A family of small-world network models built by complete graph and iteration-function

Small-world networks are popular in real-life complex systems. In the past few decades, researchers presented amounts of small-world models, in which some are stochastic and the rest are deterministic. In comparison with random models, it is not only convenient but also interesting to study the topological properties of deterministic models in some fields, such as graph theory, theorem computer sciences and so on. As another concerned darling in current researches, community structure (modular topology) is referred to as an useful statistical parameter to uncover the operating functions of network. So, building and studying such models with community structure and small-world character will be a demanded task. Hence, in this article, we build a family of sparse network space N(t) which is different from those previous deterministic models. Even though, our models are established in the same way as them, iterative generation. By randomly connecting manner in each time step, every resulting member in N(t) has no absolutely self-similar feature widely shared in a large number of previous models. This makes our insight not into discussing a class certain model, but into investigating a group various ones spanning a network space. Somewhat surprisingly, our results prove all members of N(t) to possess some similar characters: (a) sparsity, (b) exponential-scale feature P(k)∼α−k, and (c) small-world property. Here, we must stress a very screming, but intriguing, phenomenon that the difference of average path length (APL) between any two members in N(t) is quite small, which indicates this random connecting way among members has no great effect on APL. At the end of this article, as a new topological parameter correlated to reliability, synchronization capability and diffusion properties of networks, the number of spanning trees on a representative member NB(t) of N(t) is studied in detail, then an exact analytical solution for its spanning trees entropy is also obtained.