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

Scale-free networks are usually defined as the ones that have powerlaw 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’. Submitted: April 2010 Revised: Jan 2011 Revised: July 2011 Accepted: July 2011 Final: July 2011 Published: October 2011 Article type: Regular paper Communicated by: M. S. Rahman and S. Fujita Research partly supported by JSPS Global COE program “Computationism as a Foundation for the Sciences” and also partly supported by Grant-in-Aid for Scientific Research (KAKENHI), No. 23500022. A preliminary version of this paper appeared in the Proceedings of WALCOM 2010 [14]. E-mail addresses: [email protected] (Takeya Shigezumi) [email protected] (Yushi Uno) [email protected] (Osamu Watanabe) 662 T. Shigezumi et al. A New Model for Isolated Cliques