Connectivity of Large Wireless Networks Under A General Connection Model

摘要

This paper studies networks where all nodes are distributed on a unit square $A{buildrel{triangle}over{=}} [- {{1}over {2}}, {{1}over {2}}]^{2}$ following a Poisson distribution with known density $rho$ and a pair of nodes separated by an Euclidean distance $x$ are directly connected with probability $g_{r_{rho}}(x){buildrel{triangle}over{=}} g(x/r_{rho})$, independent of the event that any other pair of nodes are directly connected. Here, $g:[0,infty)rightarrow [{0,1}]$ satisfies the conditions of rotational invariance, nonincreasing monotonicity, integral boundedness, and $gleft (xright)=o(1/(x^{2}log^{2}x))$ ; further, $r_{rho}=sqrt {(log rho +b)/(Crho)}$ where $C=int_{Re^{2}}g(left Vert {mmb {x}}right Vert)d {mmb {x}}$ and $b$ is a constant. Denote the aforementioned network by ${cal G}left ({cal X}_{rho},g_{r_{rho}},Aright)$. We show that as $rho rightarrow infty$, 1) the distribution of the number of isolated nodes in ${cal G}left ({cal X}_{rho},g_{r_{rho}},Aright)$ converges to a Poisson distribution with mean $e^{-b}$ ; 2) asymptotically almost surely (a.a.s.) there - s no component in ${cal G}left ({cal X}_{rho},g_{r_{rho}},Aright)$ of fixed and finite order $k> 1$; c) a.a.s. the number of components with an unbounded order is one. Therefore, as $rho rightarrow infty$, the network a.a.s. contains a unique unbounded component and isolated nodes only; a sufficient and necessary condition for ${cal G}left ({cal X}_{rho},g_{r_{rho}},Aright)$ to be a.a.s. connected is that there is no isolated node in the network, which occurs when $brightarrow infty$ as $rho rightarrow infty$. 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.