Fractal dimensions for Iterated Graph Systems

TL;DR

This paper introduces fractal geometry into graph theory by analyzing deterministic and random iterated graph systems, establishing formulas for their fractal dimensions and exploring implications for complex networks.

Contribution

It develops a theoretical framework linking fractal dimensions with iterated graph systems, including explicit formulas and dimension analysis for both deterministic and random cases.

Findings

Minkowski and Hausdorff dimensions coincide in deterministic systems

Almost all random iterated graph systems have identical fractal dimensions

Explicit formulas relate graph properties to fractal dimensions

Abstract

Building upon [1], this study aims to introduce fractal geometry into graph theory, and to establish a potential theoretical foundation for complex networks. Specifically, we employ the method of substitution to create and explore fractal-like graphs, termed deterministic or random iterated graph systems. While the concept of substitution is commonplace in fractal geometry and dynamical systems, its analysis in the context of graph theory remains a nascent field. By delving into the properties of these systems, including diameter and distal, we derive two primary outcomes. Firstly, within the deterministic iterated graph systems, we establish that the Minkowski dimension and Hausdorff dimension align analytically through explicit formulae. Secondly, in the case of random iterated graph systems, we demonstrate that almost every graph limit exhibits identical Minkowski and Hausdorff dimensions numerically by their Lyapunov exponents. The exploration of iterated graph systems holds the potential to unveil novel directions. These findings not only, mathematically, contribute to our understanding of the interplay between fractals and graphs, but also, physically, suggest promising avenues for applications for complex networks.