Explosive percolation: A numerical analysis

Percolation is one of the most studied processes in statistical physics. A recent paper by Achlioptas et al. [Science 323, 1453 (2009)] showed that the percolation transition, which is usually continuous, becomes discontinuous (explosive) if links are added to the system according to special cooperative rules (Achlioptas processes). In this paper, we present a detailed numerical analysis of Achlioptas processes with product rule on various systems, including lattices, random networks a la Erdos-Renyi, and scale-free networks. In all cases, we recover the explosive transition by Achlioptas et al. However, the explosive percolation transition is kind of hybrid as, despite the discontinuity of the order parameter at the threshold, one observes traces of analytical behavior such as power-law distributions of cluster sizes. In particular, for scale-free networks with degree exponent lambda < 3, all relevant percolation variables display power-law scaling, just as in continuous second-order phase transitions.