Knor, Martin; Škrekovski, Riste Deterministic self-similar models of complex networks based on very symmetric graphs. (English) Zbl 1395.05159 Physica A 392, No. 19, 4629-4637 (2013). Summary: Using very symmetric graphs we generalize several deterministic self-similar models of complex networks and we calculate the main network parameters of our generalization. More specifically, we calculate the order, size and the degree distribution, and we give an upper bound for the diameter and a lower bound for the clustering coefficient. These results yield conditions under which the network is a self-similar and scale-free small world network. We remark that all these conditions are posed on a small base graph which is used in the construction. As a consequence, we can construct complex networks having prescribed properties. We demonstrate this fact on the clustering coefficient. We propose eight new infinite classes of complex networks. One of these new classes is so rich that it is parametrized by three independent parameters. Cited in 4 Documents MSC: 05C82 Small world graphs, complex networks (graph-theoretic aspects) Keywords:complex systems; small world network; scale-free network; deterministic model PDFBibTeX XMLCite \textit{M. Knor} and \textit{R. Škrekovski}, Physica A 392, No. 19, 4629--4637 (2013; Zbl 1395.05159) Full Text: DOI References: [1] Newman, M., The structure and function of complex networks, SIAM Rev., 45, 167-256, (2003) · Zbl 1029.68010 [2] Watts, D.; Strogatz, S., Collective dynamics of ‘small world’ networks, Nature, 393, 440-442, (1998) · Zbl 1368.05139 [3] Barabási, A.; Albert, R., Emergence of scaling in random networks, Science, 286, 509-512, (1999) · Zbl 1226.05223 [4] Song, C.; Havlin, S.; Makse, H., Self-similarity of complex networks, Nature, 433, 392-395, (2005) [5] Comellas, F.; Miralles, A., Modeling complex networks with self-similar outerplanar unclustered graphs, Physica A, 388, 2227-2233, (2009) [6] Miralles, A.; Comellas, F.; Chen, L.; Zhang, Z., Planar unclustered graphs to model technological and biological networks, Physica A, 389, 1955-1964, (2010) [7] Zhang, Z.; Zhou, S.; Fang, L.; Guan, J.; Zhang, Y., Maximal planar scale-free sierpinski networks with small-world effect and power-law strength-degree correlation, Europhys. Lett., 79, 38007, (2007) [8] Zhang, Z.; Zhou, S.; Zou, T.; Chen, L.; Guan, J., Incompatibility networks as models of scale-free small-world graphs, Eur. Phys. J. B, 60, 259-264, (2007) · Zbl 1189.91188 [9] Zhang, Z.; Zhou, S.; Xie, W.; Chen, L.; Lin, Y.; Guan, J., Standard random walks and trapping on the Koch network with scale-free behavior and small-world effect, Phys. Rev. E, 79, 061113, (2009) [10] Zhang, Z.; Rong, L.; Comellas, F., Evolving small-world networks with geographical attachment preference, J. Phys. A-Math. Gen., 39, 3253, (2006) · Zbl 1088.93003 [11] Zhang, Z.; Rong, L.; Guo, C., A deterministic small-world network created by edge iterations, Physica A, 363, 567-572, (2006) [12] Zhang, Z.; Rong, L.; Zhou, S., A general geometric growth model for pseudofractal scale-free web, Physica A, 377, 329-339, (2007) [13] Zhang, Z.; Zhou, S.; Chen, L., Evolving pseudofractal networks, Eur. Phys. J. B, 58, 337-344, (2007) [14] Barabási, A.; Ravasz, E.; Vicsek, T., Deterministic scale-free networks, Physica A, 299, 559-564, (2001) · Zbl 0972.57003 [15] L. Barière, F. Comellas, C. Dafó, M. Fiol, Deterministic hierarchical networks (manuscript). · Zbl 1384.90016 [16] Zhang, Z.; Zhou, S.; Chen, L.; Guan, J.; Fang, L.; Zhang, Y., Recursive weighted treelike networks, Eur. Phys. J. B, 59, 99-107, (2007) · Zbl 1189.91137 [17] Andrade, J.; Herrmann, H.; Andrade, R.; da Silva, L., Apollonian networks: simultaneously scale-free, small world, Euclidean, space filling, and with matching graphs, Phys. Rev. Lett., 94, 018702, (2005) [18] Zhang, Z.; Comellas, F., Farey graphs as models for complex networks, Theoret. Comput. Sci., 412, 865-875, (2011) · Zbl 1206.68245 [19] Comellas, F.; Zhang, Z.; Chen, L., Self-similar non-clustered planar graphs as models for complex networks, J. Phys. A-Math. Theor., 42, 045103, (2009) · Zbl 1165.05025 [20] Comellas, F.; Fertin, G.; Raspaud, A., Recursive graphs with small-world scale-free properties, Phys. Rev. E, 69, 037104, (2004) [21] Godsil, C.; Royle, G., Algebraic graph theory, (2001), Springer-Verlag New York · Zbl 0968.05002 [22] Colbourn, C. J.; Rosa, A., Triple systems, (1999), Oxford University Press Oxford · Zbl 0938.05009 [23] Gross, J. L.; Tucker, T. W., Topological graph theory, (1987), John Wiley New York · Zbl 0621.05013 [24] Grannell, M. J.; Griggs, T. S.; Knor, M.; Širáň, J., Triangulations of orientable surfaces by complete tripartite graphs, Discrete Math., 306, 600-606, (2006) · Zbl 1099.05025 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.