Enhancing Synchronizability of Complex Networks via Optimization

  • Cuili Yang
  • Qiang Jia
  • Wallace K. S. TangEmail author
Part of the Understanding Complex Systems book series (UCS)


Optimization problems are commonly encountered in the area of complex networks. Due to the high complexity of the involved networks, these problems are usually tackled with deterministic approaches. On the other hand, metaheuristic algorithms have received a lot of attentions and have been successful applied for many difficult problems. In this chapter, it is to showcase how to use the metaheuristic algorithms to provide better solutions to the optimization problems in related to complex networks. Our focus is on the synchronization of complex networks, which not only possesses its own distinct theoretical complexity but also is useful for many practical applications. Two major synchronization problems are presented. The first one is to obtain the best network that exhibits an optimal synchronizability, while the numbers of nodes and edges are fixed. A hybrid approach, combining Tabu search and a greedy local search using edge rewiring, is suggested. The second one is on pinning control. Given a network, it is to select a fraction of nodes and assign the appropriate control gains so that all the nodes in the network follow some predefined dynamics. The problem is solved by a novel genetic algorithm with hierarchical chromosome structure. In both cases, the effectiveness of the designed metaheuristic algorithms is justified with simulation results, and it is concluded that they outperform the existing methods.


Tabu Search Control Gain Laplacian Matrix Tabu List Metaheuristic Algorithm 
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.


