Connectivity of large wireless networks under a general connection model

This paper studies networks where all nodes are distributed on a unit square Aδ= [-1\2, 1\2]² following a Poisson distribution with known density rho and a pair of nodes separated by an Euclidean distance x are directly connected with probability grρ(x)= g(x/rρ), 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/x²(log²)) ; further, rρ= (log ρ +b)(Cρ) where C=f_r ²g(\left \Vert \mmb x\right \Vert)d \mmb x and b is a constant. Denote the aforementioned network by \cal G\left (\cal Xρ,g-rρ,A\right). We show that as ρ → 1) the distribution of the number of isolated nodes in \cal G\left (\cal Xρ,g-rρ,A\right) converges to a Poisson distribution with mean e^-b ; 2) asymptotically almost surely (a.a.s.) there is no component in \cal G\left (\cal Xρ,g-rρ,A\right) 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 cal G\left (cal Xρ,g rρ,A\right) to be a.a.s. connected is that there is no isolated node in the network, which occurs when b→ asρ. 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.