Distributed Consensus Algorithms in Sensor Networks: Link Failures and Channel Noise (0711.3915v2)

Abstract: The paper studies average consensus with random topologies (intermittent links) \emph{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 $\mathcal{A-ND}$ algorithm modifies conventional consensus by forcing the weights to satisfy a \emph{persistence} condition (slowly decaying to zero); and the $\mathcal{A-NC}$ algorithm where the weights are constant but consensus is run for a fixed number of iterations $\hat{\imath}$, then it is restarted and rerun for a total of $\hat{p}$ runs, and at the end averages the final states of the $\hat{p}$ runs (Monte Carlo averaging). We use controlled Markov processes and stochastic approximation arguments to prove almost sure convergence of $\mathcal{A-ND}$ to the desired average (asymptotic unbiasedness) and compute explicitly the m.s.e. (variance) of the consensus limit. We show that $\mathcal{A-ND}$ represents the best of both worlds--low bias and low variance--at the cost of a slow convergence rate; rescaling the weights...

• The paper presents two consensus algorithms (A-ND and A-NC) that achieve global data averaging in sensor networks despite link failures and channel noise.
• It applies stochastic approximation and Monte Carlo simulations to manage bias-variance tradeoffs, ensuring almost sure convergence and efficient averaging.
• Extensive numerical results validate theoretical bounds and underscore the practical implications for designing robust distributed sensor networks.

Distributed Consensus Algorithms in Sensor Networks with Imperfect Communication

The paper presents an analytical paper of distributed consensus algorithms within sensor networks, where communication is subject to link failures and noise. Two distinct approaches are proposed to navigate the inherent bias-variance tradeoff arising from the network's imperfect communications: the $\mathcal{A-ND}$ and $\mathcal{A-NC}$ algorithms.

Problem Context

The algorithmic techniques focus on achieving the global average of distributed data using localized communications, accounting for both intermittent connectivity and channel noise. This challenge is pertinent in wireless sensor networks, where link failures and noisy transmissions introduce complexity to achieving consensus efficiently.

Algorithms and Methodologies

$\mathcal{A-ND}$ Algorithm

The first approach, dubbed $\mathcal{A-ND}$, modifies conventional consensus by introducing weights that gradually decay to zero. This persistence condition ensures convergence towards an asymptotically unbiased estimate despite random topological changes and additive noise. Through stochastic approximation and controlled Markov process techniques, the authors demonstrate that $\mathcal{A-ND}$ converges almost surely to the desired average, balancing between low bias and low variance, albeit with a slower convergence rate.

$\mathcal{A-NC}$ Algorithm

Contrastingly, the $\mathcal{A-NC}$ algorithm uses constant weights across fixed iterations, incorporating Monte Carlo simulations. Each sensor runs consensus for a predetermined number of iterations and then averages the outcomes over several runs. This methodology allows for a faster convergence rate due to the constant weights but introduces different compromise points in bias-variance tradeoffs.

Theoretical and Practical Implications

The paper includes rigorous derivations, proving the almost sure convergence of $\mathcal{A-ND}$. For $\mathcal{A-NC}$, the authors derive conditions under which $(\epsilon, \delta)$-consensus can be achieved, with explicit bounds under static networks and Gaussian noise. The paper establishes that $\mathcal{A-NC}$, while achieving faster convergence, requires greater coordination among sensors due to the need for repeated averaging sessions.

Numerical Analysis

Various simulations illustrate the theoretical findings, including the tradeoffs in convergence rate and m.s.e. for different parameter settings. These simulations validate the performance guarantees for both algorithms under varied scenarios such as random topologies and different noise models.

Future Directions

Looking ahead, the insights gleaned from this research could inform the design of algorithms for other distributed systems facing similar challenges, including distributed load balancing and network flow problems. The generalizations proposed, such as extending the $\mathcal{A-NC}$ approach to random topologies and non-Gaussian noise, point towards expanding the application domain of these algorithms.

This paper contributes to the understanding of distributed consensus in impaired networking conditions, offering a solid foundation and strong analytical approach to a problem of significant interest in networked systems. The demonstrated tradeoffs and algorithmic innovations provide a basis for future studies exploring more complex network conditions or adapting these strategies to new technological needs.