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Matching Algorithms of Minimum Input Selection for Structural Controllability Based on Semi-Tensor Product of Matrices

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Abstract

In 2011, Liu, et al. investigated the structural controllability of directed networks. They proved that the minimum number of input signals, driver nodes, can be determined by seeking a maximum matching in the directed network. Thus, the algorithm for seeking a maximum matching is the key to solving the structural controllability problem of directed networks. In this study, the authors provide algebraic expressions for matchings and maximum matchings proposed by Liu, et al. (2011) via a new matrix product called semi-tensor product, based on which the corresponding algorithms are established to seek matchings and maximum matchings in digraphs, which make determining the number of driver nodes tractable in computer. In addition, according to the proposed algorithm, the authors also construct an algorithm to distinguish critical arcs, redundant arcs and ordinary arcs of the directed network, which plays an important role in studying the robust control problem. An example of a small network from Liu’s paper is used for algorithm verification.

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References

  1. Lin C T, Structural controllability, IEEE Transactions on Automatic Control, 1974, 19(3): 201–208.

    Article  MathSciNet  Google Scholar 

  2. Liu Y, Slotine J J, and Barabási A, Controllability of complex networks, Nature, 2011, 473(7346): 167–173.

    Article  Google Scholar 

  3. Edmonds J, Maximum matching and a polyhedron with (0, 1) vertices, Res. Nat. Bur. Standards, 1965, 69(1–2): 125–130.

    Article  MathSciNet  Google Scholar 

  4. Yu S S, Zhong S, and Hu S H, Matching algorithms for general graphs based on depth-first search, Computer Engineering and Science, 2008, 30: 45–48.

    Google Scholar 

  5. Madry A, Navigating central path with electrical flows: From flows to matchings, and back, Foundations of Computer Science, 2013, 253–262.

    MATH  Google Scholar 

  6. Yuan Z Z, Zhao C, Di Z R, et al., Exact controllability of complex networks, Nature Communications, 2013, 4: 2447.

    Article  Google Scholar 

  7. Olshevsky A, Minimum input selection for structural controllability, Proceedings of the American Control Conference, 2015, 2218–2223.

    Google Scholar 

  8. Yin H L and Zhang S Y, Minimum structural controllability problems of complex networks, Physica A: Statistical Mechanics and Its Applications, 2016, 443: 467–476.

    Article  MathSciNet  Google Scholar 

  9. Bai T, Li S Y, Zou Y Y, et al., Block-based minimum input design for the structural controllability of complex networks, Automatica, 2019, 107: 68–76.

    Article  MathSciNet  Google Scholar 

  10. Cheng D Z, Qi H S, and Zhao Y, An Introduction to Semi-Tensor Product of Matrices and Its Applications, Word Scientific, Beijing, 2012.

    Book  Google Scholar 

  11. Wang Y Z, Zhang C H, and Liu Z B, A matrix approach to graph maximum stable set and coloring problems with application to multi-agent systems, Automatica, 2012, 48(7): 1227–1236.

    Article  MathSciNet  Google Scholar 

  12. Meng M and Feng J E, A matrix approach to hypergraph stable set and coloring problems with its application to storing problem, Journal of Applied Mathematics, 2014, 2014(3): 1–9.

    MathSciNet  MATH  Google Scholar 

  13. Xu M and Wang Y, Robust graph coloring based on the matrix semi-tensor product with application to examination timetabling, Control Theory and Technology, 2014, 12(2): 187–197.

    Article  MathSciNet  Google Scholar 

  14. Liu Z B and Wu Y Q, L(p, q)-label coloring problem via the semi-tensor product method, Proceedings of the 2020 39th Chinese Control Conference (CCC), Shenyang, 2020.

    Google Scholar 

  15. Zhong J, Lu J Q, Huang C, et al., Finding graph minimum stable set and core via semi-tensor product approach, Neurocomputing, 2016, 174: 588–596.

    Article  Google Scholar 

  16. Yan Y Y, Yue J M, Chen Z Q, et al., Algebraic expression and construction of control sets of graphs using semi-tensor product of matrices, IEEE Access, 2019, 7: 113440–113451.

