Transport between multiple users in complex networks


We study the transport properties of model networks such as scale-free and Erdös-Rényi networks as well as a real network. We consider few possibilities for the trnasport problem. We start by studying the conductance G between two arbitrarily chosen nodes where each link has the same unit resistance. Our theoretical analysis for scale-free networks predicts a broad range of values of G, with a power-law tail distribution $\Phi_{\rm SF}(G)\sim G^{-g_G}$ , where gG=2λ-1, and λ is the decay exponent for the scale-free network degree distribution. The power-law tail in ΦSF(G) leads to large values of G, thereby significantly improving the transport in scale-free networks, compared to Erdös-Rényi networks where the tail of the conductivity distribution decays exponentially. We develop a simple physical picture of the transport to account for the results. The other model for transport is the max-flow model, where conductance is defined as the number of link-independent paths between the two nodes, and find that a similar picture holds. The effects of distance on the value of conductance are considered for both models, and some differences emerge. We then extend our study to the case of multiple sources ans sinks, where the transport is defined between two groups of nodes. We find a fundamental difference between the two forms of flow when considering the quality of the transport with respect to the number of sources, and find an optimal number of sources, or users, for the max-flow case. A qualitative (and partially quantitative) explanation is also given.

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  1. S. Havlin, D. ben-Avraham, Adv. Phys. 36, 695 (1987)

    Article  ADS  Google Scholar 

  2. D. ben-Avraham, S. Havlin, Diffusion and reactions in fractals and disordered systems (Cambridge, New York, 2000)

  3. A. Bunde, S. Havlin, edited by Fractals and Disordered Systems (Springer, New York, 1996)

  4. R. Albert, A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002); R. Pastor-Satorras, A. Vespignani, Structure and Evolution of the Internet: A Statistical Physics Approach (Cambridge University Press, Cambridge, 2004); S.N. Dorogovsetv, J.F.F. Mendes, Evolution of Networks: From Biological Nets to the Internet and WWW (Oxford University Press, Oxford, 2003)

    Article  ADS  MathSciNet  Google Scholar 

  5. G. Bonanno, G. Caldarelli, F. Lillo, R.N. Mantegna, Phys. Rev. E 68, 046130 (2003)

    Article  ADS  Google Scholar 

  6. J.-P. Onnela, A. Chakraborti, K. Kaski, J. Kertész, A. Kanto, Phys. Rev. E 68, 056110 (2003)

    Article  ADS  Google Scholar 

  7. H. Inaoka, T. Ninomiya, K. Taniguchi, T. Shimizu, H. Takayasu, Fractal Network derived from banking transaction – An analysis of network structures formed by financial institutions, Bank of Japan Working Paper Series, 04-E-04 (2004); H. Inaoka, H. Takayasu, T. Shimizu, T. Ninomiya, K. Taniguchi, Physica A 339, 62 (2004)

    Article  MathSciNet  Google Scholar 

  8. P. Erdös, A. Rényi, Publ. Math. (Debreccen) 6, 290 (1959)

    MATH  Google Scholar 

  9. B. Bollobás, Random Graphs (Academic Press, Orlando, 1985)

  10. A.-L. Barabási, R. Albert, Science 286, 509 (1999)

    Article  MathSciNet  Google Scholar 

  11. P.L. Krapivsky, S. Redner, F. Leyvraz, Phys. Rev. Lett. 85, 4629 (2000)

    Article  ADS  Google Scholar 

  12. H.A. Simon, Biometrika 42, 425 (1955)

    MATH  MathSciNet  Google Scholar 

  13. R. Cohen, S. Havlin, Phys. Rev. Lett. 90, 058701 (2003)

    Article  ADS  Google Scholar 

  14. In principle, a node can have a degree up to N-1, connecting to all other nodes of the network. The results presented here correspond to networks with upper cutoff k max=kminN1/(λ-1) imposed. We also studied networks for which kmax is not imposed, and found no significant differences in the pdf ΦSF(G)

