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Reaching a nonlinear consensus: Polynomial stochastic operators

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Abstract

We provide a general nonlinear protocol for a structured time-varying and synchronous multi-agent system. We present an opinion sharing dynamics of the multi-agent system as a trajectory of a polynomial stochastic operator associated with a multidimensional stochastic hypermatrix. We show that the multi-agent system eventually reaches to a consensus if either one of the following two conditions is satisfied: (i) every member of the group people has a positive subjective opinion on the given task after some revision steps or (ii) all entries of a multidimensional stochastic hypermatrix are positive. Numerical results are also presented.

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References

  1. M. H. DeGroot, “Reaching a consensus,” Journal of the American Statistical Association, vol. 69, no. 345, pp. 118–121, 1974.

    Article  MATH  Google Scholar 

  2. T. Vicsek, A. Czirok, E. Ben-Jacob, I. Cohen, and O. Shochet, “Novel type of phase transition in a system of self-driven particles,” Phys. Rev. Letter, vol. 75, no. 6, pp. 1226–1229, 1995.

    Article  Google Scholar 

  3. R. Olfati-Saber and R. M. Murray, “Consensus problems in networks of agents with switching topology and time-delays,” IEEE Trans. on Automatic Control, vol. 49, no. 6, pp. 1520–1533, 2004.

    Article  MathSciNet  Google Scholar 

  4. A. Tahbaz-Salehi and A. Jadbabaie, “A necessary and sufficient condition for consensus over random networks,” IEEE Trans. on Automatic Control, vol. 53, no. 3, pp. 791–795, 2008.

    Article  MathSciNet  Google Scholar 

  5. X. Liu, T. Chen, and W. Lu, “Consensus problem in directed networks of multi-agents via nonlinear protocols,” Physics Letters A, vol. 373, no. 35, pp. 3122–3127, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  6. M. Porfiri and D. J. Stilwell, “Consensus seeking over random weighted directed graphs,” IEEE Trans. on Automatic Control, vol. 52, no. 9, pp. 1767–1773, 2007.

    Article  MathSciNet  Google Scholar 

  7. Z. Guan, C. Meng, R. Liao, and D. Zhang, “Consensus of second-order multi-agent dynamic systems with quantized data,” Physics Letters A, vol. 376, no. 4, pp. 387–393, 2012.

    Article  MATH  Google Scholar 

  8. N. Crokidakis and F. Forgerini, “Consequence of reputation in the Sznajd consensus model,” Physics Letters A, vol. 374, no. 34, 3380–3383, 2010.

    Article  MATH  Google Scholar 

  9. Y. Zhang and Y. Yang, “Finite-time consensus of second-order leader-following multi-agent systems without velocity measurements,” Physics Letters A, vol. 377, no. 3–4, pp. 243–249, 2013.

    Article  MATH  Google Scholar 

  10. K. Peng and Y. Yang, “Leader-following consensus problem with a varying-velocity leader and time-varying delays,” Physica A: Statistical Mechanics and its Applications, vol. 388, no. 2, pp. 193–208, 2009.

    Article  Google Scholar 

  11. Y. Liu, F. Xionga, J. Zhu, and Y. Zhang, “External activation promoting consensus formation in the opinion model with interest decay,” Physics Letters A, vol. 377, no. 5, pp. 362–366, 2013.

    Article  Google Scholar 

  12. W. Yu, G. Chen, and M. Cao, “Consensus in directed networks of agents with nonlinear dynamics,” IEEE Trans. on Automatic Control, vol. 56, no. 6, pp. 1436–1441, 2011.

    Article  MathSciNet  Google Scholar 

  13. H. Su, G. Chen, X. Wang, and Z. Lin, “Adaptive second-order consensus of networked mobile agents with nonlinear dynamics,” Automatica, vol. 47, no. 2, pp. 368–375, 2011.

    Article  MATH  MathSciNet  Google Scholar 

  14. Q. Wang, Z. Duan, G. Chen, and Z. Feng, “Synchronization in a class of weighted complex networks with coupling delays,” Physica A: Statistical Mechanics and its Applications, vol. 387, no. 22, pp. 5616–5622, 2008.

    Article  Google Scholar 

  15. W. Ren and R. W. Beard, “Consensus seeking in multiagent systems under dynamically changing interaction topologies,” IEEE Trans. on Automatic Control, vol. 50, no. 5, pp. 655–661, 2005.

    Article  MathSciNet  Google Scholar 

  16. Q. Ma, S. Xu, and F. L. Lewis, “Second-order consensus for directed multi-agent systems with sampled data,” International Journal of Robust and Nonlinear Control vol. 23, no. 10, 2013.

    Google Scholar 

  17. Q. Ma, Z. Wang, and G. Miao, “Second-order group consensus for multi-agent systems via pinning leader-following approach,” Journal of the Franklin Institute, vol. 351, no. 3, pp. 1288–1300, 2014.

