Percolation properties of growing networks under an Achlioptas process

TL;DR

This paper investigates the percolation transition in growing networks influenced by an Achlioptas process, identifying the critical point and critical exponents through finite-size scaling and cluster analysis.

Contribution

It provides the first detailed analysis of percolation in growing networks under an Achlioptas process, including critical point determination and scaling behavior.

Findings

Percolation transition occurs at a critical probability $oldsymbol{\delta_c=0.5149(1)}$.

Critical exponents are estimated: $oldsymbol{eta o 1/2}$, $oldsymbol{ar{ u} o 5/2}$.

Cluster size distribution follows a power-law with Fisher exponent $oldsymbol{ au=2.24(1)}$.

Abstract

We study the percolation transition in growing networks under an Achlioptas process (AP). At each time step, a node is added in the network and, with the probability $\delta$, a link is formed between two nodes chosen by an AP. We find that there occurs the percolation transition with varying $\delta$ and the critical point $\delta_c=0.5149(1)$ is determined from the power-law behavior of order parameter and the crossing of the fourth-order cumulant at the critical point, also confirmed by the movement of the peak positions of the second largest cluster size to the $\delta_c$. Using the finite-size scaling analysis, we get $\beta/\bar{\nu}=0.20(1)$ and $1/\bar{\nu}=0.40(1)$, which implies $\beta \approx 1/2$ and $\bar{\nu} \approx 5/2$. The Fisher exponent $\tau = 2.24(1)$ for the cluster size distribution is obtained and shown to satisfy the hyperscaling relation.