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Algebraic Formulation and Nash Equilibrium of Competitive Diffusion Games

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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.

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Notes

  1. The reason why we choose this kind of diffusion process is that it captures the simple fact that being closer to player’s initial seeds will result in adopting that specific player’s type.

References

  1. Alon N, Feldman M, Procaccia AD, Tennenholtz M (2010) A note on competitive diffusion through social networks. Inf Process Lett 110:221–225

    Article  MathSciNet  MATH  Google Scholar 

  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 306–311

  3. Chen H, Sun J (2013) Global stability and stabilization of switched Boolean network with impulsive effects. Appl Math Comput 224:625–634

    MathSciNet  MATH  Google Scholar 

  4. Cheng D, Qi H, Li Z (2011) Analysis and control of boolean networks: a semi-tensor product approach. Springer, London

    Book  MATH  Google Scholar 

  5. Cheng D (2014) On finite potential games. Automatica 50(7):1793–1801

    Article  MathSciNet  MATH  Google Scholar 

  6. Cheng D, Xu T, Qi H (2014) Evolutionarily stable strategy of networked evolutionary games. IEEE Trans Neural Netw Learn Syst 25(7):1335–1345

    Article  Google Scholar 

  7. Cheng D, He F, Qi H, Xu T (2015) Modeling, analysis and control of networked evolutionary games. IEEE Trans Autom Control 60(9):2402–2415

    Article  MathSciNet  MATH  Google Scholar 

  8. Etesami SR, Basar T (2016) Complexity of equilibrium in competitive diffusion games on social networks. Automatica 68:100–110

    Article  MathSciNet  MATH  Google Scholar 

  9. Fornasini E, Valcher ME (2013) Observability, reconstructibility and state observers of Boolean control networks. IEEE Trans Autom Control 58(6):1390–1401

    Article  MathSciNet  MATH  Google Scholar 

  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 1982–1987

  11. Goyal S, Kearns M (2012) Competitive contagion in networks. In: Proceedings of the 44th symposium on theory of computing, pp 759–774

  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):3384–3389

    Article  MathSciNet  MATH  Google Scholar 

  13. Han M, Liu Y, Tu Y (2014) Controllability of Boolean control networks with time delays both in states and inputs. Neurocomputing 129:467–475

    Article  Google Scholar 

  14. Jadbabaie A, Lin J, Morse AS (2003) Coordination of groups of mobile autonomous agents using nearest neighbor. IEEE Trans Autom Control 48(6):1675–1675

    Article  MathSciNet  MATH  Google Scholar 

  15. Khanafer A, Basar T (2014) Information spread in networks: control, games, and equilibria. In: 2014 information theory and applications workshop, pp 1–10

  16. Laschov D, Margaliot M (2013) Minimum-time control of Boolean networks. SIAM J Control Optim 51(4):2869–2892

    Article  MathSciNet  MATH  Google Scholar 

  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):2955–2979

    Article  MathSciNet  MATH  Google Scholar 

  18. Li H, Xie L, Wang Y (2016) On robust control invariance of Boolean control networks. Automatica 68:392–396

    Article  MathSciNet  MATH  Google Scholar 

  19. Liu X, Zhu J (2016) On potential equations of finite games. Automatica 68:245–253

    Article  MathSciNet  MATH  Google Scholar 

  20. Liu Y, Chen H, Lu J, Wu B (2015) Controllability of probabilistic Boolean control networks based on transition probability matrices. Automatica 52:340–345

    Article  MathSciNet  MATH  Google Scholar 

  21. Li H, Wang Y, Xie L (2015) Output tracking control of Boolean control networks via state feedback: constant reference signal case. Automatica 59:54–59

    Article  MathSciNet  MATH  Google Scholar 

  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

    MathSciNet  Google Scholar 

  23. Meng M, Feng J (2014) Topological structure and the disturbance decoupling problem of singular Boolean networks. IET Control Theory Appl 8(13):1247–1255

    Article  MathSciNet  Google Scholar 

  24. Tang Y, Wang Z, Fang J (2009) Pinning control of fractional-order weighted complex networks. Chaos 19(1):193–204

    Article  MathSciNet  MATH  Google Scholar 

  25. Wang X, Chen G, Ching W (2002) Pinning control of scale-free dynamical networks. Phys A 310(3–4):521–531

    Article  MathSciNet  MATH  Google Scholar 

  26. Xu X, Hong Y (2013) Matrix approach to model matching of asynchronous sequential machines. IEEE Trans Autom Control 58(11):2974–2979

    Article  MathSciNet  MATH  Google Scholar 

  27. Yang M, Li R, Chu T (2013) Controller design for disturbance decoupling of Boolean control networks. Automatica 49(1):273–277

    Article  MathSciNet  MATH  Google Scholar 

  28. Young HP (2000) The diffusion of innovations in social networks. Gen Inf 413(1):2329–2334

    Google Scholar 

  29. Yu W, Chen G, Lv J (2009) On pinning synchronization of complex dynamical networks. Automatica 45:429–435

    Article  MathSciNet  MATH  Google Scholar 

  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 87:052808

    Article  Google Scholar 

  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 24:1478–1484

    Article  Google Scholar 

  32. Zhao Y, Li Z, Cheng D (2011) Optimal control of logical control networks. IEEE Trans Autom Control 56(8):1766–1776

    Article  MathSciNet  MATH  Google Scholar 

  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 25:2288–2294

    Article  Google Scholar 

  34. Zhu B, Xia X, Wu Z (2016) Evolutionary game theoretic demand-side management and control for a class of networked smart grid. Automatica 70:94–100

    Article  MathSciNet  MATH  Google Scholar 

  35. Zou Y, Zhu J (2015) Kalman decomposition for Boolean control networks. Automatica 54:65–71

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, People’s Republic of China

    Haitao Li, Xueying Ding, Qiqi Yang & Yingrui Zhou

  2. Institute of Data Science and Technology, Shandong Normal University, Jinan, 250014, People’s Republic of China

    Haitao Li

Authors
  1. Haitao Li
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  2. Xueying Ding
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  3. Qiqi Yang
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  4. Yingrui Zhou
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Corresponding author

Correspondence to Haitao Li.

Additional information

The research was supported by the National Natural Science Foundation of China under Grants 61374065 and 61503225, the Natural Science Foundation of Shandong Province under Grant ZR2015FQ003, and the Natural Science Fund for Distinguished Young Scholars of Shandong Province under Grant JQ201613.

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Li, H., Ding, X., Yang, Q. et al. Algebraic Formulation and Nash Equilibrium of Competitive Diffusion Games. Dyn Games Appl 8, 423–433 (2018). https://doi.org/10.1007/s13235-017-0228-4

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