Closed-Form Hop-Count Distributions in Random Networks with Arbitrary Routing

We contribute a new solution to the problem of establishing an analytical relationship between hop-counts under a certain routing policy and Euclidean distances in random networks, both in the linear and planar cases. The contributed solution is unified, in that hop-count distributions have similar expressions both in the 1D and the 2D cases; general in terms of routing policies, in that the effect of any given policy is accounted for by means of a single parameter; closed-form, such that hop-count probability mass functions (PMF´s) are given in terms of scaled versions of the closed-form PMF´s of the number of nodes; and mathematically tractable, since the derived hop-count distributions are in the form of a difference of the well-known Nakagami-m cumulative density functions (CDF´s). Direct and Kullback-Leibler divergence comparisons against empirical data demonstrate the high accuracy of our solution. The simplicity, accuracy and generality of the result owes partly to a self-imposed confinement to connected networks, defined formally in stochastic-geometric terms, which allows for the elimination of recursions and multivariate marginalization commonly required by existing solutions. The contributed results find application in the design and analysis of ad hoc networks, cooperative localization algorithms or latency and energy consumption analysis.