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Gaussian Networks Generated by Random Walks

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Abstract

We propose a random walks based model to generate complex networks. Many authors studied and developed different methods and tools to analyze complex networks by random walk processes. Just to cite a few, random walks have been adopted to perform community detection, exploration tasks and to study temporal networks. Moreover, they have been used also to generate networks with different topologies (e.g., scale-free). In this work, we define a random walker that plays the role of “edges-generator”. In particular, the random walker generates new connections and uses these ones to visit each node of a network. As result, the proposed model allows to achieve networks provided with a Gaussian degree distribution; moreover we found that some properties of achieved Gaussian networks, as the clustering coefficient and the assortativity, show a critical behavior. Finally, we performed numerical simulations to study the behavior and the properties of the cited model.

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References

  1. Albert, R., Jeong, H., Barabasi, A.L.: Internet: diameter of the World Wide Web. Nature 401, 130–131 (1999)

    Article  ADS  Google Scholar 

  2. Capocci, A., Servedio, V.D.P., Colaiori, F., Buriol, L.S., Donato, D., Leonardi, S., Caldarelli, G.: Preferential attachment in the growth of social networks: the internet encyclopedia Wikipedia. Phys. Rev. E 74(3), 036116 (2006)

    Article  ADS  Google Scholar 

  3. Sporns, O., Chialvo, D.R., Kaiser, M., Hilgetag, C.: Organization, development and function of complex brain networks. Trends Cogn. Sci. 8(9), 418–425 (2004)

    Article  Google Scholar 

  4. Albert, R., Barabasi, A.L.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  5. Erdos, P., Renyi, P.: On the evolution of random graphs. Publ. Math. Inst. Hungary Sci 5, 17–61 (1960)

    MathSciNet  Google Scholar 

  6. Watts, D.J., Strogatz, S.H.: Collective dynamics of “small-world” networks. Nature 393, 440–442 (1998)

    Article  ADS  Google Scholar 

  7. Albert, R., Barabasi, A.L.: Statistical mechanics of complex networks. Rev. Mod. Phys 74, 47–97 (2002)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. Bianconi, G., Barabasi, A.L.: Bose–Einstein condensation in complex networks. Phys. Rev. Lett. 86, 5632–5635 (2001)

    Article  ADS  Google Scholar 

  9. Bianconi, G.: Quantum statistics in complex networks. Phys. Rev. E 66, 056123 (2002)

    ADS  Google Scholar 

  10. Javarone, M.A., Armano, G.: Quantum-classical transitions in complex networks. J. Theory Exp. 2013(4), P04019 (2013)

    Google Scholar 

  11. Javarone, M.A.: Fermionic networks: modeling adaptive complex networks with fermionic gases. arXiv:1408.3151 (2014)

  12. Hartonen, T., Annila, A.: Natural networks as thermodynamic systems. Complexity 18(2), 53–62 (2012)

    Article  Google Scholar 

  13. Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A., Boguna, M.: Hyperbolic geometry of complex networks. Phys. Rev. E 82, 036106 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  14. Javarone, M.A., Armano, G.: Perception of similarity: a model for social network dynamics. J. Phys A 46, 455102 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  15. Lawler, G.F., Limic, V.: Random Walk: A Modern Introduction. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  16. Eguiluz, V.M., Chialvo, D.R., Cecchi, G.A., Baliki, M., Apkarian, A.V.: Scale-free brain functional networks. Phys. Rev. Lett. 94(1), 018102 (2005)

    Article  ADS  Google Scholar 

  17. Strogatz, S.H.: Exploring complex networks. Nature 410, 268–276 (2001)

    Article  ADS  Google Scholar 

  18. Zhou, H.: Distance, dissimilarity index, and network community structure. Phys. Rev. E 67(6), 061901 (2003)

    Article  ADS  Google Scholar 

  19. Tadic, B.: Exploring complex graphs by random walks. In: Proceedings of the AIP Conference , vol. 661(24) (2003).

  20. Starnini, M., Baronchelli, A., Barrat, A., Pastor-Satorras, R.: Random walks on temporal networks. Phys. Rev. E 85(5), 056115 (2012)

    Article  ADS  Google Scholar 

  21. Faccin, M., Johnson, T., Biamonte, J., Kais, S., Migdal, P.: Degree distribution in quantum walks on complex networks. Phys. Rev. X 3–4, 041007 (2013)

    Google Scholar 

  22. Saramaki, J., Kaski, K.: Scale-free networks generated by random walks. Physica A 341–1, 80–86 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  23. Gonzalez, M.C., Lind, P.G., Herrmann, H.J.: System of mobile agents to model social networks. Phys. Rev. Lett. 96(8), 088702 (2006)

    Article  ADS  Google Scholar 

  24. Noh, J.D., Rieger, H.: Random walks on complex networks. Phys. Rev. Lett. 92(11), 118701 (2004)

    Article  ADS  Google Scholar 

  25. Sood, V., Redner, S., Ben-Avraham, D.: First passage properties of the Erdös-Renyi random graph. J. Phys. A 38(1), 109 (2005)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  26. Jonasson, J.: On the cover time for random walks on random graphs. Comb. Probab. Comput. 7–3, 265–279 (1998)

    Article  MathSciNet  Google Scholar 

  27. Newman, M.E.J.: The structure and function of complex networks. SIAM Rev. 45, 167–256 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  28. Hardiman, S., Katzir, L.: Estimating clustering coefficients and size of social networks via random walk. In: Proceedings of the 22nd international conference on World Wide Web, pp. 539–550 (2013).

  29. Newman, M.E.J.: Mixing patterns in networks. Phys. Rev. E 67(2), 026126 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  30. Cohen, R., Havlin, S.: Scale-free networks are ultrasmall. Phys. Rev. Lett. 90–5, 058701 (2003)

    Article  Google Scholar 

  31. Johnson, S., Torres, J., Marro, J., Muñoz, M.A.: Entropic origin of disassortativity in complex networks. Phys. Rev. Lett. 104(10), 108702 (2010)

    Article  ADS  Google Scholar 

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Acknowledgments

The author would like to thank Fondazione Banco di Sardegna for supporting his work.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Cagliari, Cagliari, Italy

    Marco Alberto Javarone

  2. DUMAS - Department of Humanities and Social Sciences, University of Sassari, Sassari, Italy

    Marco Alberto Javarone

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  1. Marco Alberto Javarone
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Correspondence to Marco Alberto Javarone.

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Javarone, M.A. Gaussian Networks Generated by Random Walks. J Stat Phys 159, 108–119 (2015). https://doi.org/10.1007/s10955-014-1175-8

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  • DOI: https://doi.org/10.1007/s10955-014-1175-8

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