Phase transition in inhomogenous Erd\H{o}s-R\'enyi random graphs via tree counting

Abstract

Consider the complete graph \(K_n\) on \(n\) vertices where each edge \(e\) is independently open with probability \(p_n(e)\) or closed otherwise. Here \(\frac{C-\alpha_n}{n} \leq p_n(e) \leq \frac{C+\alpha_n}{n}\) where \(C > 0\) is a constant not depending on~\(n\) or~\(e\) and \(0 \leq \alpha_n \longrightarrow 0\) as \(n \rightarrow \infty.\) The resulting random graph~\(G\) is inhomogenous and we use a tree counting argument to establish phase transition in \(G.\) We also obtain that the critical value for phase transition is one in the following sense. For \(C < 1,\) all components of \(G\) are small (i.e. contain at most \(M\log{n}\) vertices) with high probability, i.e., with probability converging to one as \(n \rightarrow \infty.\) For \(C > 1,\) with high probability, there is at least one giant component (containing at least \(\epsilon n\) vertices for some \(\epsilon > 0\)) and every component is either small or giant. For \(C > 8,\) with positive probability, the giant component is unique and every other component is small. As a consequence of our method, we directly obtain the fraction of vertices present in the giant component in the form of an infinite series.