Modified box dimension of trees and hierarchical scale-free graphs
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In this paper, we introduce a new concept: the modified box 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. 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 modified Cesaro box dimension. Using rigorous proofs we determine the optimal box covering and modified box 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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