Percolation Phase Transitions from Second Order to First Order in Random Networks* *Supported by the National Natural Science Foundation of China under Grant Nos. 61172115 and 60872029, the High-Tech Research and Development Program of China under Grant No. 2008AA01Z206, the Aeronautics Foundation of China under Grant No. 20100180003, the Fundamental Research Funds for the Central Universities under Grant No. ZYGX2009J037,

Abstract

We investigate a percolation process where an additional parameter q is used to interpolate between the classical Erdös–Rényi (ER) network model and the smallest cluster (SC) model. This model becomes the ER network at q = 1, which is characterized by a robust second order phase transition. When q = 0, this model recovers to the SC model which exhibits a first order phase transition. To study how the percolation phase transition changes from second order to first order with the decrease of the value of q from 1 to 0, the numerical simulations study the final vanishing moment of the each existing cluster except the N-cluster in the percolation process. For the continuous phase transition, it is shown that the tail of the graph of the final vanishing moment has the characteristic of the convexity. While for the discontinuous phase transition, the graph of the final vanishing moment possesses the characteristic of the concavity. Just before the critical point, it is found that the ratio between the maximum of the sequential vanishing clusters sizes and the network size N can be used to decide the phase transition type. We show that when the ratio is larger than or equal to zero in the thermodynamic limit, the percolation phase transition is first or second order respectively. For our model, the numerical simulations indicate that there exists a tricritical point qc which is estimated to be between 0.2 < qc < 0.25 separating the two phase transition types.