We introduce a model of a preferential attachment based random graph which extends the family of models in which condensation phenomena can occur. Each vertex has an associated uniform random variable which we call its location. Our model evolves in discrete time by selecting r vertices from the graph with replacement, with probabilities proportional to their degrees plus a constant α. A new vertex joins the network and attaches to one of these vertices according to a given probability associated to the ranking of their locations. We give conditions for the occurrence of condensation, showing the existence of phase transitions in α below which condensation occurs. The condensation in our model differs from that in preferential attachment models with fitness in that the condensation can occur at a random location, that it can be due to a persistent hub, and that there can be more than one point of condensation.

中文翻译：

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优先附件模型中基于位置选择的冷凝

我们介绍了基于优先附件的随机图模型，该模型扩展了可能发生凝结现象的模型族。每个顶点都有一个关联的统一随机变量，我们称其为位置。通过从图中选择替换的r个顶点（概率与它们的度成正比，加上一个常数α），我们的模型在离散时间内演化。一个新的顶点加入网络，并根据与它们的位置排名相关的给定概率，将其附加到这些顶点之一。我们给出了发生凝结的条件，表明了α中存在相变低于此温度则发生凝结。我们模型中的凝结与具有适应性的优先附件模型的凝结不同，在于凝结可以发生在随机位置，这可能是由于持久的轮毂造成的，并且可能存在多个凝结点。