Topology design for fast convergence of network consensus algorithms

The quantities of coefficient of ergodicity and algebraic connectivity have been used to estimate the convergence rates of discrete-time and continuous-time network consensus algorithms respectively. Both of these two quantities are defined with respect to network topologies without the symmetry assumption, and they are applicable to the case when network topologies change with time. We present results identifying deterministic network topologies that optimize these quantities. We will also propose heuristics that can accelerate convergence in random networks by redirecting a small portion of the links assuming that the network topology is controllable.

[1]  M. Cao,et al.  A Lower Bound on Convergence of a Distributed Network Consensus Algorithm , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[2]  J.N. Tsitsiklis,et al.  Convergence in Multiagent Coordination, Consensus, and Flocking , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[3]  Brian D. O. Anderson,et al.  Reaching a Consensus in a Dynamically Changing Environment: Convergence Rates, Measurement Delays, and Asynchronous Events , 2008, SIAM J. Control. Optim..

[4]  J.N. Tsitsiklis,et al.  Convergence Rates in Distributed Consensus and Averaging , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[5]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[6]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[7]  Jie Lin,et al.  Coordination of groups of mobile autonomous agents using nearest neighbor rules , 2003, IEEE Trans. Autom. Control..

[8]  C. Tan.  On the weak ergodicity of nonhomogeneous Markov chains , 1996 .

[9]  C. Wu.  Algebraic connectivity of directed graphs , 2005 .

[10]  Soummya Kar,et al.  Topology for Distributed Inference on Graphs , 2006, IEEE Transactions on Signal Processing.

[11]  Benjamin Van Roy,et al.  Consensus Propagation , 2005, IEEE Transactions on Information Theory.

[12]  M. Bartlett,et al.  Weak ergodicity in non-homogeneous Markov chains , 1958, Mathematical Proceedings of the Cambridge Philosophical Society.

[13]  Frank Harary,et al.  Graph Theory , 2016 .

[14]  John N. Tsitsiklis,et al.  Parallel and distributed computation , 1989 .

[15]  M. Fiedler.  Algebraic connectivity of graphs , 1973 .

[16]  Seif Haridi,et al.  Distributed Algorithms , 1992, Lecture Notes in Computer Science.

[17]  Xiao Fan Wang,et al.  Synchronization in Small-World Dynamical Networks , 2002, Int. J. Bifurc. Chaos.

[18]  Richard M. Murray,et al.  Consensus problems in networks of agents with switching topology and time-delays , 2004, IEEE Transactions on Automatic Control.

[19]  C. Wu.  Perturbation of coupling matrices and its effect on the synchronizability in arrays of coupled chaotic systems , 2003, nlin/0307052.

[20]  Chai Wah Wu.  Agreement and consensus problems in groups of autonomous agents with linear dynamics , 2005, 2005 IEEE International Symposium on Circuits and Systems.

[21]  Stephen P. Boyd,et al.  Fast linear iterations for distributed averaging , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[22]  E. Seneta.  Coefficients of ergodicity: structure and applications , 1979, Advances in Applied Probability.