论文标题
无尺度随机连接模型中的泊松近似和连通性
Poisson Approximation and Connectivity in a Scale-free Random Connection Model
论文作者
论文摘要
我们研究了连通性制度中的不均匀随机连接模型。该图的顶点集是一个均质的泊松点过程$ \ MATHCAL {p} _s of强度的强度$ s> 0 $在单位立方体$ s = \ lest（ - \ frac {1} {2} {2} {2}，\ frac {1} {1} {2} {2} {2} {2} {2} \ oright $ n e n e n e n ender under nectert and under。分布为$ w $，其中$ p（w> w）= w^{ - β} 1 _ {[1，\ infty）}（w）$，$β> 0 $。给定顶点集和权重的边缘之间存在$ x，y \ in \ nathcal {p} _s $，概率$ \ left（1 - \ exp \ left（ - \ frac {ηw_xw_y} 0 $，$ d（\ cdot，\ cdot）$是$ s $上的圆环公制，$ r> 0 $是缩放参数。我们在$α，β$上得出条件，以便在缩放$ r_s（ξ）^d = \ frac {1} {c_0 s}} \ left（\ log s +s +（k-1）\ log \ log s +s +s +s +s +n log \ log \ log \ lest（ \ Mathbb {r} $，度$ k $的顶点的总变化距离与Poisson随机变量的收敛，其平均$ e^{ - ξ} $ as $ s \ to \ infty $，其中$ c_0 $是明确指定的常数，取决于$α，d $和$α，d $ and $ note $ notects $ c_0 $。特别是，对于$ k = 0 $，我们获得了隔离节点稳定数量的制度，这是建立连接阈值的先驱。我们还得出了足够的条件，可以使图形与大$ S $相连。使用Stein的方法得出泊松近似结果。
We study an inhomogeneous random connection model in the connectivity regime. The vertex set of the graph is a homogeneous Poisson point process $\mathcal{P}_s$ of intensity $s>0$ on the unit cube $S=\left(-\frac{1}{2},\frac{1}{2}\right]^{d},$ $d \geq 2$ . Each vertex is endowed with an independent random weight distributed as $W$, where $P(W>w)=w^{-β}1_{[1,\infty)}(w)$, $β>0$. Given the vertex set and the weights an edge exists between $x,y\in \mathcal{P}_s$ with probability $\left(1 - \exp\left( - \frac{ηW_xW_y}{\left(d(x,y)/r\right)^α} \right)\right),$ independent of everything else, where $η, α> 0$, $d(\cdot, \cdot)$ is the toroidal metric on $S$ and $r > 0$ is a scaling parameter. We derive conditions on $α, β$ such that under the scaling $r_s(ξ)^d= \frac{1}{c_0 s} \left( \log s +(k-1) \log\log s +ξ+\log\left(\frac{αβ}{k!d} \right)\right),$ $ξ\in \mathbb{R}$, the number of vertices of degree $k$ converges in total variation distance to a Poisson random variable with mean $e^{-ξ}$ as $s \to \infty$, where $c_0$ is an explicitly specified constant that depends on $α, β, d$ and $η$ but not on $k$. In particular, for $k=0$ we obtain the regime in which the number of isolated nodes stabilizes, a precursor to establishing a threshold for connectivity. We also derive a sufficient condition for the graph to be connected with high probability for large $s$. The Poisson approximation result is derived using the Stein's method.
