Vol.4 No.2 (February 2014)
A Kind of Deterministic Small-World Networks Model and Analysis of Their Characteristics
张 科 ：青海师范大学数学系，西宁
Deterministic small-world network is an important branch of study of complex networks. In 2008, Zhang et al. in Eur.Phys.J.B 63 have offered detailed topological characteristics of the deterministic uniform recursive tree from the viewpoint of complex network. They derived topological characteristics of the deterministic uniform recursive tree. It shows a logarithmic scaling with the size of the network; however its clustering coefficient is zero. In 2012, Lu et al. in Physica A 391, based on the deterministic uniform recursive tree and by a simple rule, added some edges and got a deterministic small-world network model. In this paper, we study the law of the structure on the basis of the model constructed in Physica A 391 from a new viewpoint, naturally, we generalize and change the rule to get an iterated class of small-world networks, and then give the analytic solutions to several important topological characteristics of these models proposed. The obtained vigorous results show that these networks have power-law cumulative degree distribution, high clustering coefficient and small diameter.
张 科 , 赵海兴 , 李发旭 , 肖玉芝 , 李 峰 (2014) 一类确定性小世界网络模型及特性分析。 计算机科学与应用， 4， 27-31. doi: 10.12677/CSA.2014.42006
 Watts, D.J. and Strogatz, S.H. (1998) Collective dynamics of “small-world” networks. Nature, 393, 440-442.
 Comellas, F., Ozon, J. and Peters, J.G. (2000) Deterministic small-world communication networks. Information Processing Letters, 76, 83-90.
 Zhang, Z.Z., Rong, L.L. and Guo, C.H. (2006) A deterministic small-world network created by edge iterations. Physica A: Statistical Mechanics and Its Applications, 363, 567-572.
 Zhang, Z., Zhou, S., Shen, Z. and Guan, J. (2007) From regular to growing small-world networks. Physica A: Statistical Mechanics and Its Applications, 385, 765-772.
 Hu, G.N., Xiao, Y.Z., Jia, H.S. and Zhao, H.X. (2013) A new class of the planar networks with high clustering and high entropy. Abstract and Applied Analysis, 2013, Article ID: 795682.
 Zhang, Z.Z., et al. (2010) Mapping Koch curves into scale-free small-world networks. Journal of Physics A: Mathematical and Theoretical, 43, Article ID: 395101.
 Zhang, Z.Z., et al. (2007) Maximal planar scale-free Sierpinski networks with small-world effect and power law strength-degree correlation. Europhysics Letters, 79, Article ID: 38007.
 Zhang, Z. and Comellas, F. (2011) Farey graphs as models for complex networks. Theoretical Computer Science, 412, 865-875.
 Zhang, Y.C., Zhang, Z.Z., Zhou, S.G. and Guan, J.H. (2010) Deterministic weighted scale-free small-world networks. Physica A: Statistical Mechanics and Its Applications, 389, 33163324.
 GovorčIn, J., Knor, M. and Škrekovski, R. (2013) Line graph operation and small worlds. Information Processing Letters, 113, 196-200.
 Smythe, R.T. and Mahmoud, H. (1995) A survey of recursive trees. Theory of Probability and Mathematical Statistics, 51, 127.
 章忠志, 周水庚, 方锦清 (2008) 复杂网络确定性模型研究的最新进展. 复杂系统与复杂性科学, 4, 29-46.
 Jung, S., Kim, S. and Kahng, B. (2002) A geometric fractal growth model scale free networks. Physical Review E, 65, Article ID: 056101.
 Zhang, Z.Z., Zhou, S.G., Qi, Y. and Guan, J.H. (2008) Topologies and Laplacian spectra of a deterministic uniform recursive tree. The European Physical Journal B, 63, 507-513.
 Lu, Z.M. and Guo, S.Z. (2012) A small-world network derived from the deterministic uniform recursive tree. Physica A: Statistical Mechanics and Its Applications, 391, 87-92.
 Xiao, Y.Z. and Zhao, H.X. (2013) New method for counting the number of spanning trees in a two-tree network. Physica A: Statistical Mechanics and Its Applications, 392, 4576-4583.