Spatial age-dependent random graphs

我们研究了一组嵌入到d维环面中的生长图，其中新顶点以恒定速率到达，随机放置在环面上，并以有利于短边和连接到旧顶点的概率独立连接到每个现有顶点。这种具有空间诱导聚类的无标度网络的简单模型称为基于年龄的空间优先依恋模型。我们对空间相关性对图拓扑的影响感兴趣，并研究攻击下的鲁棒性，典型顶点之间最短路径的长度以及网络上信息的传播。特别地，我们研究了图上接触过程的亚稳态，以及在图顶点按照独立布朗运动运动的情况下信息的传递方式。

We investigate a family of growing graphs embedded into a d-dimensional torus where new vertices arrive at constant rate, are randomly placed on the torus and connect to each existing vertex independently with a probability favouring short edges and connections to old vertices. This simple model for a scale-free network with spatially induced clustering is called the age-based spatial preferential attachment model. We are interested in the effect of the spatial correlations on the topology of the graph and study robustness under attack, length of the shortest paths between typical vertices, and spreading of information on the network. In particular, we study metastability of the contact process on the graph and the way information can be transmitted in the case that graph vertices move according to independent Brownian motions.