Shortest paths and load scaling in scale-free trees

TL;DR

This paper analyzes shortest path lengths and load distribution in scale-free trees, revealing how average distances scale logarithmically with network size and explaining the Gaussian nature of their distribution.

Contribution

It introduces a mean-field approach and a tree mapping for the Barabasi-Albert model to explain distance scaling and load distribution in scale-free networks.

Findings

Average node-to-node distance scales logarithmically with N.

Distance distribution approaches a Gaussian for large N.

Load (betweenness) distribution is discussed in the tree context.

Abstract

The average node-to-node distance of scale-free graphs depends logarithmically on N, the number of nodes, while the probability distribution function (pdf) of the distances may take various forms. Here we analyze these by considering mean-field arguments and by mapping the m=1 case of the Barabasi-Albert model into a tree with a depth-dependent branching ratio. This shows the origins of the average distance scaling and allows a demonstration of why the distribution approaches a Gaussian in the limit of N large. The load (betweenness), the number of shortest distance paths passing through any node, is discussed in the tree presentation.