The configuration model generates random graphs with any given degree distribution, and thus serves as a null model for scale-free networks with power-law degrees and unbounded degree fluctuations. For this setting, we study the local clustering c(k), i.e., the probability that two neighbors of a degree-k node are neighbors themselves. We show that c(k) progressively falls off with k and the graph size n and eventually for $$k=\varOmega (\sqrt{n})$$k=Ω(n) settles on a power law $$c(k)\sim n^{5-2\tau }k^{-2(3-\tau )}$$c(k)∼n5-2τk-2(3-τ) with $$\tau \in (2,3)$$τ∈(2,3) the power-law exponent of the degree distribution. This fall-off has been observed in the majority of real-world networks and signals the presence of modular or hierarchical structure. Our results agree with recent results for the hidden-variable model and also give the expected number of triangles in the configuration model when counting triangles only once despite the presence of multi-edges. We show that only triangles consisting of triplets with uniquely specified degrees contribute to the triangle counting.

中文翻译：

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具有无限度波动的配置模型中的三元闭包

配置模型生成具有任何给定度分布的随机图，因此用作具有幂律度和无界度波动的无标度网络的零模型。对于这个设置，我们研究局部聚类 c(k)，即一个 k 度节点的两个邻居本身就是邻居的概率。我们证明 c(k) 随着 k 和图大小 n 逐渐下降，最终对于 $$k=\varOmega (\sqrt{n})$$k=Ω(n) 建立在幂律 $$c( k)\sim n^{5-2\tau }k^{-2(3-\tau )}$$c(k)∼n5-2τk-2(3-τ) 与 $$\tau \in ( 2,3)$$τ∈(2,3) 度分布的幂律指数。在大多数现实世界的网络中都观察到了这种下降，这表明存在模块化或分层结构。我们的结果与隐藏变量模型的最新结果一致，并且在尽管存在多边的情况下仅对三角形计数一次时，还给出了配置模型中三角形的预期数量。我们表明只有由具有唯一指定度数的三元组组成的三角形才有助于三角形计数。