Structure Properties of Generalized Farey graphs based on Dynamical Systems for Networks

Farey graphs are simultaneously small-world, uniquely Hamiltonian, minimally 3-colorable, maximally outerplanar and perfect. Farey graphs are therefore famous in deterministic models for complex networks. By lacking of the most important characteristics of scale-free, Farey graphs are not a good model for networks associated with some empirical complex systems. We discuss here a category of graphs which are extension of the well-known Farey graphs. These new models are named generalized Farey graphs here. We focus on the analysis of the topological characteristics of the new models and deduce the complicated and graceful analytical results from the growth mechanism used in generalized Farey graphs. The conclusions show that the new models not only possess the properties of being small-world and highly clustered, but also possess the quality of being scale-free. We also find that it is precisely because of the exponential increase of nodes’ degrees in generalized Farey graphs as they grow that caused the new networks to have scale-free characteristics. In contrast, the linear incrementation of nodes’ degrees in Farey graphs can only cause an exponential degree distribution.

[1]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[2]  Zhongzhi Zhang,et al.  Farey graphs as models for complex networks , 2011, Theor. Comput. Sci..

[3]  Yinhu Zhai,et al.  Label-based routing for a family of small-world Farey graphs , 2016, Scientific Reports.

[4]  Yaoping Hou,et al.  Tutte polynomial of a small-world Farey graph , 2013 .

[5]  Chonghui Guo,et al.  A deterministic small-world network created by edge iterations , 2006 .

[6]  Albert-Laszlo Barabasi,et al.  Deterministic scale-free networks , 2001 .

[7]  Guanrong Chen,et al.  Small-World Topology Can Significantly Improve the Performance of Noisy Consensus in a Complex Network , 2015, Comput. J..

[8]  Lili Rong,et al.  Evolving small-world networks with geographical attachment preference , 2006 .

[9]  Joseph G. Peters,et al.  Deterministic small-world communication networks , 2000, Inf. Process. Lett..

[10]  Bin Wu,et al.  Counting spanning trees in a small-world Farey graph , 2012, 1201.4228.

[11]  J. S. Andrade,et al.  Apollonian networks: simultaneously scale-free, small world, euclidean, space filling, and with matching graphs. , 2004, Physical review letters.

[12]  Shuigeng Zhou,et al.  Mapping Koch curves into scale-free small-world networks , 2008, Journal of Physics A: Mathematical and Theoretical.

[13]  Francesc Comellas,et al.  Modeling complex networks with self-similar outerplanar unclustered graphs ☆ , 2009 .

[14]  Anjan Kumar Chandra,et al.  A small world network of prime numbers , 2005 .

[15]  B. Kahng,et al.  Geometric fractal growth model for scale-free networks. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Wenjun Xiao,et al.  Cayley graphs as models of deterministic small-world networks , 2006, Inf. Process. Lett..

[17]  Shuigeng Zhou,et al.  Self-similarity, small-world, scale-free scaling, disassortativity, and robustness in hierarchical lattices , 2007 .

[18]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[19]  Shuigeng Zhou,et al.  A general geometric growth model for pseudofractal scale-free web , 2007 .

[20]  Francesc de Paula Comellas Padró,et al.  Modeling complex networks with self-similar outerplanar unclustered graphs , 2009 .

[21]  Jonathan P K Doye,et al.  Self-similar disk packings as model spatial scale-free networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  Shuigeng Zhou,et al.  Topologies and Laplacian spectra of a deterministic uniform recursive tree , 2008, 0801.4128.

[23]  S. Boettcher,et al.  Hierarchical regular small-world networks , 2007, 0712.1259.