Abstract—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 algorithm modifies conventional consensus by forcing the weights to satisfy a persistence condition (slowly decaying to zero;) and the algorithm where the weights are constant but consensus is run for a fixed number of iterations, then it is restarted and rerun for a total of runs, and at the end averages the final states of the runs (Monte Carlo averaging). We use controlled Markov processes and stochastic approximation arguments to prove almost sure convergence of to a finite consensus limit and compute explicitly the mean square error (mse) (variance) of the consensus limit. We show that 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, , because of its constant weights, converges fast but presents a different bias-variance tradeoff. For the same number of iterations, shorter runs (smaller) lead to high bias but smaller variance (larger number of runs to average over.) For a static nonrandom network with Gaussian noise, we compute the optimal gain for to reach in the shortest number of iterations, with high probability (1), ()-consensus ( residual bias). Our results hold under fairly general assumptions on the random link failures and communication noise. Index Terms—Additive noise, consensus, sensor networks, stochastic approximation, random topology. I.

Abstract—This paper presents a distributed Kalman filter to estimate the state of a sparsely connected, large-scale,-dimensional, dynamical system monitored by a network of sensors. Local Kalman filters are implemented on-dimensional subsystems,, obtained by spatially decomposing the large-scale system. The distributed Kalman filter is optimal under an th order Gauss–Markov approximation to the centralized filter. We quantify the information loss due to this th-order approximation by the divergence, which decreases as increases. The order of the approximation leads to a bound on the dimension of the subsystems, hence, providing a criterion for subsystem selection. The (approximated) centralized Riccati and Lyapunov equations are computed iteratively with only local communication and low-order computation by a distributed iterate collapse inversion (DICI) algorithm. We fuse the observations that are common among the local Kalman filters using bipartite fusion graphs and consensus averaging algorithms. The proposed algorithm achieves full distribution of the Kalman filter. Nowhere in the network, storage, communication, or computation of-dimensional vectors and matrices is required; only dimensional vectors and matrices are communicated or used in the local computations at the sensors. In other words, knowledge of the state is itself distributed. Index Terms—Distributed algorithms, distributed estimation, information filters, iterative methods, Kalman filtering, large-scale systems, matrix inversion, sparse matrices. I.

Abstract — We consider the standard distributed average consensus algorithm under the conditions of random communication link failures, for which we analyze convergence of nodes to average consensus in the mean square sense. We first recast this problem as a discrete-time linear system with multiplicative random coefficients. We then rewrite the system equations as a nominal system in feedback with diagonally structured timevarying stochastic uncertainty; a problem for which necessary and sufficient mean square stability conditions have recently been derived. We investigate the particular instance of these conditions in the case of networked consensus with random link failures. In particular, we show that for circulant graphs, mean square convergence is guaranteed for any probability of link failure other than 1. We anticipate our particular analysis techniques to be applicable to the robust performance problem as well. I.

Abstract—A network of nodes communicate via pointto-point memoryless independent noisy channels. Each node has some real-valued initial measurement or message. The goal of each of the nodes is to acquire an estimate of a given function of all the initial measurements in the network. As the main contribution of this paper, a lower bound on computation time is derived. This bound must be satisfied by any algorithm used by the nodes to communicate and compute, so that the mean square error in the nodes ’ estimate is within a given interval around zero. The derivation utilizes information theoretic inequalities reminiscent of those used in rate distortion theory along with a novel ‘perturbation ’ technique so as to be broadly applicable. To understand the tightness of the bound, a specific scenario is considered. Nodes are required to learn a linear combination of the initial values in the network while communicating over erasure channels. A distributed quantized algorithm is developed, and it is shown that the computation time essentially scales as is implied by the lower bound. In particular, the computation time depends reciprocally on ”conductance”, which is a property of the network that captures the information-flow bottleneck. As a by-product, this leads to a quantized algorithm, for computing separable functions in a network, with minimal computation time. Index Terms—Computation time, conductance, distributed computing, noisy networks, quantized summation. I.

The thesis develops methodology and algorithms to study distributed inference problems in large scale networked systems. Typical examples that fall under the scope of this study include distributed detection, distributed field reconstruction (estimation) arising in wireless sensor network (WSN) applications, and filtering in networked dynamical (cyberphysical) systems. The systems in question operate in random environments and are constrained in terms of resources, like communication bandwidth or power. Due to the inherent randomness in sensor deployment or field sampling, often there is no center coordinating the network activity. The nodes (sensors or dynamical agents) need to collaborate with each other through local information exchange to achieve desired global network behavior. One aspect of our work involves the development of robust distributed algorithms for collaborative information processing in these networks. We study the performance of these distributed schemes in terms of their robustness to communication failures, external stochastic perturbations, and convergence to the corresponding centralized counterparts. The other aspect of the work

Consensus algorithms permit the computation of global statistics via local communications and without centralized control. We extend previous results by taking into account fading and unidirectional links in ring and random 2-D topologies. We study conditions for convergence and present simulation results to verify the analytical results in this paper. We compare the performance of consensus algorithms with a tree-based (centralized) approach. Additionally, we implement a slotted ALOHA protocol and compare its performance to that under the initial assumption of perfect scheduling. I.

We consider the weight design problem for the consensus algorithm under a finite time horizon. We assume that the underlying network is random where the links fail at each iteration with certain probability and the link failures can be spatially correlated. We formulate a family of weight design criteria (objective functions) that minimize n, n = 1,..., N (out of N possible) largest (slowest) eigenvalues of the matrix that describes the mean squared consensus error dynamics. We show that the objective functions are convex; hence, globally optimal weights (with respect to the design criteria) can be efficiently obtained. Numerical examples on large scale, sparse random networks with spatially correlated link failures show that: 1) weights obtained according to our criteria lead to significantly faster convergence than the choices available in the literature; 2) different design criteria that corresponds to different n, exhibits very interesting tradeoffs: faster transient performance leads to slower long time run performance and vice versa. Thus, n is a valuable degree of freedom and can be appropriately selected for the given time horizon. Index Terms — consensus, weight design, convex optimization, time horizon, correlated link failures 1.