Connectivity of Large Wireless Networks Under A General Connection Model
Guoqiang Mao, Brian D. O. Anderson · IEEE Transactions on Information Theory · 2012
This paper studies networks where all nodes are distributed on a unit squareA=Δ[- [1/2], [1/2]]2following a Poisson distribution with known density ρ and a pair of nodes separated by an Euclidean distancexare directly connected with probabilitygrρ(x)=Δg(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, andg(x)=o(1/(x2log2x)) ; further,rρ=√{(logρ+b)/(Cρ)} whereC=∫ℜ2g(||x||)dxandbis a constant. Denote the aforementioned network byG(Xρ,grρ,A). We show that as ρ→ ∞, 1) the distribution of the number of isolated nodes inG(Xρ,grρ,A) converges to a Poisson distribution with meane-b; 2) asymptotically almost surely (a.a.s.) there is no component inG(Xρ,grρ,A) of fixed and finite orderk>; 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 forG(Xρ,grρ,A) to be a.a.s. connected is that there is no isolated node in the network, which occurs whenb→ ∞ 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.