  1. 1.
    Banzhaf, W., Nordin, P., Keller, R.E., Francone, F.D.: Genetic Programming: An Introduction: On the Automatic Evolution of Computer Programs and Its Applications. Morgan Kaufmann, Los Altos (1998)Google Scholar
  2. 2.
    Barabsi, A.L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)Google Scholar
  3. 3.
    Barahona, M., Pecora, L.M.: Synchronization in small-world systems. Phys. Rev. Lett. 89(5), 054101 (2002)Google Scholar
  4. 4.
    Bernardo, M.D., Garofalo, F., Sorrentino, F.: Effects of degree correlation on the synchroniztion of networks of oscillators. Int. J. Bifurcat. Chaos 17, 3499–3506 (2007)Google Scholar
  5. 5.
    Beyer, H.G., Schwefel, H.P.: Evolution strategies: A comprehensive introduction. Nat. Comput.: Int. J. 1(1), 3–52 (2002)Google Scholar
  6. 6.
    Brandes, U., Pich, C.: Centrality estimation in large networks. Int. J. Bifurcat. Chaos 17, 2303–2318 (2007)Google Scholar
  7. 7.
    Buck, J.: Synchronous rhythmic flashing of fireflies. Quart. Rev. Biol. 63(3), 265–287 (1988)Google Scholar
  8. 8.
    Cancho, R.F.I., Solé, R.V.: Optimization in complex networks, arXiv:cond-mat/0111222v1 (2001)Google Scholar
  9. 9.
    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)Google Scholar
  10. 10.
    Choudhury, M., Mukherjee, A.: The structure and dynamics of linguistic networks. In: Ganguly, N., Deutsch, A., Mukherjee, A. (eds.) Dynamics on and of Complex Networks: Applications to Biology, Computer Science, Economics, and the Social Sciences, pp. 145–166. Springer, Birkhauser (2009)Google Scholar
  11. 11.
    Colorni, A., Dorigo, M., Maniezzo, V.: Distributed optimization by ant colonies. In: Proceedings of European Conference on Artificial Life, pp. 134–142 (1991)Google Scholar
  12. 12.
    Colorni, A., Dorigo, M., Maniezzo, V.: An investigation of some properties of an ant algorithm. In: Proceedings of the Conference on Parallel Problem Solving from Nature, pp. 509–520 (1992)Google Scholar
  13. 13.
    Danila, B., Yu, Y., Marsh, J.A., Bassler, K.E.: Transport optimization on complex network. Chaos 17, 026102 (2007)Google Scholar
  14. 14.
    Dehmer, M., Emmert-Streib, F.: Analysis of Complex Networks: From Biology to Linguistics. Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim (2009)Google Scholar
  15. 15.
    Donetti, L., Hurtado, P.I., Munoz, M.A.: Entangled networks, synchronization and optimal network topology. Phys. Rev. Lett. 95, 188701 (2005)Google Scholar
  16. 16.
    Donetti, L., Hurtado, P.I., Munoz, M.A.: Network synchronization: Optimal and pessimal scale-free topologies. J. Phys. A: Math. Theoret. 41, 224008 (2008)Google Scholar
  17. 17.
    Donetti, L., Neri, F., Munoz, M.A.: Optimal network topologies: Expanders, cages, Ramanujan graphs, entangled networks and all that. J. Stat. Mech. Theory Exp. 8, P08007 (2006)Google Scholar
  18. 18.
    Fogel, L.J., Owens, A.J., Walsh, M.J.: Artificial Intelligence through Simulated Evolution. Wiley, New York (1966)Google Scholar
  19. 19.
    Glover, F.: Tabu search: Part I. ORSA J. Comput. 1(3), 190–206 (1989)Google Scholar
  20. 20.
    Glover, F.: Tabu search: Part II. ORSA J. Comput. 2(1), 4–32 (1990)Google Scholar
  21. 21.
    Goldberg, D.E.: Genetic Algorithms in Search, Optimization and Machine Learning. Kluwer, Boston (1989)Google Scholar
  22. 22.
    Hagberg, A., Schult, D.A.: Rewiring networks for synchronization. Chaos 18, 037105 (2008)Google Scholar
  23. 23.
    Holland, J.H.: Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor (1975)Google Scholar
  24. 24.
    Holme, P., Kim, B.J.: Growing scale-free networks with tunable clustering. Phys. Rev. E 65, 026107 (2002)Google Scholar
  25. 25.
    Jalili, M., Rad, A.A.: Comment on “Rewiring networks for synchronization” [Chaos 18, 037105 (2008)]. Chaos 19, 028101 (2009)Google Scholar
  26. 26.
    Jeong, H., Tombor, B., Albert, R., Oltavi, Z.N., Barabasi, A.L.: The large-scale organization of metablic networks. Nature 407, 651–654 (2000)Google Scholar
  27. 27.
    Jia, Q., Tang, W.K.S., Halang, W.A.: Leader following of nonlinear agents with switching connective network and coupling delay. IEEE Trans. Circ. Syst. I 58(10), 2508–2519 (2011)Google Scholar
  28. 28.
    Jia, Z., Li, X., Rong, Z.: Pinning complex dynamical networks with local betweeness centrality information. In: Proceedings of Chinese Control Conference, pp. 5969–5974 (2011)Google Scholar
  29. 29.
    Kennedy, J., Eberhart, R.C.: Particle swarm optimization. In: Proceedings of the IEEE International Conference on Neural Networks, vol. 4, pp. 1942–1948 (1995)Google Scholar
  30. 30.
    Koza, J.: Genetic Programming: On the Programming of Computers by Means of Natural Selection. MIT, Cambridge (1992)Google Scholar
  31. 31.
    Kwoha, C.K., Nga, P.Y.: Network analysis approach for biology. Cellular Mol. Life Sci. 64, 1739–1751 (2007)Google Scholar
  32. 32.
    Liu, X.F., Tse, C.K.: A complex network perspective to volatitlity in stock markets. In: Proceedings of 2010 International Symposium on Nonlinear Theory and Its Applications, pp. 402–405 (2010)Google Scholar
  33. 33.
    Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)Google Scholar
  34. 34.
    Lu, W.: Adaptive dynamical networks via neighborhood information: Synchronization and pinning control. Chaos 17, 023122 (2007)Google Scholar
  35. 35.
    Man, K.F., Tang, K.S., Kwong, S.: Genetic Algorithms: Concepts and Designs. Springer, London (1999)Google Scholar
  36. 36.
    Marshall, J.A.: Formations of vehicles in cyclic pursuit. IEEE Trans. Automat. Contr. 49(11), 1963–1974 (2004)Google Scholar
  37. 37.
