Geodesic Length Distribution in Sparse Network Ensembles

TL;DR

This paper derives an analytic distribution of shortest path lengths in large sparse networks, providing insights into connectivity and robustness across various network models.

Contribution

It introduces a closed-form expression for the geodesic length distribution in sparse networks, applicable to multiple models and regimes, enhancing understanding of network connectivity.

Findings

Closed-form geodesic length distribution for large sparse networks.

Applicable to models like stochastic block models, geometric graphs, and graphons.

Provides insights into network connectivity and robustness.

Abstract

A key task in the study of networked systems is to derive local and global properties that impact connectivity, synchronizability, and robustness; computing shortest paths or geodesics yields measures of network connectivity that can explain such phenomena. We derive an analytic distribution of geodesic lengths on the giant component in the supercritical regime -- when the giant component exists -- or on small components in the subcritical regime, of any sparse (and possibly directed) network with conditionally independent edges, in the infinite-size limit. We provide specific results for widely used network models like stochastic block models, dot product graphs, random geometric graphs, and sparse graphons. The survival function of the geodesic length distribution possesses a simple closed-form expression which is asymptotically tight for finite lengths, has a natural interpretation of traversing independent geodesics in the network, and delivers novel insight into the aforementioned network families.