Probability of k-hop connection under random connection model

Consider a wireless sensor network with i.i.d. sensors following a homogeneous Poisson distribution in a given area A in ℜ². A sensor located at x₂ ε A is directly connected to a sensor located at {x}-{1} ε A with probability g(₂-₁), independent of any other distinct pair of sensors. In this letter, we provide a recursive formula for computing Pr(k|x), the probability that a node x ε A apart from another node is connected to that node at exactly k hops, for a generic random connection function g:ℜ² → [0,1]. The recursive formula is accurate for k=1,2 and provides an approximation for Pr(k|x) for k>2. The exact and approximate analytical results are validated by simulations. The knowledge of Pr(k|x) can be used in a number of areas in sensor networks.