The phase transition in the multi-type binomial random graph $G(\mathbf{n},P)$

TL;DR

This paper analyzes the size of the largest component in a 2-type binomial random graph near the critical point, revealing precise asymptotic behavior in the weakly supercritical regime.

Contribution

It provides a refined branching process approach to determine the asymptotic size of the largest component near criticality in multi-type random graphs.

Findings

Largest component size is approximately 2ε||n||₁ in the supercritical regime.

Other components are negligible compared to the largest, of size o(ε||n||₁).

Results hold with high probability as the graph size grows.

Abstract

We determine the asymptotic size of the largest component in the $2$-type binomial random graph $G(\mathbf{n},P)$ near criticality using a refined branching process approach. In $G(\mathbf{n},P)$ every vertex has one of two types, the vector $\mathbf{n}$ describes the number of vertices of each type, and any edge $\{u,v\}$ is present independently with a probability that is given by an entry of the probability matrix $P$ according to the types of $u$ and $v.$ We prove that in the weakly supercritical regime, i.e. if the distance to the critical point of the phase transition is given by an $\varepsilon=\varepsilon(\mathbf{n})\to0,$ with probability $1-o(1),$ the largest component in $G(\mathbf{n},P)$ contains asymptotically $2\varepsilon \|\mathbf{n}\|_1$ vertices and all other components are of size $o(\varepsilon \|\mathbf{n}\|_1).$