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A Stochastic Model for Layered Self-organizing Complex Systems

  • Yuri Dimitrov
  • Mario Lauria
Part of the Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering book series (LNICST, volume 5)

Abstract

In this paper we study a problem common to complex systems that dynamically self-organize to an optimal configuration. Assuming the network nodes are of two types, and that one type is subjected to a an upward pressure according to a preferential stochastic model , we wish to determine the distribution of the active nodes over the levels of the network. We generalize the problem to the case of layered graphs as follows. Let G be a connected graph with M vertices which are divided into d levels where the vertices of each edge of G belong to consecutive levels. Initially each vertex has a value of 0 or 1 assigned at random. At each step of the stochastic process an edge is chosen at random. Then, the labels of the vertices of this edge are exchanged with probability 1 if the vertex on the higher level has the label 0 and the lower vertex has the label 1. The labels are switched with probability λ, if the lower vertex has value of 0 and the higher vertex has the value of 1. This stochastic process has the Markov chain property and is related to random walks on graphs. We derive formulas for the steady state distribution of the number of vertices with label 1 on the levels of the graph.

Keywords

Stochastic Model Random Graph Layered Graph Overlay Network Steady State Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Albert, R., Barabási, A.-L., Jeong, H.: Scale-free characteristics of random networks: The topology of the world wide web. Physica A 281, 69–77 (2000)CrossRefGoogle Scholar
  2. 2.
    Barabási, A.-L., Ravasz, E.: Hierarchical organization in complex networks. Physical review E 67, 026112 (2003)CrossRefzbMATHGoogle Scholar
  3. 3.
    Burton, R.M., Faris, W.G.: A Self-Organizing Cluster Process. Ann. Appl. Prob. 6(4), 1232–1247 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Caldarelli, G.: Scale-Free Networks. Oxford University Press, Oxford (2007)CrossRefzbMATHGoogle Scholar
  5. 5.
    Chakravarti, A.J., Baumgartner, G., Lauria, M.: Application-specific scheduling for the Organic Grid. In: Proceedings of the 5th IEEE/ACM International Workshop on Grid Computing (GRID 2004), Pittsburgh, pp. 146–155 (2004)Google Scholar
  6. 6.
    Chakravarti, A.J., Baumgartner, G., Lauria, M.: The Organic Grid: Self-organizing computation on a peer-to-peer network. In: Proceedings of the International Conference Autonomic Computing. IEEE Computer Society, Los Alamitos (2004)Google Scholar
  7. 7.
    Chakravarti, A.J., Baumgartner, G., Lauria, M.: The Organic Grid: Self-organizing computation on a peer-to-peer network. IEEE Transactions on Systems, Man and Cybernetics, Part A 35(3), 373–384 (2005)CrossRefGoogle Scholar
  8. 8.
    Dimitrov, Y., Giovane, C., Lauria, M., Mango, G.: A Combinatorial model for self-organizing networks. In: Proceedings of 21st IEEE International Parallel and Distributed Processing Symposium (2007)Google Scholar
  9. 9.
    Dimitrov, Y., Lauria, M.: A Combinatorial model for self-organizing networks. Technical Report TR02, Department of Computer Science and Engineering, Ohio State University (2007)Google Scholar
  10. 10.
    Erdös, P., Rényi, A.: On random graphs. Publicationes Mathematicae 6, 290–297 (1959)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Erdös, P., Rényi, A.: On the evolution of random graphs. Publications of the Mathematical Institute of the Hungarian Academy of Sciences 5, 17–61 (1960)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Jannotti, J., Gifford, D.K., Johnson, K.L., Kaashoek, M.F., O’Toole Jr., J.: Overcast: Reliable Multicasting with an Overlay Network. In: Proceedings of OSDI, pp. 197–212 (2000)Google Scholar
  13. 13.
    Kostic, D., Rodriguez, A., Albrecht, J., Vahdat, A.: Bullet: High Bandwidth Data Dissemination Using an Overlay Mesh. In: Proc. of ACM SOSP (2003)Google Scholar
  14. 14.
    Krapivsky, P.L., Redner, S., Leyvraz, F.: Connectivity of growing random networks. Phys. Rev. Lett. 85 (2000)Google Scholar
  15. 15.
    Newman, M.E.J.: Random graphs as models of networks. arXiv:cond-mat/0202208v1 (2002)Google Scholar
  16. 16.
    Shi, D., Chen, Q., Liu, L.: Markov chain-based numerical method for degree distributions of growing networks. Physical review E 71 (2005)Google Scholar
  17. 17.
    Wang, C.: Stochastic Models for Self-organizing Networks and Infinite Graphs. Ph.D. thesis, Dalhousie University (2006)Google Scholar
  18. 18.
    Watts, D., Dodds, P.S., Newman, M.E.J.: Identity and Search in Social networks. Science 296, 1302–1305 (2002)CrossRefGoogle Scholar
  19. 19.
    Watts, D., Muhamad, R., Medina, D., Dodds, P.S.: Multiscale, resurgent epidemics in a hierarchical metapopulation model. PNAS 102(32), 11157–11162 (2005)CrossRefGoogle Scholar
  20. 20.
    Zhong, M., Shen, K.: Random Walk Based Node Sampling in Self-Organizing Networks. ACM SIGOPS Operating Systems Review (SPECIAL ISSUE: Self-organizing systems), 49–55 (2006)Google Scholar

Copyright information

© ICST Institute for Computer Science, Social Informatics and Telecommunications Engineering 2009

Authors and Affiliations

  • Yuri Dimitrov
    • 1
  • Mario Lauria
    • 2
  1. 1.Department of MathematicsOhio State UniversityColumbusUSA
  2. 2.Telethon Institute of Genetics and Medicine (TIGEM)NaplesItaly

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