Multiplex Network Analysis

• Multiplex network analysis is a method to represent a fixed set of nodes connected by multiple layers of distinct interactions, capturing interlayer dependencies and structural nuances.
• The multiplexon framework employs a vector of symmetric measurable functions for each non-empty subset of layers, enabling rigorous convergence of dense multiplex networks to explicit limiting laws.
• Examples such as correlated Erdős–Rényi multiplexes and block multiplexons illustrate practical applications in modeling social, industrial, and biological networks.

A multiplex network is a formalism that represents a fixed set of nodes connected by multiple distinct types of interactions, each encoded as a separate network layer. With the emergence of multiplexes as a central paradigm in modeling complex systems—ranging from social and industrial to biological domains—there is a strong need for a rigorous and general framework to describe the limiting behavior of sequences of large, dense multiplex networks. This need parallels the role of graphon theory in the paper of single-layer graph limits. The central concept developed for this purpose is the "multiplexon" (Editor's term), which generalizes the notion of graphons to multiplex networks and provides a foundation for the asymptotic analysis of their structural features and homomorphism statistics.

1. Multiplexon Limit Theory: Definition and Foundations

For a multiplex network with $r$ layers ($G_{n,1}, \ldots, G_{n,r}$) on $n$ nodes, the classical approach would be to consider a vector of $r$ graphons. However, this approach neglects the combinatorial structure arising from edges being present in multiple layer subsets. The theory of multiplexons instead prescribes a vector $\left(W_S\right)_{S\subseteq [r], S\neq\emptyset}$ of $2^r-1$ symmetric measurable functions $W_S: [0,1]^2\to[0,1]$, with each $W_S$ representing the density of edges present in precisely the layer subset $S$.

This construction enables the encoding of all interlayer dependencies in edge configurations across the multiplex. The correct notion of convergence—left convergence—prescribes that, for any fixed $r$-layer multiplex $\bm{H}$, the homomorphism densities $t(\bm{H},\bm{G}_n)$ converge as $n\to\infty$ to the multiplexon homomorphism density $t(\bm{H},\overline{\bm{W}})$ defined by

$$t(\bm{H}, \overline{\bm{W}}) = \int_{[0,1]^{V(\bm{H})}} \prod_{S\ne\emptyset} \prod_{(i,j)\in E(\hat{H}_S)} \overline{W}_S(x_i,x_j) \, dx_1\cdots dx_{|V(\bm{H})|}$$

where $E(\hat{H}_S)$ is the edge set of the simple graph corresponding to those edges present in exactly the layer subset $S$ in $\bm{H}$.

A crucial aspect of the theory is the choice between disjoint and cumulative decompositions: in the disjoint decomposition each $S$-layer is assigned edges present in exactly the subset $S$, while the cumulative decomposition groups together edges present in all at least $S$. These are related by Möbius inversion and yield equivalent homomorphism densities.

2. Compactness, Convergence, and Metric Structure

An appropriate metric over the space of multiplexons is the sum of cut norms over all layer subsets: $$\|\overline{\bm{U}} - \overline{\bm{W}}\|_\square = \sum_{S \neq \varnothing} \| \overline{U}_S - \overline{W}_S \|_\square$$ where each $\overline{W}_S$ is considered up to measure-preserving bijections acting in synchrony across components. The result is that the space of multiplexons (modulo such measure-preserving bijections) is compact under this metric, paralleling the single-layer graphon case.

Equivalence is established between left convergence in homomorphism densities, convergence in the sum-of-cuts metric, and convergence of all induced subgraph statistics. Additionally, inhomogeneous random multiplex models sampled from a multiplexon converge (in cut norm and statistics) to the underlying multiplexon.

3. Limiting Analogues of Classical Multiplex Features

Structural features of multiplex networks admit natural limiting analogues in terms of the multiplexon. For example:

• For each layer subset $S\subseteq [r]$, the limiting normalized degree of node $x$ in "layer-set" $S$ is

$$d_{\overline{W}_S}(x) := \int_0^1 \overline{W}_S(x,y) \, dy$$

The joint degree distribution over all subsets $S$ for a random node converges to the law of $(d_{\overline{W}_S}(\eta))_{S}$ for $\eta\sim\mathit{Unif}[0,1]$.

