Simple Random Walks on Radio Networks (Simple Random Walks on Hyper-Graphs)

TL;DR

This paper extends random walk theory to hyper-graphs, introducing the concept of radio cover time and providing bounds and algorithms relevant for wireless network modeling.

Contribution

It formalizes random walks on hyper-graphs, defines radio cover time, and offers bounds and algorithms, advancing understanding of network processes in hyper-graph models.

Findings

Radio cover time on hyper-graphs is O(mnr)

Polynomial algorithm for radio hitting times

Results on hyper-graphs modeling wireless networks

Abstract

In recent years, protocols that are based on the properties of random walks on graphs have found many applications in communication and information networks, such as wireless networks, peer-to-peer networks and the Web. For wireless networks (and other networks), graphs are actually not the correct model of the communication; instead hyper-graphs better capture the communication over a wireless shared channel. Motivated by this example, we study in this paper random walks on hyper-graphs. First, we formalize the random walk process on hyper-graphs and generalize key notions from random walks on graphs. We then give the novel definition of radio cover time, namely, the expected time of a random walk to be heard (as opposed to visit) by all nodes. We then provide some basic bounds on the radio cover, in particular, we show that while on graphs the radio cover time is O(mn), in hyper-graphs it is O(mnr) where n, m and r are the number of nodes, the number of edges and the rank of the hyper-graph, respectively. In addition, we define radio hitting times and give a polynomial algorithm to compute them. We conclude the paper with results on specific hyper-graphs that model wireless networks in one and two dimensions.