Monochromatic Subgraphs in Multiplex Networks

Loading...
Thumbnail Image
Degree type
PhD
Graduate group
Statistics and Data Science
Discipline
Mathematics
Statistics and Probability
Subject
Combinatorial probability
Graph limit theory
Multiplex networks
Stochastic integrals
Funder
Grant number
License
Copyright date
01/01/2025
Distributor
Related resources
Author
Daros Andrade, Mauricio
Contributor
Abstract

Given a sequence of graphs ${G_n}_{n \geq 1}$ and fixed graph $H$, denote by $T(H, G_n)$ the number of monochromatic copies of the graph $H$ in $G_n$ in a uniformly random $c_n$-coloring of the vertices of $G_n$. In this dissertation we study the joint distribution of monochromatic subgraphs for dense multiplex networks, that is, networks with multiple layers. Specifically, given a finite collection of graphs $H_1,\ldots,H_d$, we derive the asymptotic joint distribution of $\bm T_n:= (T(H_1, G_n^{(1)}),\ldots, T(H_d, G_n^{(d)}))$, where $\bm G_n=(G_n^{(1)},\ldots, G_n^{(d)})$ is a collection of graphs on the same vertex set converging in the joint cut-metric. Under a notion of joint convergence of $\bm G_n$ in the cut metric, we show that when the number of colors $c_n=c$ is fixed, then the limiting distribution of $\bm T_n$ is the sum of two independent components, one of which is a multivariate Gaussian and the other is a sum of bivariate stochastic integrals. On the other hand, when the number of colors $c_n \rightarrow \infty$ (such that $\mathbb E[\bm T_n] \rightarrow \infty$), then the asymptotic distribution of $\bm T_n$ is a multivariate normal. This generalizes the classical birthday problem, which involves understanding the asymptotics of $T(K_s,K_n)$, the number of monochromatic $s$-cliques in a complete graph $K_n$ ($s$-matching birthdays among a group of $n$ friends), to general monochromatic subgraphs in multiplex networks. This also extends previous results on the marginal convergence of $T(H, G_n)$ and is useful in establishing the joint convergence of various subgraph counting statistics that arise from random vertex coloring of graphs. Several applications and examples are discussed.

Advisor
Bhattacharya, Bhaswar, B
Date of degree
2025
Date Range for Data Collection (Start Date)
Date Range for Data Collection (End Date)
Digital Object Identifier
Series name and number
Volume number
Issue number
Publisher
Publisher DOI
Journal Issue
Comments
Recommended citation