Convergence speed of unsteady distributed consensus: Decay estimate along the settling spanning-trees

SIAM Journal on Control and Optimization

Volumn 48, Issue 1, 2009, Pages 1-32

  Angeli, David ^a,b   Bliman, Pierre Alexandre ^c  

^a UNIVERSITY OF FLORENCE   (Italy)

^b IMPERIAL COLLEGE LONDON   (United Kingdom)

^c INRIA ROCQUENCOURT   (France)

Author keywords

Convergence rate; Distributed consensus; Graph theory; Multiagent systems

Indexed keywords

GRAPH THEORY; MULTI AGENT SYSTEMS;

COMMUNICATION GRAPHS; CONVERGENCE RATES; CONVERGENCE SPEED; DECAY ESTIMATE; DISTRIBUTED CONSENSUS; NONSTATIONARY; QUANTITATIVE INFORMATION; SPANNING TREE;

TREES (MATHEMATICS);

EID: 60349121213     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/060673527     Document Type: Article

References (34)

1. M. AKAR AND R. SHORTEN, Time synchronization for wireless sensors networks, in Proceedings of the 17th Symposium on Mathematical Theory of Networks and Systems, Kyoto, Japan, 2006, pp. 221-225.
2. H. ANDO, Y. OASA, I. SUZUKI, AND M. YAMASHITA, Distributed memoryless point convergence algorithm for mobile robots with limited visibility, IEEE Trans. Robotics and Automation, 15 (1999), pp. 818-828.
3. D. ANGELI AND P.-A. BLIMAN, Stability of leaderless discrete-time multi-agent systems, Math. Control Signals Systems, 18 (2006), pp. 293-322.
4. D. ANGELI AND P.-A. BLIMAN, Convergence Speed of Unsteady Distributed Consensus: Decay Estimate Along the Settling Spanning-Trees, http://arxiv.org/abs/math.OC/0610854 (2006).
5. A. BERMAN, T. LAFFEY, T. LEIZAROWITZ, AND R. SHORTEN, On the second eigenvalues of matrices associated with TCP, Linear Algebra Appl., 416 (2006), pp. 175-183.
6. D. P. BERTSEKAS AND J. N. TSITSIKLIS, Parallel and Distributed Computation, Prentice-Hall International, London, 1989; also available online at https://dspace.mit.edu/handle/1721.1/3719.
7. V. D. BLONDEL, J. M. HENDRICKX, A. OLSHEVSKY, AND J. N. TSITSIKLIS, Convergence in multiagent coordination, consensus, and flocking, in Proceedings of the Joint European Control Conference/44th Annual IEEE Conference on Decision and Control, Sevilla, Spain, 2005, pp. 2996-3000.
8. S. BOYD, A. GHOSH, B. PRABHAKAR, AND D. SHAH, Randomized gossip algorithms, IEEE Trans. Inform. Theory (special issue of IEEE Trans. Inform. Theory and IEEE/ACM Trans. Networking), 52 (2006), pp. 2508-2530.
9. M. CAO, A. S. MORSE, AND B. D. O. ANDERSON, Reaching a consensus in a dynamically changing environment: Convergence rates, measurement delays, and asynchronous events, SIAM J. Control Optim., 47 (2008), pp. 601-623.
10. M. CAO, D. A. SPIELMAN, AND A. S. MORSE, A lower bound on convergence of a distributed network consensus algorithm, in Proceedings of the Joint European Control Conference/44th Annual IEEE Conference on Decision and Control, Sevilla, Spain, 2005, pp. 2356-2361.
11. J. CHEEGER, A lower bound for the smallest eigenvalue of the Laplacian, in Problems in Analysis. Papers Dedicated to Salomon Bochner, Princeton University Press, Princeton, NJ, 1969, pp. 195-199.
12. H. COHN, Products of stochastic matrices and applications, Int. J. Math. Math. Sci., 12 (1989), pp. 209-233.
13. A. CZIRÓK AND T. VICSEK, Collective motion, in Statistical Mechanics of Biocomplexity, D. Reguera, M. Rubi, and J. Vilar, eds., Lecture Notes in Phys. 527, Springer-Verlag, Berlin, 1999, pp. 152-164.
14. P. DIACONIS AND D. STROOCK, Geometric bounds for eigenvalues of Markov chains, Ann. Appl. Probab., 1 (1991), pp. 36-61.
