Connectivity of Large Wireless Networks Under A General Connection Model.

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 gr\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 g\left (x\right)=o(1/(x^2\log^2x)); further, r\rho=\sqrt {(\log \rho +b)/(C\rho)} 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,gr\rho,A\right). We show that as \rho \rightarrow \infty, 1) the distribution of the number of isolated nodes in \cal G\left (\cal X\rho,gr\rho,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 X\rho,gr\rho,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 \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,gr\rho,A\right) to be a.a.s. connected is that there is no isolated node in the network, which occurs when b\rightarrow \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. [ABSTRACT FROM AUTHOR]