Connectivity of Large Wireless Networks Under A General Connection Model

This paper studies networks where all nodes are distributed on a unit square A=^Δ [- [1/2], [1/2]]² following a Poisson distribution with known density ρ and a pair of nodes separated by an Euclidean distance x are directly connected with probability gr_ρ(x)=^Δg(x/r_ρ), independent of the event that any other pair of nodes are directly connected. Here, g:[0,∞)→ [0,1] satisfies the conditions of rotational invariance, nonincreasing monotonicity, integral boundedness, and g(x)=o(1/(x²log²x)) ; further, r_ρ=√{(logρ+b)/(Cρ)} where C=∫_ℜ²g(||x||)dx and b is a constant. Denote the aforementioned network by G(X_ρ,gr_ρ,A). We show that as ρ→ ∞, 1) the distribution of the number of isolated nodes in G(X_ρ,gr_ρ,A) converges to a Poisson distribution with mean e^-b ; 2) asymptotically almost surely (a.a.s.) there is no component in G(X_ρ,gr_ρ,A) of fixed and finite order k >; 1; c) a.a.s. the number of components with an unbounded order is one. Therefore, as ρ→ ∞, the network a.a.s. contains a unique unbounded component and isolated nodes only; a sufficient and necessary condition for G(X_ρ,gr_ρ,A) to be a.a.s. connected is that there is no isolated node in the network, which occurs when b→ ∞ as ρ→ ∞. These results expand recent results obtained for- connectivity of random geometric graphs from the unit disk model and the fewer results from the log-normal model to the more general and more practical random connection model.