Phase transitions in a network with range dependent connection probability

TL;DR

This paper investigates phase transitions in a one-dimensional network with distance-dependent connection probabilities, revealing a continuous transition at a critical exponent and demonstrating that these phenomena are observable across dimensions through average bond length behavior.

Contribution

It identifies a continuous phase transition at a specific exponent in a range-dependent network and shows this transition can be detected via average bond length in any dimension.

Findings

Transition to small world behavior for  < 2

Transition to a random network at  = 1

Phase transitions are non-trivial across all dimensions

Abstract

We consider a one-dimensional network in which the nodes at Euclidean distance $l$ can have long range connections with a probabilty $P(l) \sim l^{-\delta}$ in addition to nearest neighbour connections. This system has been shown to exhibit small world behaviour for $\delta < 2$ above which its behaviour is like a regular lattice. From the study of the clustering coefficients, we show that there is a transition to a random network at $\delta = 1$. The finite size scaling analysis of the clustering coefficients obtained from numerical simulations indicate that a continuous phase transition occurs at this point. Using these results, we find that the two transitions occurring in this network can be detected in any dimension by the behaviour of a single quantity, the average bond length. The phase transitions in all dimensions are non-trivial in nature.