Connectivity of Large Scale Networks: Emergence of Unique Unbounded Component

Abstract

This paper studies networks where all nodes are distributed on a unit square $A\triangleq[(-1/2,1/2)^{2}$ following a Poisson distribution with known density $\rho$ and a pair of nodes separated by an Euclidean distance $x$ are directly connected with probability $g(\frac{x}{r_{\rho}})$, 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, non-increasing monotonicity, integral boundedness and $g(x)=o(\frac{1}{x^{2}\log^{2}x})$; further, $r_{\rho}=\sqrt{\frac{\log\rho+b}{C\rho}}$ where $C=\int_{\Re^{2}}g(\Vert \boldsymbol{x}\Vert)d\boldsymbol{x}$ and $b$ is a constant. Denote the above network by\textmd{}$\mathcal{G}(\mathcal{X}_{\rho},g_{r_{\rho}},A)$. We show that as $\rho\rightarrow\infty$, asymptotically almost surely a) there is no component in $\mathcal{G}(\mathcal{X}_{\rho},g_{r_{\rho}},A)$ of fixed and finite order $k>1$; b) the number of components with an unbounded order is one. Therefore as $\rho\rightarrow\infty$, the network asymptotically almost surely contains a unique unbounded component and isolated nodes only; a sufficient condition for $\mathcal{G}(\mathcal{X}_{\rho},g_{r_{\rho}},A)$ to be asymptotically almost surely connected is that there is no isolated node in the network.{\normalsize{}}The contribution of these results, together with results in a companion paper on the asymptotic distribution of isolated nodes in \textmd{\normalsize $\mathcal{G}(\mathcal{X}_{\rho},g_{r_{\rho}},A)$}, is to expand recent results obtained for connectivity of random geometric graphs from the unit disk model to the more generic and more practical random connection model.