论文标题
软随机几何图的巨型组件
Giant component of the soft random geometric graph
论文作者
论文摘要
考虑一个二维软的随机几何图$ g（λ，s，ϕ）$，通过将泊松（$λs^2 $）在侧面$ s $的正方形中均匀地放置在随机的边缘上，将每个对$ x，y $ $ x，y $ nimagibalition $ x y $ x（\ x-y bf | x-y \ | [0,1] $是一个有限范围的连接功能。本文涉及图形$ g（λ，s，ϕ）$的渐近行为，并以$（λ，ϕ）$固定为$ s $限制。我们证明，最大组件中顶点的比例以相应的随机连接模型的概率收敛到渗透概率，这是整个平面上泊松过程类似定义的随机图。我们不涵盖$λ$等于关键值$λ_c（ϕ）$的情况。
Consider a 2-dimensional soft random geometric graph $G(λ,s,ϕ)$, obtained by placing a Poisson($λs^2$) number of vertices uniformly at random in a square of side $s$, with edges placed between each pair $x,y$ of vertices with probability $ϕ(\|x-y\|)$, where $ϕ: {\bf R}_+ \to [0,1]$ is a finite-range connection function. This paper is concerned with the asymptotic behaviour of the graph $G(λ,s,ϕ)$ in the large-$s$ limit with $(λ,ϕ)$ fixed. We prove that the proportion of vertices in the largest component converges in probability to the percolation probability for the corresponding random connection model, which is a random graph defined similarly for a Poisson process on the whole plane. We do not cover the case where $λ$ equals the critical value $λ_c(ϕ)$.
