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Monochromatic Subgraphs in Randomly Colored Dense Multiplex Networks

Mauricio Daros Andrade, Bhaswar B. Bhattacharya

TL;DR

The paper addresses the joint fluctuations of monochromatic subgraph counts in dense multiplex networks under uniform random vertex colorings. It develops a graph-limit framework with a general invariance principle to show that, for a finite collection of fixed graphs across $d$ layers, the appropriately standardized vector of monochromatic counts converges to the sum of a multivariate Gaussian and a collection of independent Wiener-Itô stochastic integrals driven by a common Brownian motion. The result relies on joint convergence in the cut-metric to a $d$-graphon and on explicit kernels given by the 2-point conditional kernels, with the covariance structure determined by overlaps of these kernels. The work extends known marginal results to the joint setting, provides corollaries for identical layers and edge cases like $H_i=K_2$, and includes illustrative examples such as correlated Erdős–Rényi multiplexes to demonstrate applicability and necessity of the assumptions.

Abstract

Given a sequence of graphs $G_n$ and a fixed graph $H$, denote by $T(H, G_n)$ the number of monochromatic copies of the graph $H$ in a uniformly random $c$-coloring of the vertices of $G_n$. In this paper we study the joint distribution of a finite collection of monochromatic graph counts in networks with multiple layers (multiplex networks). Specifically, given a finite collection of graphs $H_1, H_2, \ldots, H_d$ we derive the joint distribution of $(T(H_1, G_n^{(1)}), T(H_2, G_n^{(2)}), \ldots, T(H_d, G_n^{(d)}))$, where $\boldsymbol{G}_n = (G_n^{(1)}, G_n^{(2)}, \ldots, G_n^{(d)})$ is a collection of dense graphs on the same vertex set converging in the joint cut-metric. The limiting distribution is the sum of 2 independent components: a multivariate Gaussian and a sum of independent bivariate stochastic integrals. This extends previous results on the marginal convergence of monochromatic subgraphs in a sequence of graphs to the joint convergence of a finite collection of monochromatic subgraphs in a sequence of multiplex networks. Several applications and examples are discussed.

Monochromatic Subgraphs in Randomly Colored Dense Multiplex Networks

TL;DR

The paper addresses the joint fluctuations of monochromatic subgraph counts in dense multiplex networks under uniform random vertex colorings. It develops a graph-limit framework with a general invariance principle to show that, for a finite collection of fixed graphs across layers, the appropriately standardized vector of monochromatic counts converges to the sum of a multivariate Gaussian and a collection of independent Wiener-Itô stochastic integrals driven by a common Brownian motion. The result relies on joint convergence in the cut-metric to a -graphon and on explicit kernels given by the 2-point conditional kernels, with the covariance structure determined by overlaps of these kernels. The work extends known marginal results to the joint setting, provides corollaries for identical layers and edge cases like , and includes illustrative examples such as correlated Erdős–Rényi multiplexes to demonstrate applicability and necessity of the assumptions.

Abstract

Given a sequence of graphs and a fixed graph , denote by the number of monochromatic copies of the graph in a uniformly random -coloring of the vertices of . In this paper we study the joint distribution of a finite collection of monochromatic graph counts in networks with multiple layers (multiplex networks). Specifically, given a finite collection of graphs we derive the joint distribution of , where is a collection of dense graphs on the same vertex set converging in the joint cut-metric. The limiting distribution is the sum of 2 independent components: a multivariate Gaussian and a sum of independent bivariate stochastic integrals. This extends previous results on the marginal convergence of monochromatic subgraphs in a sequence of graphs to the joint convergence of a finite collection of monochromatic subgraphs in a sequence of multiplex networks. Several applications and examples are discussed.
Paper Structure (10 sections, 15 theorems, 135 equations, 3 figures)

This paper contains 10 sections, 15 theorems, 135 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Convergence of Graphs and Multiplexes
  3. Statement of the Results
  4. Asymptotic Notation
  5. A General Invariance Principle
  6. Proof of Theorem \ref{['thm:jointTHGn']}
  7. Examples
  8. Continuity of the 2-Point Conditional Kernel
  9. Independence of Bivariate Stochastic Integrals
  10. A CLT for Weighted Sum of Centered $\chi^2$ Random Variables

Key Result

Theorem 1.1

Suppose $\bm G_n$ is a sequence of $d$-multiplexes converging to a $d$-graphon $\bm{W}$. Let $\mathcal{H}= (H_1, H_2, \ldots, H_d)$ be a collection of fixed graphs and $\bm{\Gamma}(\mathcal{H}, \bm{G}_n)$ be as defined in eq:GammaHGnd. Assume, for $1 \leq i \ne j \leq d$, there exists $\rho_{ij} \g where $W_{H_i}^{G_n^{(i)}}$ is the 2-point conditional kernel of the empirical graphon $W^{G_n^{(i)

Figures (3)

  • Figure 1: The $(a, b), (a', b')$- join of the graphs $H_1$ and $H_2$.
  • Figure 2: The graphs $G_n^{(1)}$, $G_n^{(2)}$, and $G_n^{(3)}$ in Example \ref{['ex:joint_convergence_necessary']}.
  • Figure 3: The graphons $W_{1234}$ and $W_{23}$ in Example \ref{['ex:joint_convergence_necessary']}.

Theorems & Definitions (38)

  • Example 1.1: Birthday problem
  • Example 1.2: Graph-based nonparametric 2-sample tests
  • Example 1.3: Quadratic Rademacher chaos
  • Definition 1.1
  • Definition 1.2: 2-point conditional kernel
  • Definition 1.3
  • Theorem 1.1
  • Remark 1.1
  • Corollary 1.2
  • Corollary 1.3
  • ...and 28 more