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.
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References
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)
Barabási, A.-L., Ravasz, E.: Hierarchical organization in complex networks. Physical review E 67, 026112 (2003)
Burton, R.M., Faris, W.G.: A Self-Organizing Cluster Process. Ann. Appl. Prob. 6(4), 1232–1247 (1996)
Caldarelli, G.: Scale-Free Networks. Oxford University Press, Oxford (2007)
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)
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)
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)
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)
Dimitrov, Y., Lauria, M.: A Combinatorial model for self-organizing networks. Technical Report TR02, Department of Computer Science and Engineering, Ohio State University (2007)
Erdös, P., Rényi, A.: On random graphs. Publicationes Mathematicae 6, 290–297 (1959)
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)
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)
Kostic, D., Rodriguez, A., Albrecht, J., Vahdat, A.: Bullet: High Bandwidth Data Dissemination Using an Overlay Mesh. In: Proc. of ACM SOSP (2003)
Krapivsky, P.L., Redner, S., Leyvraz, F.: Connectivity of growing random networks. Phys. Rev. Lett. 85 (2000)
Newman, M.E.J.: Random graphs as models of networks. arXiv:cond-mat/0202208v1 (2002)
Shi, D., Chen, Q., Liu, L.: Markov chain-based numerical method for degree distributions of growing networks. Physical review E 71 (2005)
Wang, C.: Stochastic Models for Self-organizing Networks and Infinite Graphs. Ph.D. thesis, Dalhousie University (2006)
Watts, D., Dodds, P.S., Newman, M.E.J.: Identity and Search in Social networks. Science 296, 1302–1305 (2002)
Watts, D., Muhamad, R., Medina, D., Dodds, P.S.: Multiscale, resurgent epidemics in a hierarchical metapopulation model. PNAS 102(32), 11157–11162 (2005)
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)
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© 2009 ICST Institute for Computer Science, Social Informatics and Telecommunications Engineering
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Dimitrov, Y., Lauria, M. (2009). A Stochastic Model for Layered Self-organizing Complex Systems. In: Zhou, J. (eds) Complex Sciences. Complex 2009. Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02469-6_29
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DOI: https://doi.org/10.1007/978-3-642-02469-6_29
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