Explosive Percolation in Erd"os-Renyi-Like Random Graph Processes

The evolution of the largest component has been studied intensely in a variety of random graph processes, starting in 1960 with the Erd"os-R'enyi process. It is well known that this process undergoes a phase transition at n/2 edges when, asymptotically almost surely, a linear-sized component appears. Moreover, this phase transition is continuous, i.e., in the limit the function f(c) denoting the fraction of vertices in the largest component in the process after cn edge insertions is continuous. A variation of the Erd"os-R'enyi process are the so-called Achlioptas processes in which in every step a random pair of edges is drawn, and a fixed edge-selection rule selects one of them to be included in the graph while the other is put back. Recently, Achlioptas, D’Souza and Spencer (2009) gave strong numerical evidence that a variety of edge-selection rules exhibit a discontinuous phase transition. However, Riordan and Warnke (2011) very recently showed that all Achlioptas processes have a continuous phase transition. In this work we prove discontinuous phase transitions for a class of Erd"os-R'enyi-like processes in which in every step we connect two vertices, one chosen randomly from all vertices, and one chosen randomly from a restricted set of vertices.

💡 Research Summary

The paper investigates a new family of random graph processes that are closely related to the classic Erdős‑Rényi (ER) model but incorporate an asymmetric vertex‑selection rule. In each step a vertex u is chosen uniformly from the whole vertex set V of size n, while a second vertex v is drawn uniformly from a predetermined “restricted” subset R⊂V of size |R|=αn (0<α<1). An edge (u,v) is added if it does not already exist. This “restricted‑vertex” process differs from the standard ER process (where both endpoints are chosen uniformly from V) and from Achlioptas processes (where a pair of candidate edges is presented and a deterministic rule selects one).

The authors’ main contribution is a rigorous proof that this process exhibits a discontinuous (explosive) percolation transition. Specifically, they show that there exists a critical average degree

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