Hop distances in homogeneous ad hoc networks

Given is the network-level view of a wireless multi-hop network with n uniformly distributed nodes, each of them with radio transmission range r₀, on a rectangular area. This paper investigates the discrete probability distribution of the minimum number of wireless hops H between a random source and destination node. This topology attribute has significant impact on the network performance, e.g., on route discovery delay and message delivery. We derive closed form expressions for the probability that two nodes can communicate within H = 1 hop (i.e., via a direct link) or H = 2 hops (i.e., over one relay node). Connection paths with H > 2 hops and the expected hop distance E{H} are studied by analytical bounds and extensive simulations.