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Synchronization and Control of Hyper-Networks and Colored Networks

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Complex Systems and Networks

Part of the book series: Understanding Complex Systems ((UCS))

Abstract

In this chapter, hyper-networks and colored networks corresponding to hyper-graphs and colored graphs in mathematics are presented, which can be used to model real large-scale systems, such as neuronal networks, metabolic networks, social relationship networks, scientific collaboration networks, and so on. Firstly, similarly to the BA scale-free network, both growth and preferential attachment mechanisms are adopted to generate some evolving hyper-network models, including uniform and nonuniform hyper-networks. Secondly, a uniform dynamical hyper-network model is built and its synchronization is investigated using joint-degree matrix. Thirdly, a colored network with same-dimensional node dynamics is presented. The synchronization and control of both edge-colored and colored networks are studied. Finally, a general colored network with different-dimensional node dynamics is presented and its generalized matrix projective synchronization is achieved by applying open-plus-closed-loop control.

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References

  1. Berge, C.: Graphs and Hypergraphs, vol. 6. Elsevier, New York (1973)

    Book  MATH  Google Scholar 

  2. Berge, C.: Hypergraphs: Combinatorics of Finite Sets, Vol. 45. North-Holland Holl, Amsterdam (1989)

    Google Scholar 

  3. Yang, G.Y., Hu, Z.L., Liu, J.G.: Knowledge diffusion in the collaboration hypernetwork. Physica A 419, 429–436 (2015)

    Article  Google Scholar 

  4. Hu, F., Zhao, H.X., He, J.B., Li, F.X., Li, S.L., Zhang, Z.K.: An evolving model for hypergraph-structure-based scientific collaboration networks. Acta Phys. Sin. 62, 198901 (2013) (in Chinese)

    Google Scholar 

  5. Wang, J.W., Rong, L.L., Deng, Q.H., Zhang, J.Y.: Evolving hypernetwork model. Eur. Phys. J. B 77, 493–498 (2010)

    Google Scholar 

  6. Barabási, A.-L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)

    Article  MathSciNet  Google Scholar 

  7. Wu, Z.Y., Duan, J.Q., Fu, X.C.: Synchronization of an evolving complex hyper-network. Appl. Math. Model. 38, 2961–2968 (2014)

    Article  MathSciNet  Google Scholar 

  8. Guo, J.L., Zhu, X.Y.: Emergence of scaling in hypernetworks. Acta Phys. Sin. 63, 090207 (2014) (in Chinese)

    Google Scholar 

  9. Guo, J.L.: Emergence of scaling in non-uniform hypernetworks-does the rich get richer lead to a power-law distribution? Acta Phys. Sin. 63, 208901 (2014) (in Chinese)

    Google Scholar 

  10. Yang, G.Y., Liu, J.G.: A local-world evolving hypernetwork model. Chin. Phys. B 23, 018901 (2014)

    Google Scholar 

  11. Hu, F., Zhao, H.X., Ma, X.J.: An evolving hypernetwork model and its properties. Sin. China. Phys. Mech. Astron. 43, 16–22 (2013) (in Chinese)

    Google Scholar 

  12. Chen, G.R., Wang, X.F., Li, X.: Introduction to Complex Networks: Models, Structure and Dynamics. High Education Press, Beijing (2012)

    Google Scholar 

  13. Wang, X.F., Chen, G.R.: Synchronization in scale-free dynamical networks: robustness and fragility. IEEE Trans. Circuits Syst. I (49), 54–62 (2002)

    Article  Google Scholar 

  14. Chua, L.O., Wu, C.W., Huang, A., Zhong, G.Q.: A universal circuit for studying and generating chaos—part I: Routes to chaos. IEEE Trans. Circuits Syst. I(40), 732–742 (1993)

    Google Scholar 

  15. Krawiecki, A.: Chaotic synchronization on complex hypergraphs. Chaos Solitons and Fractals 65, 44–50 (2014)

    Article  MathSciNet  Google Scholar 

  16. Chavez, M., Hwang, D.-U., Amann, A., Hentschel, H.G.E., Boccaletti, S.: Synchronization is enhanced in weighted complex networks. Phys. Rev. Lett. 94, 218701 (2005)

