Universality class of explosive percolation in Barab\'{a}si-Albert networks

Abstract

In this work, we study explosive percolation (EP) in Barab\'{a}si-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 $\nu, \alpha, \beta$ and $\gamma$ 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 Erd\"{o}s-R\'{e}nyi network. Besides, the critical exponents obey the Rushbrooke inequality in the form $\alpha+2\beta+\gamma=2+\epsilon$ with $0<\epsilon<<1$.