Impact of Single Links in Competitive Percolation -- How complex networks grow under competition

TL;DR

This paper investigates how individual link additions influence the abruptness of network connectivity transitions, revealing that single links can cause observable effects in finite systems even if the transition is continuous in the limit.

Contribution

It demonstrates that in competitive percolation, single links can produce observable gaps in finite systems, clarifying their role in seemingly discontinuous transitions.

Findings

Single links do not cause a discontinuous gap in the thermodynamic limit.

Gaps scale weakly algebraically with system size in finite systems.

Multiple macroscopic clusters coexist before the transition.

Abstract

How a complex network is connected crucially impacts its dynamics and function. Percolation, the transition to extensive connectedness upon gradual addition of links, was long believed to be continuous but recent numerical evidence on "explosive percolation" suggests that it might as well be discontinuous if links compete for addition. Here we analyze the microscopic mechanisms underlying discontinuous percolation processes and reveal a strong impact of single link additions. We show that in generic competitive percolation processes, including those displaying explosive percolation, single links do not induce a discontinuous gap in the largest cluster size in the thermodynamic limit. Nevertheless, our results highlight that for large finite systems single links may still induce observable gaps because gap sizes scale weakly algebraically with system size. Several essentially macroscopic clusters coexist immediately before the transition, thus announcing discontinuous percolation. These results explain how single links may drastically change macroscopic connectivity in networks where links add competitively.