Flooding and Diameter in General Weighted Random Graphs

TL;DR

This paper analyzes the asymptotic behavior of weighted diameter and flooding time in the configuration model of random graphs with exponential tail weight distributions, establishing their limits as the graph size grows.

Contribution

It introduces the concepts of weighted diameter and flooding time for the first passage percolation on the configuration model and characterizes their asymptotic behavior for exponential tail weight distributions.

Findings

Asymptotic behavior of diameter and flooding time described

Exponential tail distributions are maximal for the asymptotic results

Results apply as the number of vertices tends to infinity

Abstract

We study in this paper, the first passage percolation on a random graph model, the configuration model. We first introduce, the notions of weighted diameter, which is the maximum of the weighted lengths of all optimal paths between any two vertices in the graph, and the flooding time, which represents the time (weighted length) needed to reach all the vertices in the graph starting from a uniformly chosen vertex. Our result consists of describing the asymptotic behavior of the diameter and the flooding time, as the number of vertices n tends to infinity, in the case where the weight distribution G has an exponential tail behavior, and proving that this category of distributions is the largest possible for which the asymptotic behavior holds.