A New Model for a Scale-Free Hierarchical Structure of Isolated Cliques

Abstract

Scale-free networks are usually defined as the ones that have power-law degree distributions. Since many of real world networks such as the World Wide Web, the Internet, citation networks, biological networks, and so on, have this property in common, scale-free networks have attracted interests of researchers so far. They also revealed that such networks have some typical properties such as high cluster coefficient and small diameter as well, and a lot of network models have been proposed to explain those properties. Recently, it is reported that the following new properties about self-similar structures of a real world network are observed [Uno and Oguri, FAW and AAIM, 2011]. For a special kind of cliques in a network, 1. the size distributions of these cliques show a power-law, 2. the degree distribution of the network after contracting these cliques show a power-law, and 3. by regarding the contracted network as the original, 1 and 2 are observed repeatedly. In this paper, we propose a new network model constructed by a `clique expansion' procedure, and show that it can explain this `hierarchical structure of cliques'.