Distances in random graphs with finite variance degrees

TL;DR

This paper analyzes the typical distances in a large random graph with finite variance degrees, confirming a logarithmic growth pattern and characterizing fluctuations, extending understanding of network topology in heavy-tailed degree distributions.

Contribution

It proves that graph distances grow logarithmically with network size in a finite variance heavy-tailed degree model, and characterizes the fluctuations around this growth.

Findings

Graph distance grows like  N, with = expected degree moments ratio.

Fluctuations around the mean are bounded and converge in distribution along exponential subsequences.

Confirms heuristic predictions for distances in heavy-tailed random graphs.

Abstract

In this paper we study a random graph with $N$ nodes, where node $j$ has degree $D_j$ and $\{D_j\}_{j=1}^N$ are i.i.d. with $\prob(D_j\leq x)=F(x)$. We assume that $1-F(x)\leq c x^{-\tau+1}$ for some $\tau>3$ and some constant $c>0$. This graph model is a variant of the so-called configuration model, and includes heavy tail degrees with finite variance. The minimal number of edges between two arbitrary connected nodes, also known as the graph distance or the hopcount, is investigated when $N\to \infty$. We prove that the graph distance grows like $\log_{\nu}N$, when the base of the logarithm equals $\nu=\expec[D_j(D_j -1)]/\expec[D_j]>1$. This confirms the heuristic argument of Newman, Strogatz and Watts \cite{NSW00}. In addition, the random fluctuations around this asymptotic mean $\log_{\nu}{N}$ are characterized and shown to be uniformly bounded. In particular, we show convergence in distribution of the centered graph distance along exponentially growing subsequences.