Load distribution in weighted complex networks

TL;DR

This paper investigates how load distributes in weighted networks, revealing power-law behavior in different network types and showing that global transport properties are not directly linked to local structure.

Contribution

It introduces a method to measure load distribution based on optimal paths and analyzes the effects of disorder and network structure on load behavior.

Findings

Load distribution follows a power law in ER and SF networks.

Load distribution correlates with the minimum spanning tree structure.

Disorder affects the correlation between vertex degree and load.

Abstract

We study the load distribution in weighted networks by measuring the effective number of optimal paths passing through a given vertex. The optimal path, along which the total cost is minimum, crucially depend on the cost distribution function $p_c(c)$. In the strong disorder limit, where $p_c(c)\sim c^{-1}$, the load distribution follows a power law both in the Erd\H{o}s-R\'enyi (ER) random graphs and in the scale-free (SF) networks, and its characteristics are determined by the structure of the minimum spanning tree. The distribution of loads at vertices with a given vertex degree also follows the SF nature similar to the whole load distribution, implying that the global transport property is not correlated to the local structural information. Finally, we measure the effect of disorder by the correlation coefficient between vertex degree and load, finding that it is larger for ER networks than for SF networks.