On the scale-freeness of random colored substitution networks

Abstract

Extending previous results in the literature, random colored substitution networks are introduced and are proved to be scale-free under natural conditions. Furthermore, the asymptotic node degrees, arc cardinalities and node cardinalities for these networks are derived. These results are achieved by proving stronger results regarding stochastic substitution processes, which form a new stochastic model that is here introduced.Many real-life phenomena are fractal in nature, including growth networks found in biology, brain connections and in social interactions. To study the properties of these networks, previous researchers introduced a mathematical model called substitution networks, to simulate the growth of the networks by iteratively replacing each arcs of a network by smaller networks. This model was later expanded by the introduction of arc colors to allow more types of arc replacements. However, these models are deterministic and do not allow for the randomness that real-life growth networks can exhibit. To capture this randomness, we expand the model to what we call random colored substitution networks, by allowing each arc to be replaced by a random choice of network. We describe properties of the randomly resulting networks, including their number of nodes and arcs and their node degrees. Our main result shows that these random colored substitution networks are scale-free and that they therefore have a particular type of structure.