Universal behavior of optimal paths in weighted networks with general disorder

TL;DR

This paper reveals a universal behavior in the statistics of optimal paths across various weighted networks with different disorder types, governed by a single parameter that determines the path length distribution in both strong and weak disorder regimes.

Contribution

It introduces a universal scaling parameter that captures the distribution of optimal path lengths in weighted networks with general disorder, unifying different disorder regimes.

Findings

Universal behavior observed across different disorder types.

A single parameter determines the distribution of optimal path lengths.

Crossover from weak to strong disorder depends on network and disorder parameters.

Abstract

We study the statistics of the optimal path in both random and scale free networks, where weights $w$ are taken from a general distribution $P(w)$. We find that different types of disorder lead to the same universal behavior. Specifically, we find that a single parameter ($S \equiv AL^{-1/\nu}$ for $d$-dimensional lattices, and $S\equiv AN^{-1/3}$ for random networks) determines the distributions of the optimal path length, including both strong and weak disorder regimes. Here $\nu$ is the percolation connectivity exponent, and $A$ depends on the percolation threshold and $P(w)$. For $P(w)$ uniform, Poisson or Gaussian the crossover from weak to strong does not occur, and only weak disorder exists.