The putative scale-free nature of real-world networks has generated a lot of interest in the past 20 years: if networks from many different fields share a common structure, then perhaps this suggests some underlying ‘network law’. Testing the degree distribution of networks for power-law tails has been a topic of considerable discussion. Ad hoc statistical methodology has been used both to discredit power-laws as well as to support them. This paper proposes a statistical testing procedure that considers the complex issues in testing degree distributions in networks that result from observing a finite network, having dependent degree sequences and suffering from insufficient power. We focus on testing whether the tail of the empirical degrees behaves like the tail of a de Solla Price model, a two-parameter power-law distribution. We modify the well-known Kolmogorov–Smirnov test to achieve even sensitivity along the tail, considering the dependence between the empirical degrees under the null distribution, while guaranteeing sufficient power of the test. We apply the method to many empirical degree distributions. Our results show that power-law network degree distributions are not rare, classifying almost 65% of the tested networks as having a power-law tail with at least 80% power.

中文翻译：

---

幂律网络到底有多罕见？

在过去的 20 年里，现实世界网络的假定无标度性质引起了很多兴趣：如果来自许多不同领域的网络共享一个共同的结构，那么这可能暗示了一些潜在的“网络定律”。测试幂律尾网络的度分布一直是一个备受关注的话题。特设统计方法既被用来诋毁幂律，也被用来支持它们。本文提出了一种统计测试程序，该程序考虑了测试网络中度分布的复杂问题，这些问题是由于观察有限网络、具有相关度序列和功率不足而导致的。我们专注于测试经验度的尾部是否表现得像 de Solla Price 模型的尾部，这是一种双参数幂律分布。我们修改了著名的 Kolmogorov-Smirnov 检验，以实现沿尾部的均匀灵敏度，考虑到零分布下经验度之间的依赖性，同时保证检验的足够功效。我们将该方法应用于许多经验度分布。我们的结果表明，幂律网络度分布并不罕见，将近 65% 的测试网络归类为具有至少 80% 功率的幂律尾。