Geodesic Length Distribution in Sparse Network Ensembles
This work offers a theoretical tool for analyzing connectivity in networked systems, but it is incremental as it builds on existing network theory.
The authors derived an analytic distribution of geodesic lengths for sparse networks with conditionally independent edges, applicable to models like stochastic block models and random geometric graphs, providing a closed-form expression that is asymptotically tight for finite lengths.
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.