A growth model of community graph with a degree distribution consisting of two distinct parts

Many complex systems can be represented as complex networks. Among them there has been much interest on community structure recently, and many studies focus attention on it, in particular community detection. While community detection can provide us much information, the community structure implies another feature, hierarchy of the system. Coarse-graining of complex networks can lead us to the definition of community graph. The empirical degree distribution of community graph has a unique nature, where it consists of two distinct parts, exponential and power law distribution. In this paper, we propose a modified model of community graph [Pollner et al. Europhys. Lett. 73 (2006) 478.] that mimics the empirical features of it. The growth mechanism of the model is a combination of preferential and non-preferential attachment in a higher level. We show that the model can reproduce the unique degree distribution by theoretical and numerical analysis. While such features might stem from some other reasons, we would expect to provide a unique aspect of community graph.