We study three preferential attachment models where the parameters are such that the asymptotic degree distribution has infinite variance. Every edge is equipped with a nonnegative i.i.d. weight. We study the weighted distance between two vertices chosen uniformly at random, the typical weighted distance, and the number of edges on this path, the typical hopcount. We prove that there are precisely two universality classes of weight distributions, called the explosive and conservative class. In the explosive class, we show that the typical weighted distance converges in distribution to the sum of two i.i.d. finite random variables. In the conservative class, we prove that the typical weighted distance tends to infinity, and we give an explicit expression for the main growth term, as well as for the hopcount. Under a mild assumption on the weight distribution the fluctuations around the main term are tight.

中文翻译：

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无标度优先附件模型中的加权距离

我们研究了三个优先附着模型，其中参数使得渐近度分布具有无限方差。每个边缘都配备有非负Iid重量。我们研究了随机选择的两个顶点之间的加权距离，典型的加权距离以及该路径上的边数（典型的跳数）。我们证明，权重分布恰好有两个通用类，称为爆炸性类和保守类。在爆炸类中，我们证明了典型的加权距离在分布上收敛于两个iid有限项之和随机变量。在保守类中，我们证明了典型的加权距离趋于无穷大，并且给出了主要增长项以及跳数的明确表示。在权重分配的温和假设下，主要项附近的波动很小。