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New Deterministic Model of Evolving Trinomial Networks

  • Alexander Goryashko
  • Leonid Samokhine
  • Pavel Bocharov
Conference paper
Part of the Studies in Computational Intelligence book series (SCI, volume 812)

Abstract

We provide a new model of attributed networks where label of each vertex is a partition of integer n into at most m integer parts and all labels are different. The metric in the space of (nm)-partitions is introduced. First, we investigate special class of the trinomial \((m^2, m)\)-partitions as a base for synthesis of networks G(m). It turns out that algorithmic complexity (the shortest computer program that produces G(m) upon halting) of these networks grows with m as \(\log m\) only. Numerical simulations of simple graphs for trinomial \((m^2, m)\)-partition families \((m= 3, 4, \ldots , 9)\) allows to estimate topological parameters of the graphs—clustering coefficients, cliques distribution, vertex degree distribution—and to show existence of such effects as scale-free and self-similarity for evolving networks. Since the model under consideration is completely deterministic, these results put forward new mode of thought about mechanisms of similarity, preferential attachment and popularity of complex networks. In addition, we obtained some numerical results relating robust behavior of the networks to disturbances like deleting nodes or cliques.

Keywords

Partition Simple graph Perfect graph Algorithmic complexity Trinomial coefficients Predictability Topological parameters Clustering coefficient Connectivity Emergency Robust 

References

  1. 1.
    Andrews, G.E.: Number Theory (1971)Google Scholar
  2. 2.
    Andrews, G.E.: Euler’s “Exemplum memorabile inductionis fallacis" and q-trinomial coefficients. J. Am. Math. Soc. 3(3), 653–669 (1990).  https://doi.org/10.2307/1990932Google Scholar
  3. 3.
    Barabási, A.L., Ravasz, E., Vicsek, T.: Deterministic scale-free networks. Physica A: Stat. Mech. Appl. 299(3–4), 559–564 (2001).  https://doi.org/10.1016/S0378-4371(01)00369-7Google Scholar
  4. 4.
    Bocharov, P.: Partition Games Research Toolbox (2015). https://github.com/pbo/partition-games
  5. 5.
    Bocharov, P., Goryashko, A.: Evolutionary dynamics of partition games. In: 2015 International Conference "Stability and Control Processes” in Memory of V.I. Zubov (SCP), pp. 225–228. IEEE (2015).  https://doi.org/10.1109/SCP.2015.7342108. http://ieeexplore.ieee.org/document/7342108/
  6. 6.
    Bollobás, B.: Random graphs. In: Modern Graph Theory, pp. 215–252. Springer, New York (1998)Google Scholar
  7. 7.
    Bollobás, B., Riordan, O.M.: Mathematical results on scale-free random graphs. In: Handbook of Graphs and Networks (2004).  https://doi.org/10.1002/3527602755.ch1
  8. 8.
    Chaitin, G.J.: On the length of programs for computing finite Binary Sequences: statistical considerations. J. ACM 13(4), 547–569 (1969).  https://doi.org/10.1145/321495.321506Google Scholar
  9. 9.
    Cvetkovic, D., Rowlinson, P., Simic, S., Beineke, W., Godsil, C., Royle, G., Biggs, N., Doob, M., Wilson, R.J.: Algebraic Graph Theory (2001).  https://doi.org/10.1007/978-1-4613-0163-9
  10. 10.
    Goryashko, A.: On suboptimal solution of antagonistic matrix games. ITM Web of Conferences, vol. 1(03001) (2017).  https://doi.org/10.1051/itmconf/20171003001
  11. 11.
    Knuth, D.E.: The Art of Computer Programming, Combinatorial Algorithms, Part 1, vol. 4A. Addison-Wesley Professional (2011)Google Scholar
  12. 12.
    Kolmogorov, A.N.: Three approaches to the quantitative definition of information. Probl. Inf. Trans. 1(1), 1–7 (1965)Google Scholar
  13. 13.
    Leskovec, J., Chakrabarti, D., Kleinberg, J., Faloutsos, C., Ghahramani, Z.: Kronecker graphs: an approach to modeling networks. J. Mach. Learn. Res. 11, 985–1042 (2010)Google Scholar
  14. 14.
    Levandowsky, M., Winter, D.: Distance between sets. Nature 234(5323), 34 (1971)Google Scholar
  15. 15.
    Li, M., Vitányi, P.M.B.: Kolmogorov complexity and its applications. Centre for Mathematics and Computer Science (1989)Google Scholar
  16. 16.
    Lovász, L.: Perfect graphs. Sel. Top. Graph Theory 2, 55–87 (1983)Google Scholar
  17. 17.
    Morzy, M., Kajdanowicz, T., Kazienko, P.: On measuring the complexity of networks: kolmogorov complexity versus entropy. Complexity (2017)Google Scholar
  18. 18.
    Newman, M.E.J.: The structure and function of complex networks. SIAM Rev. 45(2), 167–256 (2003).  https://doi.org/10.1137/S003614450342480Google Scholar
  19. 19.
    OEIS Foundation Inc.: The On-Line Encyclopedia of Integer Sequences (2018). https://oeis.org
  20. 20.
    Samokhine, L.: Trinomial Family Research Toolbox (2017). https://github.com/samokhine/gory
  21. 21.
    Tardos, E., Wexler, T.: Network Formation Games and the Potential Function Method (2007).  https://doi.org/10.1145/1785414.1785439
  22. 22.
    Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature 393(6684), 440 (1998).  https://doi.org/10.1038/30918Google Scholar
  23. 23.
    Zenil, H., Kiani, N.A., Tegnér, J.: Low-algorithmic-complexity entropy-deceiving graphs. Phys. Rev. E (2017).  https://doi.org/10.1103/PhysRevE.96.012308

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Alexander Goryashko
    • 1
  • Leonid Samokhine
    • 2
  • Pavel Bocharov
    • 3
  1. 1.Moscow Technological InstituteMoscowRussia
  2. 2.Qcue Inc.AustinUSA
  3. 3.WheelyMoscowRussia

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