  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.
