Fractality and the small-world property of generalised (u, v)-flowers

So-called (u, v)-flowers are recursive networks which produce self-similar structures with fractality or the small-world property. This paper generalises (u, v)-flowers by introducing probabilities into the realisations of u and v, which enables the study of intermediate states between small and non-small worlds in fractal networks. We obtain the analytical relation between the diameter of the graph L and the graph size N, L∼N1/dL, and the degree distribution with a power-law form. We show that the difference between the fractal cluster dc and the fractal box db dimensions reflects different behaviour of the mean path length 〈l〉 and L. There seems to be an apparent contradiction between fractality and the small-world property. However, the small-world property can be reconciled with fractality of the graph by size-dependent fractal dimensions where db shows a size-dependent increase with an upper limit dL. The invariance and equivalence of dc, db and dL are maintained only when both 〈l〉 and L are subject to the same non-small-world behaviour. Our investigation provides useful information for interpreting empirical fractal data and basic tools for studying the various dynamics that occur in networks.