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

Abstract

• We generalise ( u, v )-flowers by introducing probabilities into u and v . • Small-world property provides different values of different fractal dimensions. • Value of fractal cluster dimension reflects the behaviour of the mean path length. • Value of fractal box dimension reflects the behaviour of the diameter of the graph. 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 ∼ N 1 / d L , and the degree distribution with a power-law form. We show that the difference between the fractal cluster d c and the fractal box d b 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 d b shows a size-dependent increase with an upper limit d L . The invariance and equivalence of d c , d b and d L 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.