Phase transitions for Erdos-Renyi graphs

TL;DR

This paper investigates phase transitions in Erdős-Rényi graphs using a component counting approach, providing estimates on the size of the giant component and non-giant components for large values of the parameter C.

Contribution

It introduces an alternative component counting method to analyze phase transitions and size estimates in Erdős-Rényi graphs, complementing existing techniques.

Findings

Establishes phase transition thresholds using component counting.

Provides bounds on the size of the giant component for large C.

Shows non-giant components are small with high probability.

Abstract

Consider the complete graph on \(n\) vertices where each edge is independently open with probability \(p,\) or closed otherwise. Phase transitions for such graphs for \(p = \frac{C}{n}\) have previously been studied using techniques like branching processes and random walks. In this paper, we use an alternate component counting argument for establishing phase transition and obtaining estimates on the sum size of the non-giant components. As a corollary, we also obtain estimates on the size of the giant component for \(C\) large: If \(C\) is sufficiently large, there is a positive constant \(M_0 = M_0(C)\) so that with probability at least \(1-e^{-C/100},\) there is a giant component containing at least \(n - ne^{-C/8}\) vertices and every other component contains less than \(M_0 \log{n}\) vertices.