    Article  Google Scholar 

  18. J. Luo and C. Cao, “Consensus in multi-agent systems with nonlinear uncertainties under a fixed undirected graph,” International Journal of Control, Automation, and Systems, vol. 12, no. 2, pp. 231–240, 2014.

    Article  MathSciNet  Google Scholar 

  19. L. Xiao and X. Liao, “Periodic intermittent consensus of second-order agents networks with nonlinear dynamics,” International Journal of Control, Automation, and Systems, vol. 12, no. 1, pp. 23–28, 2014.

    Article  MathSciNet  Google Scholar 

  20. H. Li, X. Liao, and G. Chen, “Leader-following finite-time consensus in second-order multi-agent networks with nonlinear dynamics,” International Journal of Control, Automation, and Systems, vol. 11, no. 2, pp. 422–426, 2013.

    Article  Google Scholar 

  21. C. Castellano, S. Fortunato, and V. Loreto, “Statistical physics of social dynamics,” Rev. Mod. Phys., vol. 81, no. 2, pp. 591–646, 2009.

    Article  Google Scholar 

  22. V. Sood and S. Redner, “Voter model on heterogeneous graphs,” Phys. Rev. Lett., vol. 94, no. 17, (178701), pp. 1–4, 2005.

    Article  Google Scholar 

  23. S. Galam, “Minority opinion spreading in random geometry,” Eur. Phys. J. B, vol. 25, pp. 403–406, 2002.

    Google Scholar 

  24. P. L. Krapivsky and S. Redner, “Dynamics of majority rule in two-state interacting spin systems,” Phys. Rev. Letters, vol. 90, no. 23, (238701), pp. 1–4, 2003.

    Article  Google Scholar 

  25. A. Nowak, M. Kus, J. Urbaniak, and T. Zarycki, “Simulating the coordination of individual economic decisions,” Physica A, vol. 287, no. 3–4, pp. 613–630, 2000.

    Article  Google Scholar 

  26. R. Albert and A.-L. Barabasi, “Statistical mechanics of complex networks,” Rev. Mod. Phys. vol. 74, no. 1, pp. 47–97, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  27. M. J. de Oliveira, “Isotropic majority-vote model on a square lattice,” J. Stat. Phys. vol. 66, no. 1–2, pp. 273–281, 1992.

    Article  MATH  Google Scholar 

  28. A. D. Sanchez, J. M. Lopez, and M. A. Rodriguez, “Non-equilibrium phase transitions in directed small-world networks,” Phys. Rev. Letters, vol. 88, no. 4, (048701), pp. 1–4, 2002.

    Article  Google Scholar 

  29. P. Chen and S. Redner, “Majority rule dynamics in finite dimensions,” Phys. Rev. E, vol. 71, no. 3, (036101), pp. 1–7, 2005.

    Google Scholar 

  30. P.-P. Li, D.-F. Zheng, and P.M. Hui, “Dynamics of opinion formation in a small-world network,” Phys. Rev. E, vol. 73, no. 5, (056128), pp. 1–5, 2006.

    Google Scholar 

  31. F. Vazquez, V. M. Eguiluz, and M. S. Miguel, “Generic absorbing transition in coevolution dynamics,” Phys. Rev. Letters, vol. 100, no. 10, (108702), pp. 1–4, 2008.

    Article  Google Scholar 

  32. C. Nardini, B. Kozma, and A. Barrat, “Who’s talking first? consensus or lack thereof in coevolving opinion formation models,” Phys. Rev. Letters, vol. 100, no. 15, (158701), pp. 1–4, 2008.

    Article  Google Scholar 

  33. J. Shao, S. Havlin, and H. E. Stanley, “Dynamic opinion model and invasion percolation,” Phys. Rev. Letters, vol. 103, no. 1, (018701), pp. 1–4, 2009.

    Article  Google Scholar 

  34. Z.-X. Wu and P. Holme, “Majority-vote model on hyperbolic lattices,” Phys. Rev. E, vol. 81, no. 1, (011133), pp. 1–7, 2010.

    Google Scholar 

  35. C. M. Schneider-Mizell and L. M. Sander, “A generalized voter model on complex networks,” J. Stat. Phys., vol. 136, no. 1, pp. 59–71, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  36. Z.-X. Wu and P. Holme, “Local interaction scale controls the existence of a nontrivial optimal critical mass in opinion spreading,” Phys. Rev. E, vol. 82, no. 2, (022102), pp. 1–4, 2010.

    Article  Google Scholar 

  37. S. Galam, “Contrarian deterministic effects on opinion dynamics: the hung elections scenario,” Physica A: Statistical Mechanics and its Applications, vol. 333, no. 15, pp. 453–460, 2004.

    Article  MathSciNet  Google Scholar 

  38. S. Galam, “Heterogeneous beliefs, segregation, and extremism in the making of public opinions,” Phys. Rev. E, vol. 71, no. 4, (046123), pp. 1–5, 2005.

