Distance distribution in random graphs and application to networks exploration

TL;DR

This paper analyzes the distribution of distances and edge discovery in Erdős-Rényi graphs, revealing oscillatory behaviors and phase transitions, with a new computational method that aligns well with simulations.

Contribution

Introduces a novel method for computing distance distributions in random graphs, improving accuracy over previous approaches and enabling detailed analysis of phase transitions.

Findings

Oscillatory behavior in edge discovery proportions

New method matches numerical simulations closely

Characterization of phase transitions in connectivity

Abstract

We consider the problem of determining the proportion of edges that are discovered in an Erdos-Renyi graph when one constructs all shortest paths from a given source node to all other nodes. This problem is equivalent to the one of determining the proportion of edges connecting nodes that are at identical distance from the source node. The evolution of this quantity with the probability of existence of the edges exhibits intriguing oscillatory behavior. In order to perform our analysis, we introduce a new way of computing the distribution of distances between nodes. Our method outperforms previous similar analyses and leads to estimates that coincide remarkably well with numerical simulations. It allows us to characterize the phase transitions appearing when the connectivity probability varies.