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Thresholds and Fluctuations of Submultiplexes in Random Multiplex Networks

Bhaswar B. Bhattacharya, Sanchayan Bhowal, Karambir Das, Laura Eslava, Shaibal Karmakar

TL;DR

This work develops an asymptotic theory for small submultiplexes in a two-layer correlated Erdős-Rényi multiplex. It derives a precise threshold for the appearance of a fixed submultiplex $\bm H$ via a minimization over its submultiplexes, yielding a polyhedral satisfiability region in $\mathbb{R}^3$ and establishing regimes where infinitely many copies of $\bm H$ exist. Inside the satisfiability region, counts are asymptotically normal with an explicit Wasserstein convergence rate; on the threshold boundary, the limiting behavior is governed by whether $\bm H$ is balanced or unbalanced, with Poisson limits in strictly balanced cases and reductions to a balanced core in the unbalanced case. Overall, the paper extends classical small-subgraph theory to correlated multiplex networks, providing a rigorous asymptotic framework and guiding directions for further multiplex generalizations and boundary phenomena.

Abstract

In a multiplex network a common set of nodes is connected through different types of interactions, each represented as a separate graph (layer) within the network. In this paper, we study the asymptotic properties of submultiplexes, the counterparts of subgraphs (motifs) in single-layer networks, in the correlated Erdős-Rényi multiplex model. This is a random multiplex model with two layers, where the graphs in each layer marginally follow the classical (single-layer) Erdős-Rényi model, while the edges across layers are correlated. We derive the precise threshold condition for the emergence of a fixed submultiplex $\boldsymbol{H}$ in a random multiplex sampled from the correlated Erdős-Rényi model. Specifically, we show that the satisfiability region, the regime where the random multiplex contains infinitely many copies of $\boldsymbol{H}$, forms a polyhedral subset of $\mathbb{R}^3$. Furthermore, within this region the count of $\boldsymbol{H}$ is asymptotically normal, with an explicit convergence rate in the Wasserstein distance. We also establish various Poisson approximation results for the count of $\boldsymbol{H}$ on the boundary of the threshold, which depends on a notion of balance of submultiplexes. Collectively, these results provide an asymptotic theory for small submultiplexes in the correlated multiplex model, analogous to the classical theory of small subgraphs in random graphs.

Thresholds and Fluctuations of Submultiplexes in Random Multiplex Networks

TL;DR

This work develops an asymptotic theory for small submultiplexes in a two-layer correlated Erdős-Rényi multiplex. It derives a precise threshold for the appearance of a fixed submultiplex via a minimization over its submultiplexes, yielding a polyhedral satisfiability region in and establishing regimes where infinitely many copies of exist. Inside the satisfiability region, counts are asymptotically normal with an explicit Wasserstein convergence rate; on the threshold boundary, the limiting behavior is governed by whether is balanced or unbalanced, with Poisson limits in strictly balanced cases and reductions to a balanced core in the unbalanced case. Overall, the paper extends classical small-subgraph theory to correlated multiplex networks, providing a rigorous asymptotic framework and guiding directions for further multiplex generalizations and boundary phenomena.

Abstract

In a multiplex network a common set of nodes is connected through different types of interactions, each represented as a separate graph (layer) within the network. In this paper, we study the asymptotic properties of submultiplexes, the counterparts of subgraphs (motifs) in single-layer networks, in the correlated Erdős-Rényi multiplex model. This is a random multiplex model with two layers, where the graphs in each layer marginally follow the classical (single-layer) Erdős-Rényi model, while the edges across layers are correlated. We derive the precise threshold condition for the emergence of a fixed submultiplex in a random multiplex sampled from the correlated Erdős-Rényi model. Specifically, we show that the satisfiability region, the regime where the random multiplex contains infinitely many copies of , forms a polyhedral subset of . Furthermore, within this region the count of is asymptotically normal, with an explicit convergence rate in the Wasserstein distance. We also establish various Poisson approximation results for the count of on the boundary of the threshold, which depends on a notion of balance of submultiplexes. Collectively, these results provide an asymptotic theory for small submultiplexes in the correlated multiplex model, analogous to the classical theory of small subgraphs in random graphs.

Paper Structure

This paper contains 14 sections, 13 theorems, 91 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Statements of Results
  3. Thresholds for Submultiplex Containment
  4. Asymptotic Normality in the Satisfiability Region
  5. Distribution on the Boundary Surface
  6. Proof of Theorem \ref{['thm:threshold']}
  7. Proof of Theorem \ref{['thm:ZHGn']}
  8. Proofs from Section \ref{['sec:distributionH']}
  9. Proof of Theorem \ref{['thm:balancedH']}
  10. Proof of Theorem \ref{['thm:unbalancedH']}
  11. Proof of \ref{['eq:doublepresent']}
  12. Proof of \ref{['eq:modular']}
  13. Completing the Proof of Theorem \ref{['thm:unbalancedH']}
  14. Discussions and Future Directions

Key Result

Theorem 2.2

Suppose $\bm H = (V(\bm{H}), H^{(1)}, H^{(2)})$ is a fixed multiplex with at least one edge (that is, $|E(\bm{H})| \geq 1$). Then for $X(\bm{H}, \bm G_n)$ as defined above, the following holds: where

Figures (3)

  • Figure 1: (a) A sample $\bm{G}_n$ from the correlated Erdős-Rényi multiplex with $n=15$ and $p_1=p_2=0.15$, $p_{12}=0.05$. The edges in layers 1 are colored in blue and the edges in layer 2 are colored in red. (b) A fixed multiplex $\bm{H}$ on three vertices. $X(\bm H, \bm G_n)$ counts the number of injective multiplex homomorphism densities from $\bm H$ to $\bm G_n$ (see Definition \ref{['defintion:multiplexcount']}). For example, the function $\phi: V(\bm H) \rightarrow V(\bm G_n)$ defined as $\phi(1) = 7, \phi(2)=6, \phi(3)=9$ is an injective multiplex homomorphism.
  • Figure 2: (a) The edge-triangle multiplex $\bm{\mathcal{R}}$ defined in \ref{['eq:H123']}, (b) the full phase diagram in 3D, and (b) the 2D phase diagram assuming $\theta_1=\theta_2= \theta$.
  • Figure 3: (a) The multiplex $\bm{\mathcal{P}}$ defined in \ref{['eq:H']}, (b) the full phase diagram in 3D, (b) the 2D phase diagram assuming $\theta_1=\theta_2= \theta$.

Theorems & Definitions (34)

  • Definition 1.1
  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Definition 2.4
  • Example 2.5
  • Example 2.6
  • Remark 2.7: Boundedness of $\Delta_{\bm H}$
  • Definition 2.8
  • Theorem 2.9
  • ...and 24 more