• Multiplex clustering coefficients, including cross-layer triangle densities, have explicit limiting laws. For example, the normalized two-layer triangle clustering at node $i$ converges (in empirical distribution) to

$$\frac{ \sum_{s_1\ne s_2} \int_{[0,1]^2} \overline{W}_{\{s_1\}}(\eta, y) \overline{W}_{\{s_2\}}(y, z) \overline{W}_{\{s_1\}}(\eta, z) \, dy dz } { (r-1) \sum_{s_1} d_{\overline{W}_{\{s_1\}}}(\eta)^2 }$$

where $\eta\sim\mathrm{Unif}(0,1)$.

4. Key Examples: Correlated Multiplex Models

Illustrative examples concretize the power of the multiplexon theory:

• Correlated Erdős–Rényi multiplexes: For $r=2$ and edge probabilities $p_1, p_2, p_{12}$, the multiplexon consists of

$$\overline{W}_{\{1\}} = p_1 - p_{12}, \quad \overline{W}_{\{2\}} = p_2 - p_{12}, \quad \overline{W}_{\{1,2\}} = p_{12}$$

• Block multiplexons: Generalization of stochastic block models to $r$-layer settings, where the edge probabilities for any pair of node blocks and any subset of layers can be specified.
• Dynamic multiplexes: Where the $r$ layers are time snapshots, the limit object encodes correlations of edge presence across time.

These cases demonstrate that many classical random graph models, when generalized to the multiplex setting, correspond to explicit multiplexons whose analytic and statistical properties are tractable.

5. Context: Multiplex Networks within Decorated Graph Limit Theory

Multiplex networks form a special instance of the general decorated graph frameworks, notably those studied by Lovász and Szegedy. In these broader contexts, edges are "decorated" by values from a finite set (such as the power set of layers) and graphon limits map $[0,1]^2$ to probability distributions over decorations. While the decorated graphon theory is universal, working directly with multiplexons and the layer-set decomposition provides more transparent and practically computable formulas for the main statistics of interest in multiplex analysis.

6. Main Theoretical Results and Practical Implications

The development of multiplexon theory achieves several critical outcomes:

• The space of multiplexons is shown to be compact under the natural sum-of-cuts metric, establishing a solid foundation for limit theorems.
• Sequences of dense multiplex networks converge in (joint) subgraph statistics and degree distributions to explicit multiplexon-derived laws.
• Limit formulas for network statistics—degree, clustering, motif counts—are available and coincide with their graphon analogues when $r=1$.
• The framework validates the use of inhomogeneous random multiplexes as practical null models for large systems with specified multiplexon structure.

This suggests that the multiplexon paradigm provides a rigorous, tractable toolset for both descriptive and inferential statistics on large multiplex systems. Applications include model selection, estimation, and hypothesis testing for multiplex network data, as well as the analytic calculation of limiting properties in theoretical models.

7. Future Directions and Open Problems

The multiplexon framework lays the groundwork for further generalizations and research directions:

• Extension to sparse (non-dense) multiplex networks, where analogues of $L^p$-graphon theory may be required.
• Development of right-convergence and large deviation theories, connecting to statistical physics and large network fluctuations.
• Further exploration of multiplex dynamics, sampling, and functional inference in the limit.
• Connections to multilayer networks with general (not necessarily node-aligned) structure via decorated and probability graphons.

The presented theory is thus a key development in the mathematical analysis of multiplex networks, parallel to the role of graphons for simple and decorated networks, with broad consequences for both theoretical research and real-world data analysis in complex systems (Ganguly et al., 8 Oct 2025).

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Ordinary Graphs (graphon theory)	Multiplex Networks (multiplexon theory)
$W: [0,1]^2 \to [0,1]$	$(W_S)_{S \subset [r], S\neq\emptyset}$
Edge probability per pair	Probability of edge in each subset of layers
Subgraph homomorphism density	Multiplex homomorphism density
Cut norm metric	Sum of cut norms over all subsets
Limiting degree distribution	Limiting joint distribution of $d_{W_S}$ for all $S$
Limiting clustering, motif densities	Limiting multiplex motif densities