Title: Phase Transition in Inhomogenous Erdős-Rényi Random Graphs via Tree Counting

Author(s): Ghurumuruhan Ganesan
Issue: Volume 80 Series A Part 1 Year 2018
Pages: 1 -- 27
Consider the complete graph $K_n$ on $n$ vertices where each edge $e$ is independently open with probability $ p_n (e) $ or closed otherwise. The edge probabilities are not necessarily same but are close to some positive constant $C$. The resulting random graph $G$ is in general inhomogenous and we use a tree counting argument to establish phase transition in the following sense: For $C<1$, all components of $G$ are small with high probability. For $C>1$, with high probability, there is at least one giant component 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. An important consequence of our method is that we directly obtain the fraction of vertices present in the giant component in the form of an infinite series.
AMS (2000) subject classification. Primary: 60J10, 60K35; Secondary: 60C05, 62E10, 90B15, 91D30.
Keywords and phrases: Inhomogenous Erdos-Renyi random graphs, Phase transition, Tree counting argument
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