Fractal dimensions for iterated graph systems

Building upon Li & Britz (Li NZ, Britz T. 2024 On the scale-freeness of random colored substitution networks. Proc. Amer. Math. Soc . 152 , 1377–1389 (doi: 10.1090/proc/16604 )), 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 iterated graph systems (IGS) for the first time. 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 distance and diameter, we derive two primary outcomes. Firstly, within the deterministic IGS, we establish that the Minkowski dimension and Hausdorff dimension align through explicit formulae. Secondly, in the case of random IGS, we demonstrate that almost every Gromov–Hausdorff scaling limit exhibits identical Minkowski and Hausdorff dimensions analytically by their Lyapunov exponents. The exploration of IGS 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.