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, $L\sim N^{1/d_{\mathrm{L}}},$ 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.

所谓的（u，v）花是递归网络，它们产生具有分形或小世界性质的自相似结构。本文通过将概率引入u和v的实现中来概括（u，v）-花，这使得能够研究分形网络中小世界与非小世界之间的中间状态。我们获得该曲线图的直径之间的关系的分析大号和图形尺寸Ñ，$\mathrm{大号}〜{ñ}^{\mathrm{1个}/d_{\mathrm{大号}}}，$以及具有幂律形式的度分布。我们表明，分形簇d _c和分形盒d _b尺寸之间的差异反映了平均路径长度〈l〉和L的不同行为。在分形性和小世界性质之间似乎存在明显的矛盾。但是，小世界属性可以通过与尺寸有关的分形维数与图的分形相协调，其中d _b显示随上限d _L的与尺寸有关的增加 。d _c，d _b和d _L的不变性和等价性仅当< l >和L都经受相同的非小世界行为时，才维护它们。我们的研究为解释经验分形数据提供了有用的信息，并为研究网络中发生的各种动力学提供了基本工具。