Statistics of remote regions of networks

TL;DR

This paper investigates the statistical properties of remote regions in complex networks, confirming the inverse square law distribution in many real-world networks and identifying classes where it does not hold.

Contribution

It extends previous theoretical results by analyzing empirical data and simulations, revealing the law's validity in various network types and its limitations in finite-dimensional networks.

Findings

Inverse square law holds in many real-world networks.

Long chains in networks limit the observable range of the law.

Certain finite-dimensional and embedded networks do not follow the law.

Abstract

We delve into the statistical properties of regions within complex networks that are distant from vertices with high centralities, such as hubs or highly connected clusters. These remote regions play a pivotal role in shaping the asymptotic behaviours of various spreading processes and the features of associated spectra. We investigate the probability distribution $P_{\geq m}(s)$ of the number $s$ of vertices located at distance $m$ or beyond from a randomly chosen vertex in an undirected network. Earlier, this distribution and its large $m$ asymptotics $1/s^2$ were obtained theoretically for undirected uncorrelated networks [S. N. Dorogovtsev, J. F. F. Mendes, A. N. Samukhin, Nucl. Phys. B 653 (2003) 307]. Employing numerical simulations and analysing empirical data, we explore a wide range of real undirected networks and their models, including trees and loopy networks, and reveal that the inverse square law is valid even for networks with strong correlations. We observe this law in the networks demonstrating the small-world effect and containing vertices with degree $1$ (so-called leaves or dead ends). We find the specific classes of networks for which this law is not valid. Such networks include the finite-dimensional networks and the networks embedded in finite-dimensional spaces. We notice that long chains of nodes in networks reduce the range of $m$ for which the inverse square law can be spotted. Interestingly, we detect such long chains in the remote regions of the undirected projection of a large Web domain.