Percolation transitions with nonlocal constraint

TL;DR

This paper studies how nonlocal constraints affect percolation transitions in networks, revealing different universality classes and a unique phase transition at a critical nonlocality parameter.

Contribution

It introduces a nonlocal network model with an adjustable parameter controlling the nonlocal effect, and characterizes the resulting diverse percolation transition behaviors.

Findings

Mean field universality class for r<0.5

Quasi-critical phase with G ~ N^0.74 for r>0.5

Non-mean-field transition at r=0.5

Abstract

We investigate percolation transitions in a nonlocal network model numerically. In this model, each node has an exclusive partner and a link is forbidden between two nodes whose $r$-neighbors share any exclusive pair. The $r$-neighbor of a node $x$ is defined as a set of at most $N^r$ neighbors of $x$, where $N$ is the total number of nodes. The parameter $r$ controls the strength of a nonlocal effect. The system is found to undergo a percolation transition belonging to the mean field universality class for $r< 1/2$. On the other hand, for $r>1/2$, the system undergoes a peculiar phase transition from a non-percolating phase to a quasi-critical phase where the largest cluster size $G$ scales as $G \sim N^{\alpha}$ with $\alpha = 0.74 (1)$. In the marginal case with $r=1/2$, the model displays a percolation transition that does not belong to the mean field universality class.