Unusual percolation in simple small-world networks

TL;DR

This paper provides an exact solution for percolation in a class of small-world networks, revealing a non-classical critical point and three distinct regimes with different critical behaviors, useful for understanding network connectivity.

Contribution

It introduces an exact analytical framework for percolation in Watts-Strogatz networks, uncovering novel critical phenomena and regimes based on long-range bond proportions.

Findings

Identification of a non-classical critical point with discontinuous percolation probability

Discovery of three distinct critical regimes with different exponents

Development of a unified scaling function around the critical point

Abstract

We present an exact solution of percolation in a generalized class of Watts-Strogatz graphs defined on a 1-dimensional underlying lattice. We find a non-classical critical point in the limit of the number of long-range bonds in the system going to zero, with a discontinuity in the percolation probability and a divergence in the mean finite-cluster size. We show that the critical behavior falls into one of three regimes depending on the proportion of occupied long-range to unoccupied nearest-neighbor bonds, with each regime being characterized by different critical exponents. The three regimes can be united by a single scaling function around the critical point. These results can be used to identify the number of long-range links necessary to secure connectivity in a communication or transportation chain. As an example, we can resolve the communication problem in a game of "telephone".