Scaling of optimal-path-lengths distribution in complex networks

TL;DR

This paper investigates how the distribution of optimal path lengths in complex networks with random weights depends on disorder strength, revealing a universal form controlled by network parameters and supported by numerical simulations.

Contribution

It introduces a universal scaling relation for the distribution of optimal path lengths in disordered networks, bridging strong and weak disorder regimes.

Findings

Distribution follows a universal form depending on network parameters.

Numerical simulations confirm the proposed scaling relation.

Transition between strong and weak disorder regimes is explicitly shown.

Abstract

We study the distribution of optimal path lengths in random graphs with random weights associated with each link (``disorder''). With each link $i$ we associate a weight $\tau_i = \exp(ar_i)$ where $r_i$ is a random number taken from a uniform distribution between 0 and 1, and the parameter $a$ controls the strength of the disorder. We suggest, in analogy with the average length of the optimal path, that the distribution of optimal path lengths has a universal form which is controlled by the expression $\frac{1}{p_c}\frac{\ell_{\infty}}{a}$, where $\ell_{\infty}$ is the optimal path length in strong disorder ($a \to \infty$) and $p_c$ is the percolation threshold. This relation is supported by numerical simulations for Erd\H{o}s-R\'enyi and scale-free graphs. We explain this phenomenon by showing explicitly the transition between strong disorder and weak disorder at different length scales in a single network.