Computer Science – Networking and Internet Architecture
Scientific paper
2010-12-28
Computer Science
Networking and Internet Architecture
9 pages, to appear in IEEE INFOCOM 2011, Shanghai, China
Scientific paper
Consider a network where all nodes are distributed on a unit square following a Poisson distribution with known density $\rho$ and a pair of nodes separated by an Euclidean distance $x$ are directly connected with probability $g(\frac{x}{r_{\rho}})$, where $g:[0,\infty)\rightarrow[0,1]$ satisfies three conditions: rotational invariance, non-increasing monotonicity and integral boundedness, $r_{\rho}=\sqrt{\frac{\log\rho+b}{C\rho}}$, $C=\int_{\Re^{2}}g(\Vert \boldsymbol{x}\Vert)d\boldsymbol{x}$ and $b$ is a constant, independent of the event that another pair of nodes are directly connected. In this paper, we analyze the asymptotic distribution of the number of isolated nodes in the above network using the Chen-Stein technique and the impact of the boundary effect on the number of isolated nodes as $\rho\rightarrow\infty$. On that basis we derive a necessary condition for the above network to be asymptotically almost surely connected. These results form an important link in expanding recent results on the connectivity of the random geometric graphs from the commonly used unit disk model to the more generic and more practical random connection model.
Anderson Brian Do
Mao Guoqiang
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