It is well known that many random graphs with infinite variance degrees are ultra-small. More precisely, for configuration models and preferential attachment models where the proportion of vertices of degree at least k is approximately k -(τ - 1) with τ ∈ (2,3), typical distances between pairs of vertices in a graph of size n are asymptotic to 2 log log n | log ( τ - 2 ) | and 4 log log n | log ( τ - 2 ) | , respectively. In this paper, we investigate the behavior of the diameter in such models. We show that the diameter is of order log log n precisely when the minimal forward degree d fwd of vertices is at least 2. We identify the exact constant, which equals that of the typical distances plus 2 / log d fwd . Interestingly, the proof for both models follows identical steps, even though the models are quite different in nature.

中文翻译：

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超小无标度随机图中的直径。

众所周知，许多方差度无限的随机图都是超小的。更准确地说，对于配置模型和优先附着模型，其中度数至少为 k 的顶点比例大约为 k -(τ - 1) 且 τ ε (2,3)，大小为 n 的图中顶点对之间的典型距离渐近至 2 log log n | 日志 ( τ - 2 ) | 和 4 log log n | 日志 ( τ - 2 ) | ， 分别。在本文中，我们研究了此类模型中直径的行为。我们证明，当顶点的最小前向度 d fwd 至少为 2 时，直径的数量级恰好为 log log n。我们确定了精确的常数，它等于典型距离的常数加上 2 / log d fwd 。有趣的是，尽管模型本质上有很大不同，但两个模型的证明都遵循相同的步骤。