Critical exponents of the explosive percolation transition

Abstract

In a new type of percolation phase transition, which was observed in a set of non-equilibrium models, each new connection between vertices is chosen from a number of possibilities by an Achlioptas-like algorithm. This causes preferential merging of small components and delays the emergence of the percolation cluster. First simulations led to a conclusion that a percolation cluster in this irreversible process is born discontinuously, by a discontinuous phase transition, which results in the term "explosive percolation transition". We have shown that this transition is actually continuous (second-order) though with an anomalously small critical exponent of the percolation cluster. Here we propose an efficient numerical method enabling us to find the critical exponents and other characteristics of this second order transition for a representative set of explosive percolation models with different number of choices. The method is based on gluing together the numerical solutions of evolution equations for the cluster size distribution and power-law asymptotics. For each of the models, with high precision, we obtain critical exponents and the critical point.

7 pages, 4 figures

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• Percolation Transitions in Growing Networks Under Achlioptas Processes: Analytic Solutions