15. S. FRIEDLAND AND R. NABBEN, On the second real eigenvalue of nonnegative and Z-matrices, Linear Algebra Appl., 255 (1997), pp. 303-313.
16. P. M. GADE, Feedback control in coupled map lattices, Phys. Rev. E, 57 (1998), pp. 7309-7312.
17. S. T. GARREN AND R. L. SMITH, Estimating the second largest eigenvalue of a Markov transition matrix, Bernoulli, 6 (2000), pp. 215-242.
18. O. GROSS AND U. G. ROTHBLUM, Approximations of the spectral radius, corresponding eigenvector, and second largest modulus of an eigenvalue for square, nonnegative, irreducible matrices, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 15-32.
19. Y. HATANO AND M. MESBAHI, Agreement over random networks, IEEE Trans. Automat. Control, 50 (2005), pp. 1867-1872.
20. M. W. HIRSCH, Convergent activation dynamics in continuous time networks, Neural Networks, 2 (1989), pp. 331-349.
21. A. JADBABAIE, N. MOTEE, AND M. BARAHONA, On the stability of the Kuramoto model of coupled nonlinear oscillators, in Proceedings of the IEEE American Control Conference, Boston, MA, 2004, pp. 4296-4301.
22. H. J. LANDAU AND A. M. ODLYZKO, Bounds for eigenvalues of certain stochastic matrices, Linear Algebra Appl., 38 (1981), pp. 5-15.
23. N. E. LEONARD AND E. FIORELLI, Virtual leaders, artificial potentials, and coordinated control of groups, in Proceedings of the 40th Annual IEEE Conference on Decision and Control, Orlando, FL, 2001, pp. 2968-2973.
24. J. LORENZ, Consensus strikes back in the Hegselmann-Krause model of continuous opinion dynamics under bounded confidence, J. Artificial Societies and Social Simulation, 9 (2006); available online at http://jasss.soc.surrey.ac.Uk/9/l/8.html
25. S. C. MANRUBIA AND A. S. MIKHAILOV, Mutual synchronization and clustering in randomly coupled chaotic dynamical networks, Phys. Rev. E, 60 (1999), pp. 1579-1589; also available online at http://arxiv.org/abs/cond-mat/9905083.
26. L. MOREAU, Time-Dependent Unidirectional Communication in Multi-Agent Systems, http://arxiv.org/abs/math/0306426 (2003).
27. L. MOREAU, A note on leaderless communication via bidirectional and unidirectional timedependent communication, in Proceedings of the MTNS 2004, Leuven, Belgium, 2004.
28. L. MOREAU, Stability of multi-agent systems with time-dependent communication links, IEEE Trans. Automat. Control, 50 (2005), pp. 169-182.
29. A. NEDICH AND A. OZDAGLAR, Convergence Rate for Consensus with Delays, LIDS Technical Report 2774, Laboratory for Information and Decision Systems, MIT, Boston, MA, 2007 (submitted for publication).
30. A. OLSHEVSKY AND J. N. TSITSIKLIS, Convergence speed in distributed consensus and averaging, in Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, CA, 2006, pp. 3387-3392.
31. R. SEPULCHRE, D. PALEY, AND N. LEONARD, Collective motion and oscillator synchronization, in Proceedings of the 2003 Block Island Workshop on Cooperative Control, Springer-Verlag, New York, 2005, pp. 189-228.
32. J. N. TSITSIKLIS, Problems in Decentralized Decision Making and Computation, Ph.D. thesis, Department of Electrical Engineering and Computer Science, MIT, Boston, MA, also available online at http://hdl.handle. net/172Ll/15254, 1984.
33. J. N. TSITSIKLIS, D. P. BERTSEKAS, AND M. ATHANS, Distributed asynchronous deterministic and stochastic gradient optimization algorithms, IEEE Trans. Automat. Control, 31 (1986), pp. 803-812.
34. L. YANG, K. M. PASSINO, AND M. M. POLYCARPOU, Stability analysis of m-dimensional asynchronous swarms with a fixed communication topology, IEEE Trans. Automat. Control, 48 (2003), pp. 76-95.