    Article  Google Scholar 

  39. K. Suchecki, V. M. Eguiluz, and M. S. Miguel, “Voter model dynamics in complex networks: Role of dimensionality, disorder, and degree distribution,” Phys. Rev. E, vol. 72, no. 3, (036132), pp. 1–8, 2005.

    Article  Google Scholar 

  40. C. Castellano and R. Pastor-Satorras, “Irrelevance of information outflow in opinion dynamics models,” Phys. Rev. E, vol. 83, no. 1, (016113), pp. 1–4, 2011.

    Article  Google Scholar 

  41. P. Erdos and A. Renyi, “On random graphs I,” Publicationes Mathematicae Debrecen, vol. 6, pp. 290–297, 1959.

    MathSciNet  Google Scholar 

  42. M. E. J. Newman and D. J. Watts, “Renormalization group analysis of the small-world network model,” Phys. Letters A, vol. 263, no. 4–6, pp. 341–346, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  43. A.-L. Barabasi and R. Albert, “Emergence of scaling in random networks,” Science, vol. 286, no. 5439, pp. 509–512, 1999.

    Article  MathSciNet  Google Scholar 

  44. A. Papachristodoulou and A. Jadbabaie, “Synchronization in oscillator networks: switching topologies and non-homogeneous delays,” Proc. of 44th IEEE Decision and Control, pp. 5692–5697, 2005.

    Chapter  Google Scholar 

  45. R. L. Berger, “A necessary and sufficient condition for reaching a consensus using DeGroot’s method,” J. Amer. Stat. Assoc., vol. 76, no. 374, pp. 415–418, 1981.

    Article  MATH  Google Scholar 

  46. S. Chatterjee and E. Seneta, “Towards consensus: some convergence theorems on repeated averaging,” J. App. Prob., vol. 14, no. 1, pp. 89–97, 1977.

    Article  MATH  MathSciNet  Google Scholar 

  47. Y. I. Lyubich, Mathematical Structures in Population Genetics, Springer, 1992.

    Book  MATH  Google Scholar 

  48. N. Ganikhodjaev, M. Saburov, and U. Jamilov, “Mendelian and non-mendelian quadratic operators,” Applied Mathematics & Information Sciences, vol. 7, no. 5, pp. 1721–1729, 2013.

    Article  MathSciNet  Google Scholar 

  49. R. Ganikhodzhaev, F. Mukhamedov, and U. Rozikov, “Quadratic stochastic operators and processes: results and open problems,” Infinite Dimensional Analysis, Quantum Probability and Related Topics, vol. 14, no. 2, pp. 279–335, 2011.

    Article  MATH  MathSciNet  Google Scholar 

  50. F. Mukhamedov and M. Saburov, “On homotopy of volterrian quadratic stochastic operator,” Applied Mathematics & Information Sciences, vol. 4, no. 1, pp. 47–62, 2010.

    MATH  MathSciNet  Google Scholar 

  51. M. Saburov, “Some strange properties of quadratic stochastic volterra operators,” World Applied Sciences Journal, vol. 21, pp. 94–97, 2013.

    Google Scholar 

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Authors and Affiliations

  1. Computational & Theoretical Sciences, Kulliyyah of Science, International Islamic University Malaysia, Jalan Bandar Indera Mahkota, 25200, Kuantan, Pahang, Malaysia

    Mansoor Saburov

  2. Pramenni 20/3, 301 00, Pilsen, Czech Republic

    Khikmat Saburov

Authors
  1. Mansoor Saburov
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  2. Khikmat Saburov
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Correspondence to Mansoor Saburov.

Additional information

Recommended by Editor Ju Hyun Park.

This work has been supported in part by the grant IIUM-EDW B 13-019-0904. The first author is grateful to the Junior Associate scheme of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

Mansoor Saburov received his B.S. and M.S. degrees in Pure Mathematics from National University Uzbekistan, in 2005 and 2007, respectively. He received his Ph.D. in Pure Mathematics from International Islamic University Malaysia in 2011. He completed his Post Doc during the period 2011-2012. Since 2012, He is an Assistant Professor at International Islamic University Malaysia. His research interests include nonlinear control, dynamical system, ergodic theory, functional analysis.

Khikmat Saburov received his B.S. degree in Pure Mathematics from National University Uzbekistan in 2003. He received his M.S. and Ph.D. in Applied Mathematics from University of West Bohemia, Czech Republic, in 2006 and 2011, respectively. His research interests include graph theory, discrete mathematics, nonlinear control.

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Saburov, M., Saburov, K. Reaching a nonlinear consensus: Polynomial stochastic operators. Int. J. Control Autom. Syst. 12, 1276–1282 (2014). https://doi.org/10.1007/s12555-014-0061-0

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  • DOI: https://doi.org/10.1007/s12555-014-0061-0

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