Monochromatic Subgraphs in Randomly Colored Dense Multiplex Networks

TL;DR

This paper investigates the joint distribution of monochromatic subgraph counts in dense multiplex networks with multiple layers, extending previous results to a multivariate setting and describing the limiting distribution as a combination of Gaussian and stochastic integral components.

Contribution

It derives the joint distribution of monochromatic subgraph counts in dense multiplex networks, extending marginal results to a multivariate framework with explicit limiting distribution characterization.

Findings

Limiting distribution combines Gaussian and stochastic integral components.

Joint convergence results extend previous marginal analyses.

Applications demonstrate the theoretical findings in network analysis.

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.