van der Hofstad Remco, van Leeuwaarden Johan S H, Stegehuis Clara
Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands.
J Stat Phys. 2018;173(3):746-774. doi: 10.1007/s10955-018-1952-x. Epub 2018 Jan 25.
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 (), i.e., the probability that two neighbors of a degree- node are neighbors themselves. We show that () progressively falls off with and the graph size and eventually for settles on a power law with 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.
配置模型生成具有任何给定度分布的随机图，因此可作为具有幂律度和无界度波动的无标度网络的零模型。对于这种情况，我们研究局部聚类（），即度为节点的两个邻居本身也是邻居的概率。我们表明（）随着和图大小逐渐下降，最终对于，稳定在幂律上，其中是度分布的幂律指数。这种下降在大多数现实世界网络中都有观察到，表明存在模块化或层次结构。我们的结果与隐藏变量模型的近期结果一致，并且在存在多重边的情况下仅计算一次三角形时，也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的三元组组成的三角形才对三角形计数有贡献。
