Explosive percolation on a scale-free multifractal weighted planar stochastic lattice

Universality class of explosive percolation in Barabási-Albert networks

In this work, we study explosive percolation (EP) in Barabási-Albert (BA) network, in which nodes are born with degree k = m, for both product rule (PR) and sum rule (SR) of the Achlioptas process. For m = 1 we find that the critical point t_c = 1 which is the maximum possible value of the relative link density t; Hence we cannot have access to the other phase like percolation in one dimension. However, for m > 1 we find that t_c decreases with increasing m and the critical exponents ν, α, β and γ for m > 1 are found to be independent not only of the value of m but also of PR and SR. It implies that they all belong to the same universality class like EP in the Erdös-Rényi network. Besides, the critical exponents obey the Rushbrooke inequality α + 2β + γ ≥ 2 but always close to equality.

    PACS numbers: 61.43.Hv, 64.60.Ht, 68.03.Fg, 82.70.Dd.

Product-sum universality and Rushbrooke inequality in explosive percolation

We study explosive percolation (EP) on an Erdös-Rényi network for product rule (PR) and sum rule (SR). Initially, it was claimed that EP describes discontinuous phase transition; now it is well accepted as a probabilistic model for thermal continuous phase transition (CPT). However, no model for CPT is complete unless we know how to relate its observable quantities with those of thermal CPT. To this end, we define entropy and specific heat, redefine susceptibility, and show that they behave exactly like their thermal counterparts. We obtain the critical exponents ν,α,β, and γ numerically and find that both PR and SR belong to the same universality class and obey Rushbrooke inequality.