Percolation Transitions of Random Networks under a Weight Probability Function

Based on the clusters growth mechanisms, we study a percolation model where the clusters are assigned to a weight probability function and the intracluster edges are excluded. The weight probability function includes a tunable parameter α. The model can realize the phase transition from continuous to multiple discontinuous and discontinuous as the value of α is tuned. According to the properties of the weight probability function, three typical cases which correspond to different clusters growth mechanisms are analyzed. When the system size N is equal to 1/α, probability modulation effect indicates that the percolation process generates a continuous phase transition which is similar to the classical Erd?s–Rényi (ER) network model. At α=1, it is shown that the lower pseudotransition point is converging to 1 in the thermodynamic limit and the cluster size distribution at the lower pseudotransition point does not obey the power-law behavior, indicating a first-order phase transition. For α=N^?1/2, the order parameter exhibits multiple jumps and the magnitude of the jumps are randomly distributed. The numerical simulations find that the relative variance of the order parameter is nonzero on an extended interval. It indicates that the order parameter is non-self-averaging. The cluster size heterogeneity decreases oscillatorily from some moment, which also implies the phenomenon.