分形复杂网络的标度理论
Scaling theory of fractal complex networks
AI总结：
本文基于相变标度理论和重整化群，提出分形复杂网络的标度理论，引入微观与宏观两类标度指数并揭示其关系，在真实网络和模型中验证了局部自相似性与全局标度不变性的联系。
摘要：
我们证明，复杂网络中的分形性源于其内在层次化类社区结构的几何自相似性，该结构在数学上由覆盖网络以确定其盒维数时所用盒质量的标度不变方程描述。这一方法建立在相变标度理论和重整化群理论之上，从而形成了一致的分形复杂网络标度理论，该理论用若干新的标度指数补充了现有的标度指数集合，并揭示了它们之间的各种关系。我们建议引入两类指数：微观指数和宏观指数，分别刻画分形复杂网络的局部结构和全局性质。有趣的是，这两类指数彼此相关，并且其中只有少数几个（七个中的三个）是独立的，从而在分形网络中架起了局部自相似性与全局标度不变性之间的桥梁。我们在多个领域的真实网络（信息领域——万维网、生物领域——人脑、社会领域——科学合作网络）以及若干分形网络模型中成功验证了我们的发现。
英文摘要：
We show that fractality in complex networks arises from the geometric self-similarity of their built-in hierarchical community-like structure, which is mathematically described by the scale-invariant equation for the masses of the boxes with which we cover the network when determining its box dimension. This approach - grounded in both scaling theory of phase transitions and renormalization group theory - leads to the consistent scaling theory of fractal complex networks, which complements the collection of scaling exponents with several new ones and reveals various relationships between them. We propose the introduction of two classes of exponents: microscopic and macroscopic, characterizing the local structure of fractal complex networks and their global properties, respectively. Interestingly, exponents from both classes are related to each other and only a few of them (three out of seven) are independent, thus bridging the local self-similarity and global scale-invariance in fractal networks. We successfully verify our findings in real networks situated in various fields (information - the World Wide Web, biological - the human brain, and social - scientific collaboration networks) and in several fractal network models.
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