    Article  Google Scholar 

  17. Liu X Y and Zhu J D, On potential equations of finite games, Automatica, 2016, 68: 245–253.

    Article  MathSciNet  Google Scholar 

  18. Guo P L and Wang Y Z, The computation of Nash equilibrium in fashion games via semi-tensor product method, Journal of Systems Science & Complexity, 2016, 29(4): 881–896.

    Article  MathSciNet  Google Scholar 

  19. Chen L, Networked evolutionary model of snow-drift game based on semi-tensor product, Journal of Applied Mathematics and Physics, 2019, 7(3): 726–737.

    Article  Google Scholar 

  20. Dou W H, Li H T, and Alsaadi F E, Semi-tensor product approach to controllability, reachability, and stabilizability of probabilistic finite automata, Mathematical Problems in Engineering, 2019, 2019: 1–7.

    MATH  Google Scholar 

  21. Yue J M, Yan Y Y, and Chen Z Q, Three matrix conditions for the reduction of finite automata based on the theory of semi-tensor product of matrices, Science China Information Sciences, 2020, 63(2): 1–3.

    Article  MathSciNet  Google Scholar 

  22. Lu J Q, Li M L, Liu Y, et al., Nonsingularity of Grain-like cascade FSRs via semi-tensor product, Science China Information Sciences, 2018, 61(1): 1–12.

    Article  MathSciNet  Google Scholar 

  23. Wang X Y and Gao S, Image encryption algorithm for synchronously updating Boolean networks based on matrix semi-tensor product theory, Information Sciences, 2020, 507: 16–36.

    Article  MathSciNet  Google Scholar 

  24. Wang B and Feng J E, On detectability of probabilistic Boolean networks, Information Sciences, 2019, 483: 383–395.

    Article  MathSciNet  Google Scholar 

  25. Meng M, James L, Feng J E, et al., Stability and stabilization of Boolean networks with stochastic delays, IEEE Transactions on Automatic Control, 2019, 64(2): 790–796.

    MathSciNet  MATH  Google Scholar 

  26. Kong X S, Wang S L, Li H T, et al., New developments in control design techniques of logical control networks, Frontiers of Information Technology and Electronic Engineering, 2020, 21: 220–233.

    Article  Google Scholar 

  27. Wen W Y, Hong Y K, Fang Y M, et al. A visually secure image encryption scheme based on semi-tensor product compressed sensing, Signal Processing, 2020, 173: 107580.

    Article  Google Scholar 

  28. Liu Y C and Zhang W, Boolean Methodology, Shanghai Technology Literature, Shanghai, 1993.

    Google Scholar 

  29. Bondy J A and Murty U S R, Graph Theory, Springer, New York, 2008.

    Book  Google Scholar 

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Authors and Affiliations

  1. College of Intelligent Systems Science and Engineering, Harbin Engineering University, Harbin, 150001, China

    Naqi Fan & Lijun Zhang

  2. School of Marine Technology, Northwestern Polytechnical University, Xi’an, 710072, China

    Lijun Zhang

  3. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710129, China

    Shenggui Zhang

  4. College of Big Data Statistics, Guizhou University of Finance and Economics, Guiyang, 550025, China

    Jiuqiang Liu

Authors
  1. Naqi Fan
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  2. Lijun Zhang
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  3. Shenggui Zhang
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  4. Jiuqiang Liu
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Corresponding author

Correspondence to Lijun Zhang.

Additional information

This research was supported by the National Natural Science Foundation of China under Grant Nos. 61573288, 12071370, U1803263, 71973103 and Key Programs in Shaanxi Province of China under Grant No. 2021JZ-12.

This paper was recommended for publication by Editor QI Hongsheng.

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Fan, N., Zhang, L., Zhang, S. et al. Matching Algorithms of Minimum Input Selection for Structural Controllability Based on Semi-Tensor Product of Matrices. J Syst Sci Complex 35, 1808–1823 (2022). https://doi.org/10.1007/s11424-022-1178-5

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  • DOI: https://doi.org/10.1007/s11424-022-1178-5

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