随机连接模型下随机网络的渐近连通性
On the Asymptotic Connectivity of Random Networks under the Random Connection Model
摘要：
考虑一个网络，其中所有节点都分布在一个单位正方形上，遵循具有已知密度$ρ$的泊松分布，并且由欧几里得距离$x$分开的一对节点以概率$g(\frac{x}{r_ρ})$直接连接，其中$g:[0,\infty)\rightarrow[0,1]$满足三个条件：旋转不变性、非增单调性和积分有界性，$r_ρ=\sqrt{\frac{\logρ+b}{Cρ}}$、$C=\int_{\Re^{2}}g(\Vert \boldsymbol{x}\Vert)d\boldsymbol{x}$和$b$是常数，与另一对节点直接连接的情况无关。本文利用Chen-Stein技巧分析了上述网络中孤立节点数的渐近分布，并将边界效应对孤立节点数的影响作为$ρ\rightarrow\infty$.在此基础上，我们得到了上述网络渐近几乎必然连通的一个必要条件。这些结果形成了一个重要的环节，在扩大最近的结果连接的随机几何图形从常用的单位盘模型，更通用和更实用的随机连接模型。
英文摘要：
Consider a network where all nodes are distributed on a unit square following a Poisson distribution with known density $ρ$ and a pair of nodes separated by an Euclidean distance $x$ are directly connected with probability $g(\frac{x}{r_ρ})$, where $g:[0,\infty)\rightarrow[0,1]$ satisfies three conditions: rotational invariance, non-increasing monotonicity and integral boundedness, $r_ρ=\sqrt{\frac{\logρ+b}{Cρ}}$, $C=\int_{\Re^{2}}g(\Vert \boldsymbol{x}\Vert)d\boldsymbol{x}$ 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 $ρ\rightarrow\infty$. 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.
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