Stratified structure of fractal scale-free networks generated by local rules

Fractal networks are a special class of self-organized networks in the real world. Several network models have been proposed to explain fractality, but most of them are intentionally designed to generate self-similar structures. However, considering that most real-world networks appear to be self-organized by random and local events, random phenomena occurring in networks are expected to be more relevant for explaining fractality than intentional design. In this study, we focus on the process of the formation of stratified structures to clarify the mechanism by which local rules generate fractal properties. First, we numerically show that fractal networks are generated by local and random events by investigating two model networks, the n-step random walk (RW) model and the intermediary process (IP) model. Second, we investigate stratified structures, including the power-law forms of frequency distributions of offspring at each depth l, where the depth l is defined as the distance from the root vertices to each vertex in a family tree embedded in the resulting networks. As a result, we find that the local structures defined on each layer, such as the birth rate factor Bl, characterize the growth of stratified structures of the model networks. Finally, to demonstrate a crucial role of Bl in the emergence of fractality, we showed that a general algorithm based on the concept of birth rate factor can certainly produce fractal small-world scale-free trees. Surprisingly, random processes such as random walks induce local structures that support stratified, fractal, and small-world (or non-small-world) networks.