Distributed Consensus Algorithms in Sensor Networks With Imperfect Communication: Link Failures and Channel Noise

The paper studies average consensus with random topologies (intermittent links) and noisy channels. Consensus with noise in the network links leads to the bias-variance dilemma-running consensus for long reduces the bias of the final average estimate but increases its variance. We present two different compromises to this tradeoff: the A-ND algorithm modifies conventional consensus by forcing the weights to satisfy a persistence condition (slowly decaying to zero;) and the A-NC algorithm where the weights are constant but consensus is run for a fixed number of iterations [^(iota)], then it is restarted and rerun for a total of [^(p)] runs, and at the end averages the final states of the [^(p)] runs (Monte Carlo averaging). We use controlled Markov processes and stochastic approximation arguments to prove almost sure convergence of A-ND to a finite consensus limit and compute explicitly the mean square error (mse) (variance) of the consensus limit. We show that A-ND represents the best of both worlds-zero bias and low variance-at the cost of a slow convergence rate; rescaling the weights balances the variance versus the rate of bias reduction (convergence rate). In contrast, A-NC, because of its constant weights, converges fast but presents a different bias-variance tradeoff. For the same number of iterations [^(iota)][^(p)] , shorter runs (smaller [^(iota)] ) lead to high bias but smaller variance (larger number [^(p)] of runs to average over.) For a static nonrandom network with Gaussian noise, we compute the optimal gain for A-NC to reach in the shortest number of iterations [^(iota)][^(p)] , with high probability (1-delta), (epsiv, delta)-consensus (epsiv residual bias). Our results hold under fairly general assumptions on the random link failures and communication noise.

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

[2]  Mikhail Borisovich Nevelʹson,et al.  Stochastic Approximation and Recursive Estimation , 1976 .

[3]  John N. Tsitsiklis,et al.  Problems in decentralized decision making and computation , 1984 .

[4]  Craig W. Reynolds.  Flocks, herds, and schools: a distributed behavioral model , 1987, SIGGRAPH.

[5]  B. Mohar.  THE LAPLACIAN SPECTRUM OF GRAPHS y , 1991 .

[6]  David Williams,et al.  Probability with Martingales , 1991, Cambridge mathematical textbooks.

[7]  Vicsek,et al.  Novel type of phase transition in a system of self-driven particles. , 1995, Physical review letters.

[8]  Michael William Newman,et al.  The Laplacian spectrum of graphs , 2001 .

[9]  R. Murray,et al.  Consensus protocols for networks of dynamic agents , 2003, Proceedings of the 2003 American Control Conference, 2003..

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

[11]  H. Kushner,et al.  Stochastic Approximation and Recursive Algorithms and Applications , 2003 .

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

[13]  Mehran Mesbahi,et al.  Agreement over random networks , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[14]  M. Mesbahi,et al.  Agreement in presence of noise: pseudogradients on random geometric networks , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[15]  M.G. Rabbat,et al.  Generalized consensus computation in networked systems with erasure links , 2005, IEEE 6th Workshop on Signal Processing Advances in Wireless Communications, 2005..

[16]  Devavrat Shah,et al.  Computing separable functions via gossip , 2005, PODC '06.

[17]  J. Moura,et al.  Topology for Global Average Consensus , 2006, 2006 Fortieth Asilomar Conference on Signals, Systems and Computers.

[18]  Stephen P. Boyd,et al.  Randomized gossip algorithms , 2006, IEEE Transactions on Information Theory.

[19]  Alireza Tahbaz-Salehi,et al.  On consensus over random networks , 2006 .

[20]  F. Chung,et al.  Complex Graphs and Networks , 2006 .

[21]  Mehran Mesbahi,et al.  On maximizing the second smallest eigenvalue of a state-dependent graph Laplacian , 2006, IEEE Transactions on Automatic Control.

[22]  Chai Wah Wu,et al.  Synchronization and convergence of linear dynamics in random directed networks , 2006, IEEE Transactions on Automatic Control.

[23]  Reza Olfati-Saber,et al.  Flocking for multi-agent dynamic systems: algorithms and theory , 2006, IEEE Transactions on Automatic Control.

[24]  José M. F. Moura,et al.  Ramanujan Topologies for Decision Making in Sensor Networks , 2006 .

[25]  Alejandro Ribeiro,et al.  Consensus-Based Distributed Parameter Estimation in Ad Hoc Wireless Sensor Networks with Noisy Links , 2007, 2007 IEEE International Conference on Acoustics, Speech and Signal Processing - ICASSP '07.

[26]  Stephen P. Boyd,et al.  Distributed average consensus with least-mean-square deviation , 2007, J. Parallel Distributed Comput..

[27]  Jonathan H. Manton,et al.  Stochastic approximation for consensus seeking: Mean square and almost sure convergence , 2007, 2007 46th IEEE Conference on Decision and Control.

[28]  J.H. Manton,et al.  Stochastic Lyapunov Analysis for Consensus Algorithms with Noisy Measurements , 2007, 2007 American Control Conference.

[29]  Soummya Kar,et al.  Distributed Average Consensus in Sensor Networks with Random Link Failures and Communication Channel Noise , 2007, 2007 Conference Record of the Forty-First Asilomar Conference on Signals, Systems and Computers.

[30]  Reza Olfati-Saber,et al.  Consensus and Cooperation in Networked Multi-Agent Systems , 2007, Proceedings of the IEEE.

[31]  D.J. Stilwell,et al.  Stochastic consensus over weighted directed networks , 2007, 2007 American Control Conference.

[32]  George J. Pappas,et al.  Flocking in Fixed and Switching Networks , 2007, IEEE Transactions on Automatic Control.

[33]  Michael Rabbat,et al.  Distributed Average Consensus using Probabilistic Quantization , 2007, 2007 IEEE/SP 14th Workshop on Statistical Signal Processing.

[34]  Soummya Kar,et al.  Sensor Networks With Random Links: Topology Design for Distributed Consensus , 2007, IEEE Transactions on Signal Processing.

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

[36]  Soummya Kar,et al.  Distributed Consensus Algorithms in Sensor Networks: Quantized Data and Random Link Failures , 2007, IEEE Transactions on Signal Processing.