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.

Figures (9)

• Figure 1: Log-log plots of the decay of $\mathrm{RMSE}(\widehat{\mathrm{I}}_{\mathbf{W}_{1,2}}(W_1;W_2),\mathrm{I}_{\mathbf{W}_{1,2}}(W_1;W_2))$ as a function of $n \in \{2^6,2^7,\cdots,2^{11}\}$, averaged over $300$ trials.
• Figure 2: Log-log plots of the decay of $\mathrm{RMSE}(\widehat{\mathrm{TC}}_{\mathbf{W}_{1,2,3}}(W^{(1)},W^{(2)},W^{(3)}),$$\mathrm{TC}_{\mathbf{W}_{1,2,3}}(W^{(1)},W^{(2)},W^{(3)})$ as a function of $n \in \{2^6,2^7,\cdots,2^{11}\}$, averaged over $300$ trials.
• Figure 3: Log-log plots of the decay of $\mathrm{RMSE}(\widehat{\mathrm{I}}_{\mathbf{W}_{1,2,3}}(W^{(1)};W^{(2)};W^{(3)})$, $\mathrm{I}_{\mathbf{W}_{1,2,3}}(W^{(1)};W^{(2)};W^{(3)}))$ as a function of $n \in \{2^6,2^7,\cdots,2^{11}\}$, averaged over $300$ trials.
• Figure 4: Log-log plots of the decay of $\mathrm{RMSE}(\widehat{\mathrm{DTC}} _{\mathbf{W}_{1,2,3}}(W^{(1)},W^{(2)},W^{(3)}),$$\mathrm{DTC}_{\mathbf{W}_{1,2,3}}(W^{(1)},W^{(2)},W^{(3)})$ as a function of $n \in \{2^6,2^7,\cdots,2^{11}\}$, averaged over $300$ trials.
• Figure 5: A parent graph (left) alongside two percolated versions (middle and right) with $5\%$ and $10\%$ of edges removed, respectively. Edges colored orange represent those shared among the graphs, while blue unshared.

Theorems & Definitions (30)

• Definition 1: Graphon lovasz2006limits
• Definition 2: Joint exchangeability
• Theorem 1: Aldous--Hoover aldous1981representationshoover1979relations
• Example 1: Exchangeable graph
• Definition 3: Graphon entropy janson2013graphonshatami2018graph
• Definition 4
• Definition 5: Bivariate joint graphon entropy
• Corollary 1
• proof
• Definition 6: Graphon mutual information