Probability of k-Hop Connection under Random Connection Model

Consider a wireless sensor network with Ltd. sensors following a homogeneous Poisson distribution in a given area A in ℜ
²
. A sensor located at x
₂
∈ A is directly connected to a sensor located at x
₁
∈ A with probability g (x
₂
- x
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 ∈ A apart from another node is connected to that node at exactly k hops, for a generic random connection function g : ℜ
²
→ . 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.