The distribution of shortest path lengths on trees of a given size in subcritical Erdos-Renyi networks

TL;DR

This paper derives the distribution of shortest path lengths between node pairs within the same tree component of a given size in subcritical Erdős-Rényi networks, revealing a size-dependent distribution independent of the network's mean degree.

Contribution

It provides an explicit formula for the shortest path length distribution conditioned on tree size, showing independence from the mean degree in subcritical ER networks.

Findings

Shortest path length distribution depends only on tree size, not on mean degree.

Average shortest path scales as the square root of the component size.

Distribution matches that of uniformly sampled labeled trees.

Abstract

In the subcritical regime Erd\H{o}s-R\'enyi (ER) networks consist of finite tree components, which are non-extensive in the network size. The distribution of shortest path lengths (DSPL) of subcritical ER networks was recently calculated using a topological expansion [E. Katzav, O. Biham and A.K. Hartmann, Phys. Rev. E 98, 012301 (2018)]. The DSPL, which accounts for the distance $\ell$ between any pair of nodes that reside on the same finite tree component, was found to follow a geometric distribution of the form $P(L=\ell | L < \infty) = (1-c) c^{\ell - 1}$, where $0 < c < 1$ is the mean degree of the network. This result includes the contributions of trees of all possible sizes and topologies. Here we calculate the distribution of shortest path lengths $P(L=\ell | S=s)$ between random pairs of nodes that reside on the same tree component of a given size $s$. It is found that $P(L=\ell | S=s) = \frac{\ell+1}{s^{\ell}} \frac{(s-2)!}{(s-\ell-1)!}$. Surprisingly, this distribution does not depend on the mean degree $c$ of the network from which the tree components were extracted. This is due to the fact that the ensemble of tree components of a given size $s$ in subcritical ER networks is sampled uniformly from the set of labeled trees of size $s$ and thus does not depend on $c$. The moments of the DSPL are also calculated. It is found that the mean distance between random pairs of nodes on tree components of size $s$ satisfies ${\mathbb E}[L|S=s] \sim \sqrt{s}$, unlike small-world networks in which the mean distance scales logarithmically with $s$.