Growth model for fractal scale-free networks generated by a random walk

The diversity of networked systems with fractal structures suggests that knowing the underlying mechanism that generates the fractality is necessary for building a model of the development of complex networks. In the present paper, we propose a growth model of a network generated by a random walk and show that the evolving graph forms a fractal structure with various properties including the scale-free property, if the graph which provides a space where a random walk occurs by itself is formed by the random walk. The proposed model is regulated by two parameters p(v) and P-e, which define the probability of either a roundabout path via a new vertex or a shortcut being formed by the random walk, respectively. The power-law exponent gamma describing the vertex degree distribution is determined by the ratio P-e/P-v and is related to an internal factor F-I via the relation gamma = 1/F-I + 1, where F-I is a parameter that describes the local structure generated by the random walk. A sufficiently small p(v) provides the small-world property to the model network. The small-world property is usually considered to be incompatible with the fractal scaling property (M-c) similar to l(c)(dc), where < M-c > is the average number of vertices which can be reached from a randomly chosen vertex in at most I-c steps. However, we demonstrate that fractality can be reconciled with the small-world property by introducing a size-dependent fractal cluster dimension d(c). (C) 2019 Elsevier B.V. All rights reserved.