  15. E. López, S.V. Buldyrev, S. Havlin, H.E. Stanley, Phys. Rev. Lett. 94, 248701 (2005)

    Article  ADS  Google Scholar 

  16. S. Havlin, E. López, S.V. Buldyrev, H.E. Stanley, in Diffusion Fundamentals, edited by Jörg Kärger, Farida Grinberg, Paul Heitjans (Leipzig: Universitätsverlag, 2005), pp. 38–48

  17. E. Lopez, S. Carmi, S. Havlin, S. Buldyrev, H.E. Stanley, Physica D 224, 69 (2006)

    MATH  Article  ADS  MathSciNet  Google Scholar 

  18. G.R. Grimmett, H. Kesten, J. Lond. Math. Soc. 30, 171 (1984); e-print arXiv:math/0107068

    MATH  Article  MathSciNet  Google Scholar 

  19. D.-S. Lee, H. Rieger, Europhys. Lett. 73, 471 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  20. Z. Toroczkai, K. Bassler, Nature 428, 716 (2004)

    Article  ADS  Google Scholar 

  21. J.D. Noh, H. Rieger, Phys. Rev. Lett. 92, 118701 (2004)

    Article  ADS  Google Scholar 

  22. V. Sood, S. Redner, D. ben-Avraham, J. Phys. A, 38, 109 (2005)

    Google Scholar 

  23. L.K. Gallos, Phys. Rev. E 70, 046116 (2004)

    Article  ADS  Google Scholar 

  24. B. Tadic, G.J. Rodgers, Advances in Complex Systems 5, 445 (2002)

    Article  Google Scholar 

  25. The dynamical properties we study are related to transport on networks and differ from those which treat the network topology itself as evolving in time scale-Barabasi, dyn-network

  26. R.K. Ahuja, T.L. Magnanti, J.B. Orlin, Network Flows: Theory, Algorithms, and Applications (Prentice Hall, 1993)

  27. The study of community structure in social networks has led some authors (M.E.J. Newman, M. Girvan, Phys. Rev. E 69, 026113 (2004); F. Wu, B.A. Huberman, Eur. Phys. J. B 38, 331 (2004)) to develop methods in which networks are considered as electrical networks in order to identify communities. In these studies, however, transport properties have not been addressed

    Article  ADS  Google Scholar 

  28. M. Molloy, B. Reed, Random Struct. Algorithms 6, 161 (1995)

    MATH  MathSciNet  Article  Google Scholar 

  29. G. Kirchhoff, Ann. Phys. Chem. 72 497 (1847); N. Balabanian, Electric Circuits (McGraw-Hill, New York, 1994)

  30. S. Kirkpatrick, Proceedings of Inhomogeneous Superconductors Conference, Berkeley Springs, W. Va, edited by S.A. Wolf, D.U. Gubser, A.I.P. Conf. Procs. 58, 79 (1979)

    Article  ADS  Google Scholar 

  31. B.V. Cherkassky, Algorithmica 19, 390 (1997)

    MATH  Article  MathSciNet  Google Scholar 

  32. Z. Wu, L.A. Braunstein, S. Havlin, H.E. Stanley, Phys. Rev. Lett. 96, 148702 (2006)

    Article  ADS  Google Scholar 

  33. S. Carmi, S. Havlin, S. Kirkpatrick, Y. Shavitt, E. Shir, MEDUSA - New Model of Internet Topology Using k-shell Decomposition, arXiv:cond-mat/0601240

  34. Y. Shavitt, E. Shir, ACM SIGCOMM Computer Communication Review, 35, 71 (2005)

    Google Scholar 

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Carmi, S., Wu, Z., López, E. et al. Transport between multiple users in complex networks. Eur. Phys. J. B 57, 165–174 (2007).

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  • 05.60.Cd Classical transport