Fractal networks induced by movements of random walkers on a tree graph

We show that a fractal, scale-free, and small-world network can grow on a non-fractal Cayley tree through the simple random process of shortcut creation by movements of random walkers. The Cayley tree provides a stage for the walkers to diffuse into a locally one-dimensional structure with the small-world property. Self-organization of fractal graph by adding shortcut edges to the locally one-dimensional structure generated by walkers’ flow from the root vertex to the outermost shell of the Cayley tree is a novel mechanism for generating fractal graphs. Also, we show numerically that the fractal box counting dimension and a power-law exponent describing the degree distribution of the resulting graph are mainly determined by the bifurcation number of the original Cayley tree. In our model, the fractal cluster dimension must change with the size of the evolving graph. This is a general result that resolves an apparent contradiction between the fractality in the context of the cluster-growing method and the small-world property of the graph.