    Article  Google Scholar 

  17. Hell, P., Manoussakis, Y., Tuza, Z.: Packing problems in edge-colored graphs. Discrete Appl. Math. 52, 295–306 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zielonka, W.: Infinite games on finitely coloured graphs with applications to automata on infinite trees. Theor. Comput. Sci. 200, 135–183 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fujita, S., Nakamigawa, T.: Balanced decomposition of a vertex-colored graph. Discrete Appl. Math. 156, 3339–3344 (2008)

    Article  MathSciNet  Google Scholar 

  20. Becu, J.M., Dah, M., Manoussakis, Y., Mendy, G.: Links in edge-colored graphs. Eur. J. Combin. 31, 442–460 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Orlitsky, A., Venkatesh, S.S.: On edge-colored interior planar graphs on a circle and the expected number of RNA secondary structures. Discrete Appl. Math. 64, 151–178 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wang, Y., Desmedt, Y.: Edge-colored graphs with applications to homogeneous faults. Inform. Process Lett. 111, 634–641 (2011)

    Article  MathSciNet  Google Scholar 

  23. Wu, Z.Y., Xu, X.J., Chen, G.R., Fu, X.C.: Adaptive synchronization and pinning control of colored networks. Chaos 22, 043137 (2012)

    Article  Google Scholar 

  24. Song, Q., Cao, J., Liu, F.: Synchronization of complex dynamical networks with nonidentical nodes. Phys. Lett. A 374, 544–551 (2010)

    Article  MATH  Google Scholar 

  25. Lorenz, E.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)

    Article  Google Scholar 

  26. Erdös, P., Rényi, A.: On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci. 5, 17–60 (1960)

    MATH  Google Scholar 

  27. Rössler, O.E.: An equation for continuous chaos. Phys. Lett. A 57, 397–398 (1976)

    Article  Google Scholar 

  28. Newman, M.E.J., Watts, D.J.: Renormalization group analysis of the small-world network model. Phys. Lett. A 263, 341–346 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  30. Chen, G.R., Ueta, T.: Yet another chaotic attractor. Int. J. Bifur. Chaos 9, 1465–1466 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wu, Z.Y., Xu, X.J., Chen, G.R., Fu, X.C.: Generalized matrix projective synchronization of general colored networks with different-dimensional node dynamics. J. Franklin Inst. 351, 4584–4595 (2014)

    Article  MathSciNet  Google Scholar 

  32. Jia, Q.: Hyperchaos generated from the Lorenz chaotic system and its control. Phys. Lett. A 366, 217–222 (2007)

    Article  Google Scholar 

  33. Chen, A., Lu, J., Lü, J., Yu, S.: Generating hyperchaotic Lü attractor via state feedback control. Physica A 364, 103–110 (2006)

    Article  Google Scholar 

  34. Belykh, V.N., Chua, L.O.: New type of strange attractor from a geometric model of Chua’s circuit. Int. J. Bifur. Chaos 2, 697–704 (1992)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

    Xinchu Fu

  2. College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, 330022, People’s Republic of China

    Zhaoyan Wu

  3. Department of Electronic Engineering, City University of Hong Kong, Hong Kong, People’s Republic of China

    Guanrong Chen

Authors
  1. Xinchu Fu
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  2. Zhaoyan Wu
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  3. Guanrong Chen
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Corresponding author

Correspondence to Xinchu Fu .

Editor information

Editors and Affiliations

  1. Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

    Jinhu Lü

  2. School of Electrical and Computer Engineering, RMIT University, Melbourne, Australia

    Xinghuo Yu

  3. Department of Electronic Engineering, City University of Hong Kong, Hong Kong, China

    Guanrong Chen

  4. Department of Mathematics, Southeast University, Nanjing, China

    Wenwu Yu

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Fu, X., Wu, Z., Chen, G. (2016). Synchronization and Control of Hyper-Networks and Colored Networks. In: Lü, J., Yu, X., Chen, G., Yu, W. (eds) Complex Systems and Networks. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47824-0_5

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  • DOI: https://doi.org/10.1007/978-3-662-47824-0_5

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-47823-3

  • Online ISBN: 978-3-662-47824-0

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