Transfinite fractal dimension of trees and hierarchical scale-free graphs

TL;DR

This paper introduces the transfinite fractal dimension for graph sequences, especially trees, modifying existing fractal definitions to account for exponential growth, and computes this parameter for various models.

Contribution

It proposes a modified fractal dimension concept suitable for exponentially growing tree-like networks and calculates this parameter for several complex graph models.

Findings

Transfinite fractal dimension accounts for exponential growth in trees.

The parameter relates to the growth rate of hierarchical structures.

Explicit dimensions are computed for multiple graph models.

Abstract

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 Komj\'athy and Simon, Song-Havlin-Makse model, spherically symmetric trees, and supercritical Galton-Watson trees.