Distribution of Edge Load in Scale-free Trees

TL;DR

This paper provides an exact analytical study of edge betweenness distribution in evolving scale-free and non-scale-free trees, considering local in-degree properties, with solutions applicable to both finite and infinite networks.

Contribution

It introduces the first exact analytical expressions for edge betweenness distribution conditioned on local in-degree in evolving trees.

Findings

Derived exact joint distribution of cluster size and in-degree.

Calculated expectation values of edge betweenness.

Provided solutions applicable to finite and infinite networks.

Abstract

Node betweenness has been studied recently by a number of authors, but until now less attention has been paid to edge betweenness. In this paper, we present an exact analytic study of edge betweenness in evolving scale-free and non-scale-free trees. We aim at the probability distribution of edge betweenness under the condition that a local property, the in-degree of the ``younger'' node of a randomly selected edge, is known. En route to the conditional distribution of edge betweenness the exact joint distribution of cluster size and in-degree, and its one dimensional marginal distributions have been presented in the paper as well. From the derived probability distributions the expectation values of different quantities have been calculated. Our results provide an exact solution not only for infinite, but for finite networks as well.