Flooding in Weighted Random Graphs

TL;DR

This paper investigates how edge weights modeled as exponential delays affect distances in diluted random graphs, revealing that weighted flooding time and diameter grow logarithmically with graph size and outperform synchronized algorithms.

Contribution

It provides a rigorous analysis of weighted distances in random graphs with exponential delays, including exact growth rates and prefactors, and compares asynchronous and synchronized broadcast algorithms.

Findings

Weighted flooding time and diameter grow as log(n) with precise prefactors.

Asynchronous broadcast algorithms outperform synchronized ones in large graphs.

Results apply under regularity conditions on the degree sequence.

Abstract

In this paper, we study the impact of edge weights on distances in diluted random graphs. We interpret these weights as delays, and take them as i.i.d exponential random variables. We analyze the weighted flooding time defined as the minimum time needed to reach all nodes from one uniformly chosen node, and the weighted diameter corresponding to the largest distance between any pair of vertices. Under some regularity conditions on the degree sequence of the random graph, we show that these quantities grow as the logarithm of $n$, when the size of the graph $n$ tends to infinity. We also derive the exact value for the prefactors. These allow us to analyze an asynchronous randomized broadcast algorithm for random regular graphs. Our results show that the asynchronous version of the algorithm performs better than its synchronized version: in the large size limit of the graph, it will reach the whole network faster even if the local dynamics are similar on average.