Growth model for complex networks with hierarchical and modular structures

A hierarchical and modular network model is suggested by adding a growth rule along with the preferential attachment (PA) rule into Motter's modeling procedure. The proposed model has an increasing number of vertices but a fixed number of modules and hierarchical levels. The vertices form lowest-level modules which in turn constitute higher-level modules hierarchically. The creation of connections between two vertices in a single module or in two different modules of the same level obeys the PA rule. The structural characteristics of this model are investigated analytically and numerically. The results show that the degree distribution, the module size distribution, and the clustering function of the model possess a power-law property which is similar to that in many real-world networks. The model is then used to predict the growth trends of real-world networks with modular and hierarchical structures. By comparing this model with those real-world networks, an interesting conclusion is found: that many real-world networks are in their early stages of development, and when the growth time is large enough, the modules and levels of the networks will be ultimately merged.