Continuous percolation phase transitions of random networks under a generalized Achlioptas process

TL;DR

This study investigates how the percolation phase transition in evolving random networks varies continuously with a parameter p in a generalized Achlioptas process, revealing p-dependent critical exponents and universality classes.

Contribution

It introduces a generalized Achlioptas process model that interpolates between Erdős-Rényi and Achlioptas networks, demonstrating continuous phase transitions with p-dependent critical exponents.

Findings

Phase transitions are continuous for all p in [0.5, 1].

Critical exponents β and ν depend on p.

Universality class varies with p.

Abstract

Using the finite-size scaling, we have investigated the percolation phase transitions of evolving random networks under a generalized Achlioptas process (GAP). During this GAP, the edge with minimum product of two connecting cluster sizes is taken with a probability $p$ from two randomly chosen edges. This model becomes the Erd\H os-R\'enyi network at $p=0.5$ and the random network under the Achlioptas process at $p=1$. Using both the fixed point of $s_2/s_1$ and the straight line of $\ln s_1$, where $s_1$ and $s_2$ are the reduced sizes of the largest and the second largest cluster, we demonstrate that the phase transitions of this model are continuous for $0.5 \le p \le 1$. From the slopes of $\ln s_1$ and $\ln (s_2/s_1)'$ at the critical point we get the critical exponents $\beta$ and $\nu$, which depend on $p$. Therefore the universality class of this model should be characterized by $p$ also.