The cover times of random walks on hypergraphs.

Random walks in graphs have been applied to various network exploration and network maintenance problems. In some applications, however, it may be more natural, and more accurate, to model the underlying network not as a graph but as a hypergraph, and solutions based on random walks require a notion of random walks in hypergraphs. While random walks in graphs have been extensively studied, there have been very few results showing properties of random walks in hypergraphs. At each step, a random walk on a hypergraph moves from its current position v to a random vertex in a randomly selected hyperedge containing v. We consider two definitions of cover time for random walks on a hypergraph H. If the walk sees only the vertices it moves between, then the usual definition of cover time, C(H), applies. If the walk sees the complete edge during the transition, then an alternative definition of cover time, the inform time I(H) is used. The notion of inform time is a reasonable model of passive listening which fits the following types of situations. The particle is a rumor passing between friends, which is overheard by other friends present in the group at the same time. The particle is a message transmitted randomly from location to location by a directional transmission in an ad-hoc network, but all receivers within the transmission range can hear. In this paper we give an expression for C(H) which is tractable for many classes of hypergraphs, and calculate C(H) and I(H) exactly for random r-regular, s-uniform hypergraphs. We find that for such hypergraph whp C(H)/I(H) = Ω(s) for large s