IOP PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICAL J. Phys. A: Math. Theor. 42 (2009) 045103 (10pp) doi:10.1088/1751-8113/42/4/045103 Self-similar non-clustered planar graphs as models for complex networks Francesc Comellas 1 , Zhongzhi Zhang 2,3 and Lichao Chen 2,3 1 Dep. de Matem` atica Aplicada IV, EPSC, Universitat Polit` ecnica de Catalunya, Av. Canal Ol´ ımpic s/n, 08860 Castelldefels, Barcelona, Catalonia, Spain 2 School of Computer Science, Fudan University, Shanghai 200433, People’s Republic of China 3 Shanghai Key Lab of Intelligent Information Processing, Fudan University, Shanghai 200433, People’s Republic of China E-mail: comellas@ma4.upc.edu, zhangzz@fudan.edu.cn and chenlichao@gmail.com Received 26 September 2008, in final form 13 November 2008 Published 19 December 2008 Online at stacks.iop.org/JPhysA/42/045103 Abstract In this paper we introduce a family of planar, modular and self-similar graphs which has small-world and scale-free properties. The main parameters of this family are comparable to those of networks associated with complex systems, and therefore the graphs are of interest as mathematical models for these systems. As the clustering coefficient of the graphs is zero, this family is an explicit construction that does not match the usual characterization of hierarchical modular networks, namely that vertices have clustering values inversely proportional to their degrees. PACS numbers: 02.10.Ox, 89.20.Ff, 89.75.Da, 89.75.−k 1. Introduction Research and studies performed in the last few years show that many networks associated with complex systems, like the world wide web, the Internet, telephone networks, transportation systems (including power and water distribution networks), social and biological networks, belong to a class of networks now known as small-world scale-free networks, see [1, 2] and references therein. These networks exhibit a small average distance and diameter (compared to a random network with the same number of nodes and links) and, in many cases, a strong local clustering (nodes have many mutual neighbors). Another important common characteristic is that the number of links attached to the nodes usually obeys a power-law distribution (is scale- free). Moreover, a degree hierarchy in these networks is sometimes related to the modularity of the system. By introducing a new measuring technique, it has been discovered that many real networks are self-similar and fractal [3, 4]. More recently, a characterization of self-similarity versus fractality has been given in [5, 6]. 1751-8113/09/045103+10$30.00 © 2009 IOP Publishing Ltd Printed in the UK 1