Explosive Percolation in Erdős-Rényi-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 Erdős-Rényi process (ER). 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 ER 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, DʼSouza and Spencer [Achlioptas, D., R. M. DʼSouza and J. Spencer, Explosive percolation in random networks, Science 323 (2009), pp. 1453–1455] gave strong numerical evidence that a variety of edge-selection rules exhibit a discontinuous phase transition. However, Riordan and Warnke [Riordan, O. and L. Warnke, Achlioptas process phase transitions are continuous, arXiv:1102.5306v1 (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 ER-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.