Dynamic Games and Applications

, Volume 8, Issue 2, pp 423–433 | Cite as

Algebraic Formulation and Nash Equilibrium of Competitive Diffusion Games

Article

Abstract

This paper investigates the algebraic formulation and Nash equilibrium of competitive diffusion games by using semi-tensor product method, and gives some new results. Firstly, an algebraic formulation of competitive diffusion games is established via the semi-tensor product of matrices, based on which all the fixed points (the end of the diffusion process) are obtained. Secondly, using the algebraic formulation, a necessary and sufficient condition is presented for the verification of pure-strategy Nash equilibrium. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained new results.

Keywords

Competitive diffusion game Pure-strategy Nash equilibrium Algebraic formulation Semi-tensor product of matrices 

Notes

Acknowledgements

The authors would like to thank the anonymous reviewers for their constructive comments and suggestions which improved the quality of this paper.

References

  1. 1.
    Alon N, Feldman M, Procaccia AD, Tennenholtz M (2010) A note on competitive diffusion through social networks. Inf Process Lett 110:221–225MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bharathi S, Kempe D, Salek M (2007) Competitive influence maximization in social networks. In: Proceedings of international conference on internet and network economics, pp 306–311Google Scholar
  3. 3.
    Chen H, Sun J (2013) Global stability and stabilization of switched Boolean network with impulsive effects. Appl Math Comput 224:625–634MathSciNetMATHGoogle Scholar
  4. 4.
    Cheng D, Qi H, Li Z (2011) Analysis and control of boolean networks: a semi-tensor product approach. Springer, LondonCrossRefMATHGoogle Scholar
  5. 5.
    Cheng D (2014) On finite potential games. Automatica 50(7):1793–1801MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cheng D, Xu T, Qi H (2014) Evolutionarily stable strategy of networked evolutionary games. IEEE Trans Neural Netw Learn Syst 25(7):1335–1345CrossRefGoogle Scholar
  7. 7.
    Cheng D, He F, Qi H, Xu T (2015) Modeling, analysis and control of networked evolutionary games. IEEE Trans Autom Control 60(9):2402–2415MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Etesami SR, Basar T (2016) Complexity of equilibrium in competitive diffusion games on social networks. Automatica 68:100–110MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Fornasini E, Valcher ME (2013) Observability, reconstructibility and state observers of Boolean control networks. IEEE Trans Autom Control 58(6):1390–1401MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ghaderi J, Srikant R (2013) Opinion dynamics in social networks: a local interaction game with stubborn agents. In: Proccedings of 2013 American control conference, pp 1982–1987Google Scholar
  11. 11.
    Goyal S, Kearns M (2012) Competitive contagion in networks. In: Proceedings of the 44th symposium on theory of computing, pp 759–774Google Scholar
  12. 12.
    Guo P, Wang Y, Li H (2013) Algebraic formulation and strategy optimization for a class of evolutionary network games via semi-tensor product method. Automatica 49(11):3384–3389MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Han M, Liu Y, Tu Y (2014) Controllability of Boolean control networks with time delays both in states and inputs. Neurocomputing 129:467–475CrossRefGoogle Scholar
  14. 14.
    Jadbabaie A, Lin J, Morse AS (2003) Coordination of groups of mobile autonomous agents using nearest neighbor. IEEE Trans Autom Control 48(6):1675–1675MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Khanafer A, Basar T (2014) Information spread in networks: control, games, and equilibria. In: 2014 information theory and applications workshop, pp 1–10Google Scholar
  16. 16.
    Laschov D, Margaliot M (2013) Minimum-time control of Boolean networks. SIAM J Control Optim 51(4):2869–2892MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Li H, Wang Y (2015) Controllability analysis and control design for switched Boolean networks with state and input constraints. SIAM J Control Optim 53(5):2955–2979MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Li H, Xie L, Wang Y (2016) On robust control invariance of Boolean control networks. Automatica 68:392–396MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Liu X, Zhu J (2016) On potential equations of finite games. Automatica 68:245–253MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Liu Y, Chen H, Lu J, Wu B (2015) Controllability of probabilistic Boolean control networks based on transition probability matrices. Automatica 52:340–345MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Li H, Wang Y, Xie L (2015) Output tracking control of Boolean control networks via state feedback: constant reference signal case. Automatica 59:54–59MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Lu J, Li H, Liu Y, Li F (2017) A survey on semi-tensor product method with its applications in logical networks and other finite-valued systems. IET Control Theory Appl. doi: 10.1049/iet-cta.2016.1659 MathSciNetGoogle Scholar
  23. 23.
    Meng M, Feng J (2014) Topological structure and the disturbance decoupling problem of singular Boolean networks. IET Control Theory Appl 8(13):1247–1255MathSciNetCrossRefGoogle Scholar
  24. 24.
    Tang Y, Wang Z, Fang J (2009) Pinning control of fractional-order weighted complex networks. Chaos 19(1):193–204MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Wang X, Chen G, Ching W (2002) Pinning control of scale-free dynamical networks. Phys A 310(3–4):521–531MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Xu X, Hong Y (2013) Matrix approach to model matching of asynchronous sequential machines. IEEE Trans Autom Control 58(11):2974–2979MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Yang M, Li R, Chu T (2013) Controller design for disturbance decoupling of Boolean control networks. Automatica 49(1):273–277MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Young HP (2000) The diffusion of innovations in social networks. Gen Inf 413(1):2329–2334Google Scholar
  29. 29.
    Yu W, Chen G, Lv J (2009) On pinning synchronization of complex dynamical networks. Automatica 45:429–435MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Zhang J, Huang Z, Dong J, Huang L, Lai Y (2013) Controlling collective dynamics in complex minority-game resource-allocation systems. Phys Rev E 87:052808CrossRefGoogle Scholar
  31. 31.
    Zhang L, Zhang K (2013) Controllability and observability of Boolean control networks with time-variant delays in states. IEEE Trans Neural Netw Learn Syst 24:1478–1484CrossRefGoogle Scholar
  32. 32.
    Zhao Y, Li Z, Cheng D (2011) Optimal control of logical control networks. IEEE Trans Autom Control 56(8):1766–1776MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Zhong J, Lu J, Liu Y, Cao J (2014) Synchronization in an array of output-coupled Boolean networks with time delay. IEEE Trans Neural Netw Learn Syst 25:2288–2294CrossRefGoogle Scholar
  34. 34.
    Zhu B, Xia X, Wu Z (2016) Evolutionary game theoretic demand-side management and control for a class of networked smart grid. Automatica 70:94–100MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Zou Y, Zhu J (2015) Kalman decomposition for Boolean control networks. Automatica 54:65–71MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Haitao Li
    • 1
    • 2
  • Xueying Ding
    • 1
  • Qiqi Yang
    • 1
  • Yingrui Zhou
    • 1
  1. 1.School of Mathematics and StatisticsShandong Normal UniversityJinanPeople’s Republic of China
  2. 2.Institute of Data Science and TechnologyShandong Normal UniversityJinanPeople’s Republic of China

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