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Multiplexons: Limits of Multiplex Networks

Ankan Ganguly, Bhaswar B. Bhattacharya

TL;DR

This work develops a comprehensive limit theory for sequences of dense multiplex networks by introducing multiplexons as the natural limit objects, analogous to graphons for single-layer graphs. It proves that left-convergence is equivalent to convergence in the multiplex cut distance, and provides a full framework (including two decomposition schemes, compactness, and sampling) to analyze limits of multiplex features such as degree distributions and clustering. By embedding multiplexons within the broader decorated/probability graphon framework, the paper unifies dense multiplex limit theory with existing graphon models and yields concrete limiting expressions for key network statistics. The results enable principled analysis of large multiplex data and pave the way for future work on right convergence, large deviations, and sparse or dynamic multiplex regimes.

Abstract

In a multiplex network, a set of nodes is connected by different types of interactions, each represented as a separate layer within the network. Multiplexes have emerged as a key instrument for modeling large-scale complex systems, due to the widespread coexistence of diverse interactions in social, industrial, and biological domains. This motivates the development of a rigorous and readily applicable framework for studying properties of large multiplex networks. In this article, we provide a self-contained introduction to the limit theory of dense multiplex networks, analogous to the theory of graphons (limit theory of dense graphs). As applications, we derive limiting analogues of commonly used multiplex features, such as degree distributions and clustering coefficients. We also present a range of illustrative examples, including correlated versions of Erdős-Rényi and inhomogeneous random graph models and dynamic networks. Finally, we discuss how multiplex networks fit within the broader framework of decorated graphs, and how the convergence results can be recovered from the limit theory of decorated graphs. Several future directions are outlined for further developing the multiplex limit theory.

Multiplexons: Limits of Multiplex Networks

TL;DR

This work develops a comprehensive limit theory for sequences of dense multiplex networks by introducing multiplexons as the natural limit objects, analogous to graphons for single-layer graphs. It proves that left-convergence is equivalent to convergence in the multiplex cut distance, and provides a full framework (including two decomposition schemes, compactness, and sampling) to analyze limits of multiplex features such as degree distributions and clustering. By embedding multiplexons within the broader decorated/probability graphon framework, the paper unifies dense multiplex limit theory with existing graphon models and yields concrete limiting expressions for key network statistics. The results enable principled analysis of large multiplex data and pave the way for future work on right convergence, large deviations, and sparse or dynamic multiplex regimes.

Abstract

In a multiplex network, a set of nodes is connected by different types of interactions, each represented as a separate layer within the network. Multiplexes have emerged as a key instrument for modeling large-scale complex systems, due to the widespread coexistence of diverse interactions in social, industrial, and biological domains. This motivates the development of a rigorous and readily applicable framework for studying properties of large multiplex networks. In this article, we provide a self-contained introduction to the limit theory of dense multiplex networks, analogous to the theory of graphons (limit theory of dense graphs). As applications, we derive limiting analogues of commonly used multiplex features, such as degree distributions and clustering coefficients. We also present a range of illustrative examples, including correlated versions of Erdős-Rényi and inhomogeneous random graph models and dynamic networks. Finally, we discuss how multiplex networks fit within the broader framework of decorated graphs, and how the convergence results can be recovered from the limit theory of decorated graphs. Several future directions are outlined for further developing the multiplex limit theory.

Paper Structure

This paper contains 21 sections, 13 theorems, 147 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Fundamentals of Graph Limit Theory
  3. Multiplex Networks and Multiplexons
  4. Homomorphism Densities
  5. Cut Distance
  6. Left-Convergence
  7. Decomposition of Multiplexes and Multiplexons
  8. The Space of Cumulatively Decomposable Multiplexons
  9. Sampling Random Multiplexes
  10. Counting Lemma for Multiplexons
  11. Equivalence of Convergence
  12. Degree Distribution and Clustering Coefficients
  13. Examples of Convergent Multiplexes
  14. Connections to Decorated/Probability Graphons
  15. Cut Distance for Probability Graphons
  16. ...and 6 more sections

Key Result

Proposition 3.13

For any two $r$-multiplexes $\bm{H} = (V(\bm H), H_1, H_2, \ldots, H_r)$ and $\bm{G} = (V(\bm G),$$G_1,G_2, \ldots, G_r)$, where $\hat{\bm{H}}$ is the disjoint decomposition of $\bm{H}$, $\widebar{\bm{G}}$ is the cumulative decomposition of $\bm{G}$, and $\widebar{\bm{W}}^{\bm G}$ is the empirical multiplexon of $\bm{G}$.

Figures (3)

  • Figure 2: The disjoint decomposition of the 2-layered multiplex $\bm{H} = (H_1, H_2)$.
  • Figure 3: The cumulative decomposition of the 2-layered multiplex $\bm{H}=(H_1, H_2)$.
  • Figure 4: The empirical multiplexon corresponding to the multiplex $\bm H$ in \ref{['eq:H123']}.

Theorems & Definitions (48)

  • Definition 1.1
  • Definition 2.1
  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Definition 3.4
  • Example 3.5: Correlated random graphs
  • Definition 3.6
  • Definition 3.7
  • Remark 3.8: Multilinks
  • ...and 38 more