树与层次无标度图的超限分形维数
Transfinite fractal dimension of trees and hierarchical scale-free graphs
摘要：
本文在Song等人提出的复杂网络分形概念的基础上，引入了一个新的概念：图序列的超限分形维数，指出分形的定义不适用于具有“树状”结构和邻域指数增长性的网络.然而，我们表明，分形维数的定义可以修改的方式，考虑到指数增长性，并与修改后的定义，分形维数成为一个适当的参数的图序列。我们发现这个参数与树木的生长速率有关。进一步推广了盒维数的概念，引入了超限Cesaro分形维数。通过严格的证明，我们确定了各种模型的最佳盒覆盖和超限分形维数：由Komjáthy和Simon引入的层次图序列模型，Song-Havlin-Makse模型，球对称树和超临界Galton-Watson树。
英文摘要：
In this paper, we introduce a new concept: the transfinite fractal dimension of graph sequences motivated by the notion of fractality of complex networks proposed by Song et al. We show that the definition of fractality cannot be applied to networks with `tree-like' structure and exponential growth rate of neighborhoods. However, we show that the definition of fractal dimension could be modified in a way that takes into account the exponential growth, and with the modified definition, the fractal dimension becomes a proper parameter of graph sequences. We find that this parameter is related to the growth rate of trees. We also generalize the concept of box dimension further and introduce the transfinite Cesaro fractal dimension. Using rigorous proofs we determine the optimal box-covering and transfinite fractal dimension of various models: the hierarchical graph sequence model introduced by Komjáthy and Simon, Song-Havlin-Makse model, spherically symmetric trees, and supercritical Galton-Watson trees.
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