We investigate a class of growing graphs embedded into the $d$-dimensional torus where new vertices arrive according to a Poisson process in time, are randomly placed in space and connect to existing vertices with a probability depending on time, their spatial distance and their relative ages. This simple model for a scale-free network is called the age-based spatial preferential attachment network and is based on the idea of preferential attachment with spatially induced clustering. We show that the graphs converge weakly locally to a variant of the random connection model, which we call the age-dependent random connection model. This is a natural infinite graph on a Poisson point process where points are marked by a uniformly distributed age and connected with a probability depending on their spatial distance and both ages. We use the limiting structure to investigate asymptotic degree distribution, clustering coefficients and typical edge lengths in the age-based spatial preferential attachment network.

中文翻译：

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年龄相关的随机连接模型

我们研究了一类嵌入到 $d$ 维环面中的生长图，其中新顶点根据泊松过程及时到达，随机放置在空间中并以取决于时间、空间距离和它们的概率的概率连接到现有顶点。相对年龄。这种无标度网络的简单模型称为基于年龄的空间优先附着网络，它基于具有空间诱导聚类的优先附着思想。我们展示了图在局部弱收敛到随机连接模型的变体，我们称之为年龄相关的随机连接模型。这是泊松点过程的自然无限图，其中点由均匀分布的年龄标记，并根据它们的空间距离和两个年龄以概率连接。