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

Abstract

In this article, we investigate explosive bond percolation (EBP) with product rule, formally known as Achlioptas process, on a scale-free multifractal weighted planar stochastic lattice (WPSL). One of the key features of the EBP transition is the delay, compared to corresponding random bond percolation (RBP), in the onset of spanning cluster. However, when it happens, it happens so dramatically that initially it was believed, albeit ultimately proved wrong, that explosive percolation (EP) exhibits first order transition. In the case of EP, much efforts were devoted to resolving the issue of its order of transition and almost no effort being devoted to find critical point, critical exponents etc., to classify it into universality classes. This is in sharp contrast to the classical random percolation. We do not even know all the exponents of EP for regular planar lattice or for Erdös-Renyi network. We first find numerically the critical point pc and then obtain all the critical exponents β, γ, ν as well as the Fisher exponent τ and the fractal dimension d f of the spanning cluster. We also compare our results for EBP with those of the RBP and find that all the exponents of EBP obeys the same scaling relations as do the RBP. Our findings suggests that EBP is no special except the fact that the exponent β is unusually small compared to that of RBP.

Universality class of explosive percolation in Barabási-Albert networks

“…Following the procedures in refs. 37–39 we first find β / ν = 0.0468 and once again we obersve that its value is independent of the rules and the values of m except for m = 1. We now plot PN β / ν vs ( t − t c ) N 1/ ν with β / ν = 0.0468 and 1/ ν = 0.5195 and find that indeed all the distinct curves of Fig.…”

“…Once the network of desired size is grown we use that as a skeleton like we earlier used scale-free weighted planar stochastic lattice and ER network 37–39 . We assume that all the links of the BA network are temporarily frozen.…”

Percolation Analysis in a Fractal Network with Stable Opinion Dynamics

“…From the above equations we can perform a calculation of the critical exponent τ is enough to confirm the hyperscale relationship [6] Eq. (27).…”

“…In this regard, it is pertinent to note that the rapid convergence of Zk to Z∞ allowing us to observe fractal effects in Sierpiski pre-fractal carpets after a few iterations (see, for example, Refs. [13,[21][22][23][24][25][26][27][28]).…”