    Mishkovski, I., Righero, M., Biey, M., Kocarev, L.: Building synchronizable and robust networks. In: Proceedings of IEEE International Symposium on Circuits and Systems, pp. 681–684 (2010)Google Scholar
  38. 38.
    Nishikawa, T., Motter, A.E., Lai, Y.C., Hoppensteadt, F.C.: Heterogeneity in oscillator networks: Are smaller worlds easier to synchronize? Phys. Rev. Lett. 91, 014101 (2003)Google Scholar
  39. 39.
    Niwa, H.S.: Self-organizing dynamic model of fish schooling. J. Theoret. Biol. 171(2), 123–136 (1994)Google Scholar
  40. 40.
    Olfati-Saber, R., Murray, R.M.: Distributed cooperative control of multiple vehicle formations using structural potential functions. In: Proceedings of the 15th IFAC World Congress (2002)Google Scholar
  41. 41.
    Pecora, L.M., Carroll, T.L.: Master stability functions for synchronized coupled systems. Phys. Rev. Lett. 80, 2109–2112 (1998)Google Scholar
  42. 42.
    Price, K., Storn, R.: Differential evolution – A simple and efficient heuristic for global optimization over continuous spaces. J. Global Optim. 11(4), 341–359 (1997)Google Scholar
  43. 43.
    Rad, A.A., Jalili, M., Hasler, M.: Efficient rewirings for enhancing synchronizability of dynamical networks. Chaos 18, 037104 (2008)Google Scholar
  44. 44.
    Rechenberg, I.: Evolutionsstrategie ’94. Frommann-Holzboog, Stuttgart (1994)Google Scholar
  45. 45.
    Ren, W.: On Consensus algorithms for double-integrator dynamics. IEEE Trans. Automat. Contr. 53(6), 1503–1509 (2008)Google Scholar
  46. 46.
    Ren, W., Beard, R.W.: Distributed Consensus in Multi-vehicle Cooperative Control: Theory and Applications. Springer, London (2008)Google Scholar
  47. 47.
    Rong, Z.H., Li, X., Lu, W.L.: Pinning a complex network through the betweenness centrality strategy. In: Proceedings of the IEEE International Symposium on Circuits and Systems, pp. 1689–1692 (2009)Google Scholar
  48. 48.
    Song, Q., Cao, J.: On pinning synchronization of directed and undirected complex dynamical networks. IEEE Trans. Circ. Syst. I 57(3), 672–680 (2010)Google Scholar
  49. 49.
    Syswerda, G.: Uniform crossover in genetic algorithms. In: Proceedings of the 3rd International Conference on Genetic Algorithms, pp. 2–9 (1989)Google Scholar
  50. 50.
    Tang, W.K.S., Ng, K.H., Jia, Q.: A degree-based strategy for constrained pinning control of complex networks. Int. J. Bifurcat. Chaos 20(5), 1533–1539 (2010)Google Scholar
  51. 51.
    Trpevski, D., Tang, W.K.S., Kocarev, L.: Model for rumor spreading over networks. Phys. Rev. E 81, 056102 (2010)Google Scholar
  52. 52.
    Wang, B., Zhou, T., Xiu, Z.L., Kim, B.J.: Optimal synchronizability of networks. Eur. Phys. J. B 60, 89–95 (2007)Google Scholar
  53. 53.
    Wang, J., Rong, L., Guo, T.: A new measure of node importance in complex networks with tunable parameters. In: Proceedings of 4th International Conference on Wireless Communications, Networking and Mobile Computing, pp. 1–4 (2008)Google Scholar
  54. 54.
    Wang, L.F., Wang, Q.L., Kong, Z., Jing, Y.W.: Enhancing synchronizability by rewiring networks. Chin. Phys. B 19, 080207 (2010)Google Scholar
  55. 55.
    Wang, X., Lai, Y.C., Lai, C.H.: Enhancing synchronization based on complex gradient networks. Phys. Rev. E 75, 056205 (2007)Google Scholar
  56. 56.
    Wang, X.F., Chen, G.: Pinning control of scale-free dynamical networks. Physica A 310, 521–531 (2002)Google Scholar
  57. 57.
    Wang, X.F., Li, X., Lu, J.: Control and flocking of networked systems via pinning. IEEE Circ. Syst. Mag. 10, 83–91 (2010)Google Scholar
  58. 58.
    Wasserman, S., Faust, K.: Social Network Analysis: Methods and Applications. Cambridge University Press, New York (1994)Google Scholar
  59. 59.
    Watanabe, T., Masuda, N.: Enhancing the spectral gap of networks by node removal. Phys. Rev. E 82, 046102 (2010)Google Scholar
  60. 60.
    Watts, D., Strogatz, S.: Collective dynamics of “small-world” network. Nature 393, 440–442 (1998)Google Scholar
  61. 61.
    Williams, R.J., Berlow, E.L., Jennifer, A.D., Barabasi, A.L., Martinez, N.D.: Two degrees of separation in complex food webs. Proc. Nat. Acad. Sci. USA 99, 12913–12916 (2002)Google Scholar
  62. 62.
    Wu, C.: Localization of effective pinning control in complex networks of dynamical systems. In: Proceedings of the IEEE International Symposium on Circuits and Systems, pp. 2530–2533 (2008)Google Scholar
  63. 63.
    Wu, C.W.: Synchronization in an array of linearly coupled dynamical systems. IEEE Trans. Circ. Syst. I 42(8), 430–447 (1995)Google Scholar
  64. 64.
    Wu, Y., Wei, W., Li, G., Xiang, J.: Pinning control of uncertain complex networks to a homogeneous orbit. IEEE Trans. Circ. Syst. II 56(3), 235–239 (2009)Google Scholar
  65. 65.
    Yang, C.L., Tang, K.S.: Enhancing the synchronizability of networks by rewiring based on tabu search and a local greedy algorithm. Chin. Phys. B. 20(12), 128901 (2011)Google Scholar
  66. 66.
    Yang, C.L., Tang, W.K.S.: A degree-based genetic algorithm for constrained pinning control in complex network. Accepted by International Symposium on Circuits and Systems (2012)Google Scholar
  67. 67.
    Yang, C.L., Tang, W.K.S., Jia, Q.: Node selection and gain assignement in pinning control using genetic algorithm. Submitted to 38th Annual Conference of the IEEE Industrial Electronics Society (2012)Google Scholar
  68. 68.
    Yu, W., Chen, G., Lu, J.: On pinning synchronization of complex dynamical networks. Automatica 45, 429–435 (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Electronic EngineeringCity University of Hong KongHong KongChina

Personalised recommendations