Continuous percolation phase transitions of two-dimensional lattice networks under a generalized Achlioptas process

Abstract

The percolation phase transitions of two-dimensional lattice networks under a generalized Achlioptas process (GAP) are investigated. During the GAP, two edges are chosen randomly from the lattice and the edge with minimum product of the two connecting cluster sizes is taken as the next occupied bond with a probability p. At p = 0.5, the GAP becomes the random growth model and leads to the minority product rule at p = 1. Using the finite-size scaling analysis, we find that the percolation phase transitions of these systems with 0.5 ≤ p ≤ 1 are always continuous and their critical exponents depend on p. Therefore, the universality class of the critical phenomena in two-dimensional lattice networks under the GAP is related to the probability parameter p in addition.PACS. 64.60.ah 64.60. 75.Hc

Method for estimating critical exponents in percolation processes with low sampling

“…In Fig.2, we present the evolution of p c as a function of q, for 2D square lattices and ER networks. For comparison, we also plot the values estimated in [31,32]. Even though we use data averaged over realizations 2 − 4 orders of magnitude less than in [31,32], the values are almost identical to those predicted there (values of q correspond to values of p as it is defined in [31,32]).…”

“…In [31,32], the authors propose a way to produce a hybrid model. At each time step, a link is placed between two nodes (or sites) either by the Achlioptas (product rule) process (with probability 1 − q) or in random (with probability q).…”

“…This means that noise is present in our results, but as proven below, it does not significantly affect them. For comparison, data from [31,32] are also plotted (square symbols).…”

“…Using these values we obtain also the 1/ν exponent and thus define the universality class of the system. We apply this method to a hybrid percolation model which is essentially the same to that described in [31,32]. This specific model exhibits an interesting behavior, useful to display the power of the suggested method, as it interpolates between the Achioptas process and random percolation.…”

“…Plot of β/ν as a function of q for (a) ER networks and (b) 2D square lattices (circular symbols). For comparison, data from[31,32] are also plotted (square symbols). (Color Online).Plot of 1/ν as a function of q for (a) ER networks and (b) 2D square lattices (square symbols).…”