Radio cover time in hyper-graphs

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, 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 visited) 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. We conclude the paper with results on specific hyper-graphs that model wireless mesh networks in one and two dimensions and show that in both cases the radio cover time can be significantly faster than the standard cover time. In the two-dimension case, the radio cover time becomes sub-linear for an average degree larger than log2 n.