Phase transition in inhomogenous Erdős-Rényi random graphs via tree counting (1704.00458v1)

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.