Random geometric graphs consist of randomly distributed nodes (points), with pairs of nodes within a given mutual distance linked. In the usual model the distribution of nodes is uniform on a square, and in the limit of infinitely many nodes and shrinking linking range, the number of isolated nodes is Poisson distributed, and the probability of no isolated nodes is equal to the probability the whole graph is connected. Here we examine these properties for several self-similar node distributions, including smooth and fractal, uniform and nonuniform, and finitely ramified or otherwise. We show that nonuniformity can break the Poisson distribution property, but it strengthens the link between isolation and connectivity. It also stretches out the connectivity transition. Finite ramification is another mechanism for lack of connectivity. The same considerations apply to fractal distributions as smooth, with some technical differences in evaluation of the integrals and analytical arguments.

中文翻译：

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具有自相似强度测度的随机几何图中的隔离和连通性

随机几何图形由随机分布的节点（点）组成，给定相互距离内的节点对相互连接。在通常的模型中，节点在正方形上的分布是均匀的，在无限多个节点和收缩连接范围的限制下，孤立节点的数量是泊松分布的，没有孤立节点的概率等于整体的概率图是连通的。在这里，我们检查了几个自相似节点分布的这些属性，包括平滑和分形、均匀和非均匀以及有限分支或其他。我们表明不均匀性可以破坏泊松分布特性，但它加强了隔离和连通性之间的联系。它还延长了连接转换。有限分枝是缺乏连通性的另一种机制。