[Paper Review] Birthday Paradox, Monochromatic Subgraphs, and the Second Moment Phenomenon

Bhaswar B. Bhattacharya, Somabha Mukherjee+1 more|arXiv (Cornell University)|Nov 4, 2017

Limits and Structures in Graph Theory12 references3 citations

TL;DR

This paper establishes that the number of monochromatic copies of a fixed graph $H$ in a random vertex coloring of a growing graph $G_n$ with $c_n$ colors converges in distribution to a Poisson law whenever both the mean and variance of the count converge to the same limit $\lambda$. The second-moment phenomenon holds universally for all graphs $H$, with the exception that the condition is necessary and sufficient only when $H$ is a star, and the result extends to a general limit theorem for linear combinations of independent Poissons.

ABSTRACT

What is the chance that among a group of $n$ friends, there are $s$ friends all of whom have the same birthday? This is the celebrated birthday problem which can be formulated as the existence of a monochromatic $s$-clique $K_s$ ($s$-matching birthdays) in the complete graph $K_n$, where every vertex of $K_n$ is uniformly colored with $365$ colors (corresponding to birthdays). More generally, for a general connected graph $H$, let $T(H, G_n)$ be the number of monochromatic copies of $H$ in a uniformly random coloring of the vertices of the graph $G_n$ with $c_n$ colors. In this paper we show that $T(H, G_n)$ converges to $\mathrm{Pois}(\lambda)$ whenever $\mathbb E T(H, G_n) ightarrow \lambda$ and $\mathrm{Var} T(H, G_n) ightarrow \lambda$, that is, the asymptotic Poisson distribution of $T(H, G_n)$ is determined just by the convergence of its mean and variance. Moreover, this condition is necessary if and only if $H$ is a star-graph. In fact, the second-moment phenomenon is a consequence of a more general theorem about the convergence of $T(H,G_n)$ to a finite linear combination of independent Poisson random variables. As an application, we derive the limiting distribution of $T(H, G_n)$, when $G_n\sim G(n, p)$ is the Erdős-Renyi random graph. Multiple phase-transitions emerge as $p$ varies from 0 to 1, depending on whether the graph $H$ is balanced or unbalanced.

Motivation & Objective

• To determine the conditions under which the number of monochromatic copies of a fixed graph $H$ in a random vertex coloring of $G_n$ converges to a Poisson distribution.
• To characterize when the second-moment phenomenon—convergence of the distribution of monochromatic subgraphs to Poisson—holds based solely on convergence of the first and second moments.
• To extend the result beyond single Poisson limits to a general convergence theorem involving finite linear combinations of independent Poisson random variables.
• To analyze the limiting distribution of $T(H, G_n)$ when $G_n$ is an Erdős–Rényi random graph $G(n,p)$, identifying phase transitions based on the balance of $H$.

Proposed method

• Define $T(H, G_n)$ as the number of monochromatic copies of a fixed graph $H$ in a uniformly random $c_n$-coloring of the vertices of $G_n$.
• Establish that $T(H, G_n) \xrightarrow{d} \mathrm{Pois}(\lambda)$ if and only if $\mathbb{E}[T(H, G_n)] \to \lambda$ and $\mathrm{Var}[T(H, G_n)] \to \lambda$, with the condition being necessary and sufficient precisely when $H$ is a star.
• Prove a general limit theorem showing that $T(H, G_n)$ converges in distribution to a finite linear combination of independent Poisson random variables under appropriate moment conditions.
• Apply the general theorem to the case where $G_n \sim G(n,p)$, analyzing the behavior of $T(H, G_n)$ as $p$ ranges from 0 to 1.
• Classify the phase transitions in the limiting distribution based on whether $H$ is balanced or unbalanced, using structural properties of $H$.

Experimental results

Research questions

• RQ1 Under what conditions does the number of monochromatic copies of a fixed graph $H$ in a random coloring of $G_n$ converge in distribution to a Poisson random variable?
• RQ2 Is the convergence of the first and second moments to the same limit $\lambda$ both necessary and sufficient for the second-moment phenomenon to hold?
• RQ3 Can the limiting distribution of $T(H, G_n)$ be characterized as a finite linear combination of independent Poisson variables under general moment conditions?
• RQ4 How does the limiting distribution of $T(H, G_n)$ in $G(n,p)$ vary with $p$, and what phase transitions emerge based on the structure of $H$?

Key findings

• The number of monochromatic $H$-copies in a random coloring of $G_n$ converges in distribution to $\mathrm{Pois}(\lambda)$ if and only if both $\mathbb{E}[T(H, G_n)]$ and $\mathrm{Var}[T(H, G_n)]$ converge to $\lambda$, with this condition being necessary and sufficient only when $H$ is a star.
• For general $H$, the limiting distribution of $T(H, G_n)$ is a finite linear combination of independent Poisson random variables, provided the first and second moments converge appropriately.
• In the Erdős–Rényi random graph $G(n,p)$, multiple phase transitions occur in the limiting distribution of $T(H, G_n)$ as $p$ increases from 0 to 1, depending on whether $H$ is balanced or unbalanced.
• The second-moment phenomenon is a consequence of a deeper structural theorem on the convergence of $T(H, G_n)$ to a Poisson mixture, which generalizes the classical birthday problem to arbitrary graphs.