Approximate Consensus in Highly Dynamic Networks: The Role of Averaging Algorithms

  • Bernadette Charron-Bost
  • Matthias Függer
  • Thomas NowakEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9135)


We investigate the approximate consensus problem in highly dynamic networks in which topology may change continually and unpredictably. We prove that in both synchronous and partially synchronous networks, approximate consensus is solvable if and only if the communication graph in each round has a rooted spanning tree. Interestingly, the class of averaging algorithms, which have the benefit of being memoryless and requiring no process identifiers, entirely captures the solvability issue of approximate consensus in that the problem is solvable if and only if it can be solved using any averaging algorithm.

We develop a proof strategy which for each positive result consists in a reduction to the nonsplit networks. It dramatically improves the best known upper bound on the decision times of averaging algorithms and yields a quadratic time non-averaging algorithm for approximate consensus in non-anonymous networks. We also prove that a general upper bound on the decision times of averaging algorithms have to be exponential, shedding light on the price of anonymity.

Finally we apply our results to networked systems with a fixed topology and benign fault models to show that with n processes, up to \(2n-3\) of link faults per round can be tolerated for approximate consensus, increasing by a factor 2 the bound of Santoro and Widmayer for exact consensus.


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  1. 1.
    Angluin, D., Fischer, M.J., Jiang, H.: Stabilizing consensus in mobile networks. In: Gibbons, P.B., Abdelzaher, T., Aspnes, J., Rao, R. (eds.) DCOSS 2006. LNCS, vol. 4026, pp. 37–50. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  2. 2.
    Attiya, H., Lynch, N.A., Shavit, N.: Are wait-free algorithms fast? J. ACM 41(4), 725–763 (1994)zbMATHCrossRefGoogle Scholar
  3. 3.
    Attiya, H., Welch, J.: Distributed Computing. Wiley, Hoboken (2005) Google Scholar
  4. 4.
    Bertsekas, D.P., Tsitsiklis, J.N.: Parallel and Distributed Computation: Numerical Methods. Athena Scientific, Belmont (1989) zbMATHGoogle Scholar
  5. 5.
    Biely, M., Robinson, P., Schmid, U.: Agreement in directed dynamic networks. In: Even, G., Halldórsson, M.M. (eds.) SIROCCO 2012. LNCS, vol. 7355, pp. 73–84. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  6. 6.
    Blondel, V., Olshevshy, A.: How to decide consensus? A combinatorial necessary and sufficient condition and a proof that consensus is decidable but NP-hard. SIAM J. Control Optim. 52(5), 2707–2726 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Cao, M., Morse, A.S., Anderson, B.D.O.: Reaching a consensus in a dynamically changing environment: a graphical approach. SIAM J. Control Optim. 47(2), 575–600 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Cao, M., Morse, A.S., Anderson, B.D.O.: Reaching a consensus in a dynamically changing environment: convergence rates, measurement delays, and asynchronous events. SIAM J. Control Optim. 47(2), 601–623 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Charron-Bost, B.: Orientation and connectivity based criteria for asymptotic consensus (2013). arXiv:1303.2043v1 [cs.DC]
  10. 10.
    Charron-Bost, B., Schiper, A.: The Heard-Of model: computing in distributed systems with benign faults. Distrib. Comput. 22(1), 49–71 (2009)zbMATHCrossRefGoogle Scholar
  11. 11.
    Chung, F.R.: Spectral Graph Theory. AMS, Providence (1997) zbMATHGoogle Scholar
  12. 12.
    Coulouma, É., Godard, E.: A characterization of dynamic networks where consensus is solvable. In: Moscibroda, T., Rescigno, A.A. (eds.) SIROCCO 2013. LNCS, vol. 8179, pp. 24–35. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  13. 13.
    Daubechies, I., Lagarias, J.C.: Sets of matrices all infinite products of which converge. Linear Algebra Appl. 161, 227–263 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Dolev, D., Lynch, N.A., Pinter, S.S., Stark, E.W., Weihl, W.E.: Reaching approximate agreement in the presence of faults. J. ACM 33(2), 499–516 (1986)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Fekete, A.D.: Asymptotically optimal algorithms for approximate agreement. Distrib. Comput. 4(1), 9–29 (1990)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Fischer, M.J., Lynch, N.A., Paterson, M.S.: Impossibility of distributed consensus with one faulty process. J. ACM 32(2), 374–382 (1985)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Kuhn, F., Lynch, N.A., Oshman, R.: Distributed computation in dynamic networks. In: 42nd ACM Symposium on Theory of Computing, pp. 513–522. ACM, New York City (2010)Google Scholar
  18. 18.
    Kuhn, F., Moses, Y., Oshman, R.: Coordinated consensus in dynamic networks. In: 30th Annual ACM Symposium on Principles of Distributed Computing, pp. 1–10. ACM, New York City (2011)Google Scholar
  19. 19.
    Lynch, N.A.: Distributed Algorithms. Morgan Kaufmann, San Francisco (1996) zbMATHGoogle Scholar
  20. 20.
    Olshevsky, A., Tsitsiklis, J.N.: Degree fluctuations and the convergence time of consensus algorithms (2011), arXiv:1104.0454v1 [math.OC]
  21. 21.
    Santoro, N., Widmayer, P.: Time is not a healer. In: Monien, B., Cori, R. (eds.) 6th Symposium on Theoretical Aspects of Computer Science, LNCS, vol. 349, pp. 304–313. Springer, Heidelberg (1989)Google Scholar
  22. 22.
    Tsitsiklis, J.N.: Problems in Decentralized Decision Making and Computation. Ph.D. thesis, Massachusetts Institute of Technology (1984)Google Scholar
  23. 23.
    Xia, W., Cao, M.: Sarymsakov matrices and their application in coordinating multi-agent systems. In: 31st Chinese Control Conference, pp. 6321–6326. IEEE, New York City (2012)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Bernadette Charron-Bost
    • 1
  • Matthias Függer
    • 2
  • Thomas Nowak
    • 3
    Email author
  1. 1.CNRSÉcole polytechniquePalaiseauFrance
  2. 2.Max-Planck-Institut für InformatikSaarbruckenGermany
  3. 3.ENS ParisParisFrance

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