The most important structural characteristics in the study of large-scale real-world complex networks in pattern analysis are degree distribution. Empirical observations on the pattern of the real-world networks have led to the claim that their degree distributions follow, in general, a single power law. However, a closer observation, while fitting, shows that the single power-law distribution is often inadequate to meet the data characteristics properly. Since the degree distribution in the log–log scale actually displays, under inspection, two different slopes unlike what happens while fitting with the single power law. These two slopes with a transition in between closely resemble the pattern of the sigmoid function. This motivates us to derive a novel double power-law distribution for accurately modeling the real-world networks based on the sigmoid function. The proposed modeling approach further leads to the identification of a transition degree which, it has been demonstrated, may have a significant implication in analyzing the complex networks. The applicability, as well as effectiveness of the proposed methodology, is shown using rigorous experiments and also validated using statistical tests.

中文翻译：

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在现实世界中复杂网络的度分布中寻找模式：超越幂律

在模式分析中研究大型现实世界复杂网络时，最重要的结构特征是度分布。对现实世界网络模式的实证观察表明，它们的度分布通常遵循单一幂定律。但是，在进行拟合的同时仔细观察表明，单次幂律分布通常不足以正确满足数据特征。由于对数-对数刻度的度数分布实际显示，在检查中，有两个不同的斜率，与采用单一幂定律时的情况不同。这两个之间有过渡的斜率非常类似于S型函数的模式。这激励我们推导出新颖的双幂律分布，以便基于S型函数对真实世界的网络进行精确建模。所提出的建模方法进一步导致了对过渡程度的识别，已经证明，该过渡程度可能在分析复杂网络中具有重要意义。使用严格的实验显示了所提出方法的适用性和有效性，并使用统计测试对其进行了验证。