Inhomogeneous spatial networks - beyond the weak decay regime

只要研究相互作用的对象的大系统，整个科学领域就会出现复杂的网络。在这个项目中，我们研究了一类丰富的数学网络模型，它们满足三个基本原则：无标度、小世界性质和聚类性。为了生成模型，我们形成由随机分布的顶点组成的图，这些顶点配备了一个权重，然后通过边连接起来。每条边的生成既考虑了其端点之间的空间距离，也考虑了它们的权重。我们的目标是通过回答以下问题来了解这些模型的行为：何时存在大型连接组件？它们的拓扑属性是什么，例如连通性和点间图距离？它们在本质上是更像晶格，还是更像树？如果我们随机移除它们的边缘，会发生什么？

Complex Networks arise throughout the sciences wherever large systems of interacting objects are investigated. In this project we study a rich class of mathematical network models that satisfy three basic principles: scale-freeness, the small-world property and clustering. To generate the models we form graphs consisting of randomly distributed vertices equipped with a weight, which are then connected by edges. The generation of each edge takes into account both the spatial distance between its endvertices as well as their weight. Our goal is to understand the behavior of these models by answering questions such as: When are there large connected components? What are their topological properties, e.g. connectivity and interpoint graph distances? Do they qualitatively behave more like lattices or more like trees? And what happens if we randomly remove edges from them?