Monochromatic Subgraphs in Randomly Colored Graphons

TL;DR

This paper characterizes the limiting distribution of monochromatic subgraph counts in randomly colored dense graphs, revealing different behaviors depending on the number of colors and generalizing classical birthday problem results.

Contribution

It provides a complete characterization of the asymptotic distribution of monochromatic subgraph counts in converging dense graphs, extending classical results to general subgraphs.

Findings

Convergence to Poisson or normal distribution as colors grow

Convergence to a linear combination of chi-squared variables with fixed colors

Generalization of the birthday problem to arbitrary subgraphs

Abstract

Let $T(H, G_n)$ be the number of monochromatic copies of a fixed connected graph $H$ in a uniformly random coloring of the vertices of the graph $G_n$. In this paper we give a complete characterization of the limiting distribution of $T(H, G_n)$, when $\{G_n\}_{n \geq 1}$ is a converging sequence of dense graphs. When the number of colors grows to infinity, depending on whether the expected value remains bounded, $T(H, G_n)$ either converges to a finite linear combination of independent Poisson variables or a normal distribution. On the other hand, when the number of colors is fixed, $T(H, G_n)$ converges to a (possibly infinite) linear combination of independent centered chi-squared random variables. 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 a network.