The relations between network-operation and topological-property in a scale-free and small-world network with community structure

It is always an open, demanding and difficult task for generating available model to simulate dynamical functions and reveal inner principles from complex systems and networks. In this article, due to lots of real-life and artificial networks are built from series of simple and small groups (components), we discuss some interesting and helpful network-operation to generate more realistic network models. In view of community structure (modular topology), we present a class of sparse network models N(t,m). At the moment, we capture the fact the N(t,4) has not only scale-free feature, which means that the probability that a randomly selected vertex with degree k decays as a power-law, following P(k)∼k−γ, where γ is the degree exponent, but also small-world property, which indicates that the typical distance between two uniform randomly chosen vertices grows proportionally to logarithm of the order of N(t,4), namely, relatively shorter diameter and lower average path length, simultaneously displays higher clustering coefficient. Next, as a new topological parameter correlating to reliability, synchronization capability and diffusion properties of networks, the number of spanning trees over a network is studied in more detail, an exact analytical solution for the number of spanning trees of the N(t,4) is obtained. Based on the network-operation, part hub-vertex linking with each other will be helpful for structuring various network models and investigating the rules related with real-life networks.