Generating multi-scaling networks with two types of vertices

A variety of scale-free networks have been created since the pioneer work by Barabási and Albert [Science 286 (1999) 509]. Most of these models are homogeneous since they are composed of the same kind of nodes. In the realistic world, however, elements (nodes or vertices) in the network may play different roles or have different functions. In this work, we develop an alternative way of vertex classification other than the ordinary modularity method by introducing two types of vertices. The interaction between two neighbor vertices is dependent on their types. It is found that the vertex degree exhibits a multi-scaling law distribution with the scaling exponent of each types of vertex adjustable. This network model may exhibit some interesting properties concerning the dynamical processes on it.