Percolation Transitions in Scale-Free Networks under Achlioptas Process

TL;DR

This paper investigates how the nature of percolation transitions in scale-free networks under the Achlioptas Process varies with degree distribution, revealing conditions for both continuous and discontinuous transitions.

Contribution

It demonstrates that scale-free networks can exhibit either continuous or discontinuous percolation transitions depending on their degree exponent, unlike Erdős-Rényi networks.

Findings

Discontinuous transitions occur in certain scale-free networks.

Continuous transitions are possible depending on degree distribution.

A critical degree exponent separates the two transition types.

Abstract

It has been recently shown that the percolation transition is discontinuous in Erd\H{o}s-R\'enyi networks and square lattices in two dimensions under the Achlioptas Process (AP). Here, we show that when the structure is highly heterogeneous as in scale-free networks, a discontinuous transition does not always occur: a continuous transition is also possible depending on the degree distribution of the scale-free network. This originates from the competition between the AP that discourages the formation of a giant component and the existence of hubs that encourages it. We also estimate the value of the characteristic degree exponent that separates the two transition types.