Transfinite fractal dimension of trees and hierarchical scale-free graphs

Júlia Komjáthy, Roland Molontay, Károly Simon

Research output: Contribution to journalArticle

Abstract

In this article, 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 neighbourhoods. 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.

Original languageEnglish
Pages (from-to)764-791
Number of pages28
JournalJournal of Complex Networks
Volume7
Issue number5
DOIs
Publication statusPublished - Oct 1 2019

Keywords

  • fractal dimension
  • Galton-Watson tree
  • growth rate
  • hierarchical graph sequence model
  • Song-Havlin-Makse model
  • spherically symmetric tree

ASJC Scopus subject areas

  • Computer Networks and Communications
  • Management Science and Operations Research
  • Control and Optimization
  • Computational Mathematics
  • Applied Mathematics

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