加权双曲随机图和几何非齐次随机图的爆炸
Explosion in weighted Hyperbolic Random Graphs and Geometric Inhomogeneous Random Graphs
摘要：
本文研究了无标度空间网络模型：双曲随机图（HRG）、几何非齐次随机图（GIRG）和无标度逾渗（SFP）中的加权距离。在HRG中，$n=Θ(\mathrm{e}^{R/2})$顶点独立于半径为$R$的双曲盘采样，并且两个顶点在双曲距离$R$内时连接，或者以取决于双曲距离的概率独立地连接。在GIRG和SFP中，每个顶点分别从底层测量度量空间和$\mathbb{Z}^d$中获得独立的权重和位置，并且两个顶点以作为其距离和权重的函数的概率独立连接。我们分配身份证。权的随机图的边，并研究了两个均匀选择的顶点之间的加权距离。在SFP中，我们研究了范数趋于无穷大的顶点序列到原点的加权距离。特别是，我们研究的情况下，当参数是这样的图中的度分布遵循幂律指数$τ\in(2,3)$（无穷方差），和边缘权重分布是这样的，它产生一个爆炸性的年龄相关的分支过程与幂律后代分布。我们证明了在所有三个模型中，巨/无限分量内的典型距离在分布上收敛，解决了[无标度渗流中的爆炸和距离（2017）]中的一个悬而未决的问题。我们证明的主要工具是将模型耦合到无限版本，遵循最短路径到无穷大，并使用图上的权重相关渗透连接这些路径：以更高的概率删除连接到具有更高权重的顶点的边。我们使用边权重来实现这一点：只有连接到高权重顶点的短边才会保留，从而产生任意短的连接上界。
英文摘要：
In this paper we study weighted distances in scale-free spatial network models: hyperbolic random graphs (HRG), geometric inhomogeneous random graphs (GIRG) and scale-free percolation (SFP). In HRGs, $n=Θ(\mathrm{e}^{R/2})$ vertices are sampled independently from the hyperbolic disk with radius $R$ and two vertices are connected either when they are within hyperbolic distance $R$, or independently with a probability depending on the hyperbolic distance. In GIRGs and SFP, each vertex is given an independent weight and location from an underlying measured metric space and $\mathbb{Z}^d$, respectively, and two vertices are connected independently with a probability that is a function of their distance and weights. We assign i.i.d. weights to the edges of the random graphs and study the weighted distance between two uniformly chosen vertices. In SFP, we study the weighted distance from the origin of vertex-sequences with norm tending to infinity. In particular, we study the case when the parameters are so that the degree distribution in the graph follows a power law with exponent $τ\in(2,3)$ (infinite variance), and the edge-weight distribution is such that it produces an explosive age-dependent branching process with power-law offspring distribution. We show that in all three models, typical distances within the giant/infinite component converge in distribution, solving an open question in [Explosion and distances in scale-free percolation (2017)]. The main tools of our proof are to couple the models to infinite versions, to follow the shortest paths to infinity and to connect these paths using weight-dependent percolation on the graphs: delete edges attached to vertices with higher weight with higher probability. We realise this using the edge-weights: only short edges connected to high weight vertices will stay, yielding arbitrarily short upper bounds for the connections.
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