Abstract
Recently, the analytical study of the structural properties of complex networks has attracted increasing attention due to the growth of these real-world networks. Mathematical graph theory helps to understand and predict their behavior. This paper examines and compares the structural properties such as the small-world effect, the clustering coefficient and the degree distribution of Erdos-Renyi random networks with some real-world networks. Besides, we propose an algorithm to calculate the number and the entropy of spanning trees of these networks by using the electrically equivalent transformations. The result allows us to evaluate the robustness and the homogeneity of their structure. The proposed technique is efficient and more general compared to the classical ones.
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Mokhlissi, R., Lotfi, D., Debnath, J., El Marraki, M.: An innovative combinatorial approach for the spanning tree entropy in Flower Network. In: El Abbadi, A., Garbinato, B. (eds.) International Conference on Networked Systems, pp. 3–14. Springer, Cham (2017)
Mokhlissi, R., Lotfi, D., Debnath, J., El Marraki, M., El Khattabi, N.: The evaluation of the number and the entropy of spanning trees on generalized small-world networks. J. Appl. Math. 2018, 1–7 (2018)
Mokhlissi, R., Lotfi, D., Debnath, J., El Marraki, M.: Complexity analysis of “small-world networks” and spanning tree entropy. In: Cherifi, H., Gaito, S., Quattrociocchi, W., Sala, A. (eds.) International Workshop on Complex Networks and their Applications, pp. 197–208. Springer, Cham (2016)
Mokhlissi, R., Lotfi, D., El Marraki, M.: A theoretical study of the complexity of complex networks. In: 2016 7th International Conference on Sciences of Electronics, Technologies of Information and Telecommunications (SETIT), pp. 24–28. IEEE (2016)
Mokhlissi, R., Lotfi, D., El Marraki, M., Debnath, J.: The structural properties and the spanning trees entropy of the generalized Fractal Scale-Free Lattice. J. Complex Netw. cnz030 (2019)
Stauffer, D., Aharony, A., Redner, S.: Introduction to percolation theory. Phys. Today 46, 64 (1993)
Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature 393(6684), 440 (1998)
Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)
Zhang, Z., Comellas, F.: Farey graphs as models for complex networks. Theor. Comput. Sci. 412(8–10), 865–875 (2011)
Knuth, D.E.: Aztec diamonds, checkerboard graphs, and spanning trees. J. Algebraic Comb. 6(3), 253–257 (1997)
Krebs, V.: Books about US politics (2004, unpublished). http://www.orgnet.com
Girvan, M., Newman, M.E.: Community structure in social and biological networks. Proc. Nat. Acad. Sci. 99(12), 7821–7826 (2002)
Lusseau, D., Schneider, K., Boisseau, O.J., Haase, P., Slooten, E., Dawson, S.M.: The bottlenose dolphin community of Doubtful Sound features a large proportion of long-lasting associations. Behav. Ecol. Sociobiol. 54(4), 396–405 (2003)
Newman, M.E.: Finding community structure in networks using the eigenvectors of matrices. Phys. Rev. E 74(3), 036104 (2006)
Zachary, W.W.: An information flow model for conflict and fission in small groups. J. Anthropol. Res. 33(4), 452–473 (1977)
Knuth, D.E.: The Stanford GraphBase: A Platform for Combinatorial Computing, pp. 74–87. ACM Press, New York (1993)
Erdős, P.: On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci. 5, 17–61 (1960)
Erdős, P., Rényi, A.: On the strength of connectedness of a random graph. Acta Math. Hung. 12(1–2), 261–267 (1961)
Teufl, E., Wagner, S.: On the number of spanning trees on various lattices. J. Phys. A: Math. Theor. 43(41), 415001 (2010)
Garcia, A., Noy, M., Tejel, J.: The asymptotic number of spanning trees in d-dimensional square lattices. J. Comb. Math. Comb. Comput. 44, 109–114 (2003)
Bistouni, F., Jahanshahi, M.: Reliability analysis of Ethernet ring mesh networks. IEEE Trans. Reliab. 66(4), 1238–1252 (2017)
Chaiken, S., Kleitman, D.J.: Matrix tree theorems. J. Comb. Theory Series A 24(3), 377–381 (1978)
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Mokhlissi, R., Lotfi, D., Debnath, J., El Marraki, M. (2020). Assessment of the Effectiveness of Random and Real-Networks Based on the Asymptotic Entropy. In: Barbosa, H., Gomez-Gardenes, J., Gonçalves, B., Mangioni, G., Menezes, R., Oliveira, M. (eds) Complex Networks XI. Springer Proceedings in Complexity. Springer, Cham. https://doi.org/10.1007/978-3-030-40943-2_4
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