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 pv and pe, 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 γ describing the vertex degree distribution is determined by the ratio pe∕pv and is related to an internal factor FI via the relation γ=1∕FI+1, where FI is a parameter that describes the local structure generated by the random walk. A sufficiently small pv provides the small-world property to the model network. The small-world property is usually considered to be incompatible with the fractal scaling property Mc∼lcdc, where Mc is the average number of vertices which can be reached from a randomly chosen vertex in at most lc steps. However, we demonstrate that fractality can be reconciled with the small-world property by introducing a size-dependent fractal cluster dimension dc.