Two Typical Discontinuous Transitions Observed in a Generalized Achlioptas Percolation Process

We extend the Achlioptas percolation (AP) process [Achlioptas et al. Science 323 (2009) 1453] to two generalized Achlioptas percolation processes named GAP1 and GAP2. GAP1 induces a weighted probability factor alpha in the node sampling process and excludes the intracluster links. Based on GAP1, GAP2 requires m pairs of nodes sampled to add alpha candidate links that should be residing in 2m different clusters at each step. In the evolution of GAP1, the phase transition can evolve from the continuous to the 'most explosive' percolation as the value of the factor alpha is decreasing to a certain negative number. It indicates that there might be a type of discontinuous transition induced by the probability modulation effect even in the thermodynamic limit, and the most explosive percolation is only one of its extreme cases. We analyze the characteristics of the evolving process of the two-nodes-clusters and the cluster-size distribution at the transformation point for different alpha; the numerical results suggest that there might be a critical value alpha(0) and the phase transition should be discontinuous (alpha <=alpha(0)) or continuous (alpha>alpha(0)). In the evolution of GAP2, twice phase transitions are observed successively and the time duration between them becomes shorter till they amalgamate into the 'most explosive' percolation. The first transition is induced by the probability modulation effect analyzed in GAP1, the second transition, induced by the three coexisting giant clusters, is always discontinuous and the maximum jump of order parameter approaches N/3 while the value of the factor alpha is increasing to 1.4 approximately. In this work, two typical discontinuous transitions induced respectively by the probability modulation and the extended local competition are exhibited in GAP2, which might provide references to analyze the discontinuous phase transition in networks further.