Automation and Remote Control

, Volume 78, Issue 1, pp 88–99 | Cite as

Models of latent consensus

  • R. P. AgaevEmail author
  • P. Yu. Chebotarev
Intellectual Control Systems


The paper studies the problem of achieving consensus in multi-agent systems in the case where the dependency digraph Γ has no spanning in-tree. We consider the regularization protocol that amounts to the addition of a dummy agent (hub) uniformly connected to the agents. The presence of such a hub guarantees the achievement of an asymptotic consensus. For the “evaporation” of the dummy agent, the strength of its influences on the other agents vanishes, which leads to the concept of latent consensus. We obtain a closed-form expression for the consensus when the connections of the hub are symmetric; in this case, the impact of the hub upon the consensus remains fixed. On the other hand, if the hub is essentially influenced by the agents, whereas its influence on them tends to zero, then the consensus is expressed by the scalar product of the vector of column means of the Laplacian eigenprojection of Γ and the initial state vector of the system. Another protocol, which assumes the presence of vanishingly weak uniform background links between the agents, leads to the same latent consensus.

Key words

consensus multi-agent system decentralized control regularization eigenprojection DeGroot’s iterative pooling PageRank Laplacian matrix of a digraph 


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  1. 1.
    Olfati-Saber, R. and Murray, R.M., Consensus Problems in Networks of Agents with Switching Topology and Time-delays, IEEE Trans. Automat. Control, 2004, vol. 49, no. 9, pp. 1520–1533.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Mesbahi, M. and Egerstedt, M., Graph Theoretic Methods in Multiagent Networks, Princeton: Princeton Univ. Press, 2010.CrossRefzbMATHGoogle Scholar
  3. 3.
    Agaev, R.P. and Chebotarev, P.Yu., The ProjectionMethod for Reaching Consensus and the Regularized Power Limit of a Stochastic Matrix, Autom. Remote Control, 2011, vol. 72, no. 12, pp. 2458–2476.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Agaev, R.P. and Chebotarev, P.Yu., The Projection Method for Continuous-Time Consensus Seeking, Autom. Remote Control, 2015, vol. 76, no. 8, pp. 1436–1445.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Lin, Z., Low Gain Feedback, London: Springer, 1999.zbMATHGoogle Scholar
  6. 6.
    Seo, J.H., Shim, H., and Back, J., Consensus of High-Order Linear Systems Using Dynamic Output Feedback Compensator: Low Gain Approach, Automatica, 2009, vol. 45, no. 11, pp. 2659–2664.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Su, H., Chen, M.Z., Lam, J., and Lin, Z., Semi-Global Leader-Following Consensus of Linear Multi- Agent Systems with Input Saturation via Low Gain Feedback, IEEE Trans. Circuits Syst. I: Regular Papers, 2013, vol. 60, no. 7, pp. 1881–1889.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ren, W. and Cao, Y., Distributed Coordination of Multi-Agent Networks: Emergent Problems, Models, and Issues, London: Springer, 2011.CrossRefzbMATHGoogle Scholar
  9. 9.
    Chebotarev, P. and Agaev, R., Forest Matrices around the Laplacian Matrix, Linear Algebra Appl., 2002, vol. 356, pp. 253–274.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chebotarev, P. and Agaev, R., The Forest Consensus Theorem, IEEE Trans. Automat. Control, 2014, vol. 59, no. 9, pp. 2475–2479.MathSciNetCrossRefGoogle Scholar
  11. 11.
    Agaev, R.P. and Chebotarev, P.Yu., The Matrix of Maximum Out Forests of a Digraph and Its Applications, Autom. Remote Control, 2000, vol. 61, no. 9, pp. 1424–1450.MathSciNetzbMATHGoogle Scholar
  12. 12.
    Chebotarev, P.Yu. and Shamis, E.V., The Matrix-Forest Theorem and Measuring Relations in Small Social Groups, Autom. Remote Control, 1997, vol. 58, no. 9, pp. 1505–1514.zbMATHGoogle Scholar
  13. 13.
    DeGroot, M.H., Reaching a Consensus, J. Am. Stat. Ass., 1974, vol. 69, no. 345, pp. 118–121.CrossRefzbMATHGoogle Scholar
  14. 14.
    Brin, S. and Page, L., The Anatomy of a Large-Scale Hypertextual Web Search Engine, Comput. Networks ISDN Syst., 1998, vol. 30, pp. 107–117.CrossRefGoogle Scholar
  15. 15.
    Langville, A.N. and Meyer, C.D., Google’s PageRank and Beyond: The Science of Search Engine Rankings, Princeton: Princeton Univ. Press, 2006.zbMATHGoogle Scholar
  16. 16.
    Polyak, B.T. and Tremba, A.A., Regularization-Based Solution of the PageRank Problem for Large Matrices, Autom. Remote Control, 2012, vol. 73, no. 11, pp. 1877–1894.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ishii, H. and Tempo, R., The PageRank Problem, Multiagent Consensus, and Web Aggregation: A Systems and Control Viewpoint, IEEE Control Syst. Mag., 2014, vol. 34, no. 3, pp. 34–53.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Agaev, R.P. and Chebotarev, P.Yu., Spanning Forests of a Digraph and Their Applications, Autom. Remote Control, 2001, vol. 62, no. 3, pp. 443–466.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gantmacher, F.R., The Theory of Matrices, New York: Chelsea, 1959.zbMATHGoogle Scholar
  20. 20.
    Meyer, C.D., Jr., The Role of the Group Generalized Inverse in the Theory of Finite Markov Chains, SIAM Rev., 1975, vol. 17, no. 3, pp. 443–464.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rothblum, U.G., Computation of the Eigenprojection of a Nonnegative Matrix at Its Spectral Radius, in Stochastic Systems: Modeling, Identification and Optimization II, ser. Mathematical Programming Study, Wets, R.J.-B., Ed., Amsterdam: North-Holland, 1976, vol. 6, pp. 188–201.CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Trapeznikov Institute of Control SciencesRussian Academy of SciencesMoscowRussia

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