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.

We study the convergence speed of distributed iterative algorithms for the consensus and averaging problems, with emphasis on the latter. We first consider the case of a fixed communication topology. We show that a simple adaptation of a consensus algorithm leads to an averaging algorithm. We prove lower bounds on the worst-case convergence time for various classes of linear, time-invariant, distributed consensus methods, and provide an algorithm that essentially matches those lower bounds. We then consider the case of a time-varying topology, and provide a polynomial-time averaging algorithm.

Number: CR-PRL-2008-08-0001

Abstract — We examine the use of gossip protocols for continuous monitoring of network-wide aggregates. Aggregates are computed from local management variables using functions such as AVERAGE, MIN, MAX, or SUM. A particular challenge is to develop a gossip-based aggregation protocol that is robust against node failures. In this paper, we present G-GAP, a gossip protocol for continuous monitoring of aggregates, which is robust against discontiguous failures (i.e., under the constraint that neighboring nodes do not fail within a short period of each other). We formally prove this property, and we evaluate the protocol through simulation using real traces. The simulation results suggest that the design goals for this protocol have been met. For instance, the tradeoff between estimation accuracy and protocol overhead can be controlled, and a high estimation accuracy (below some 5 % error in our measurements) is achieved by the protocol, even for large networks and frequent node failures. Further, we perform a comparative assessment of G-GAP against a tree-based aggregation protocol using simulation. Surprisingly, we find that the tree-based aggregation protocol consistently outperforms the gossip protocol for comparative overhead, both in terms of accuracy and robustness.

Gossip algorithms have recently received significant attention, mainly because they constitute simple and robust algorithms for distributed information processing over networks. However for many topologies that are realistic for wireless ad-hoc and sensor networks (like grids and random geometric graphs), the standard nearest-neighbor gossip converges very slowly. A recently proposed algorithm called geographic gossip improves gossip efficiency by a p n / log n factor for random geometric graphs, by exploiting geographic information of node locations. In this paper we prove that a variation of geographic gossip that averages along routed paths, improves efficiency by an additional p n / log n factor and is order optimal for grids and random geometric graphs. Our analysis provides some general techniques and can be used to provide bounds on the performance of randomized message passing algorithms operating over various graph topologies.

Abstract—Gossip algorithms for distributed computation are attractive due to their simplicity, distributed nature, and robustness in noisy and uncertain environments. However, using standard gossip algorithms can lead to a significant waste of energy by repeatedly recirculating redundant information. For realistic sensor network model topologies like grids and random geometric graphs, the inefficiency of gossip schemes is related to the slow mixing times of random walks on the communication graph. We propose and analyze an alternative gossiping scheme that exploits geographic information. By utilizing geographic routing combined with a simple resampling method, we demonstrate substantial gains over previously proposed gossip protocols. For regular graphs such as the ring or grid, our algorithm improves standard gossip by factors of and, respectively. For the more challenging case of random geometric graphs, our algorithm computes the true average to accuracy using 1 5 1 ( ( log) log) radio transmissions, which yields a log factor improvement over standard gossip algorithms. We illustrate these theoretical results with experimental comparisons between our algorithm and standard methods as applied to various classes of random fields. Index Terms—Aggregation problems, consensus problems, distributed signal processing, gossip algorithms, message-passing algorithms, random geometric graphs, sensor networks. I.

Gossip-based algorithms were first introduced for reliably disseminating data in large-scale distributed systems. However, their simplicity, robustness, and flexibility make them attractive for more than just pure data dissemination alone. In particular, gossiping has been applied to data aggregation, overlay maintenance, and resource allocation. Gossiping applications more or less fit the same framework, with often subtle differences in algorithmic details determining divergent emergent behavior. This divergence is often difficult to understand, as formal models have yet to be developed that can capture the full design space of gossiping solutions. In this paper, we present a brief introduction to the field of gossiping in distributed systems, by providing a simple framework and using that framework to describe solutions for various application domains.

We consider and analyze a new algorithm for balancing indivisible loads on a distributed network with n processors. The aim is minimizing the discrepancy between the maximum and minimum load. In every time-step paired processors balance their load as evenly as possible. The direction of the excess token is chosen according to a randomized rounding of the participating loads. We prove that in comparison to the corresponding model of Rabani, Sinclair, and Wanka (1998) with arbitrary roundings, the randomization yields an improvement of roughly a square root of the achieved discrepancy in the same number of time-steps on all graphs. For the important case of expanders we can even achieve a constant discrepancy in O(log n(log log n) 3) rounds. This is optimal up to log log n-factors while the best previous algorithms in this setting either require Ω(log 2 n) time or can only achieve a logarithmic discrepancy. This result also demonstrates that with randomized rounding the difference between discrete and continuous load balancing vanishes almost completely.

In this paper we propose a lightweight algorithm for constructing multi-resolution data representations for sensor networks. We compute, at each sensor node u, O(log n) aggregates about exponentially enlarging neighborhoods centered at u. The ith aggregate is the aggregated data among nodes approximately within 2 i hops of u. We present a scheme, named the hierarchical spatial gossip algorithm, to extract and construct these aggregates, for all sensors simultaneously, with a total communication cost of O(npolylog n). The hierarchical gossip algorithm adopts atomic communication steps with each node choosing to exchange information with a node distance d away with probability 1/d 3. The attractiveness of the algorithm attributes to its simplicity, low communication cost, distributed nature and robustness to node failures and link failures. Besides the natural applications of multi-resolution data summaries in data validation and information mining, we also demonstrate the application of the pre-computed spatial multi-resolution data summaries in answering range queries efficiently.

Average consensus and gossip algorithms have recently received significant attention, mainly because they constitute simple and robust algorithms for distributed information processing over networks. Inspired by heat diffusion, they compute the average of sensor networks measurements by iterating local averages until a desired level of convergence. Confronted with the diversity of these algorithms, the engineer may be puzzled in his choice for one of them. As an answer to his/her need, we develop precise mathematical metrics, easy to use in practice, to characterize the convergence speed and the cost (time, message passing, energy...) of each of the algorithms. In contrast to other works focusing on time-invariant scenarios, we evaluate these metrics for ergodic time-varying networks. Our study is based on Oseledec’s theorem, which gives an almost-sure description of the convergence speed of the algorithms of interest. We further provide upper bounds on the convergence speed. Finally, we use these tools to make some experimental observations illustrating the behavior of the convergence speed with respect to network topology and reliability in both average consensus and gossip algorithms. A. Problem statement. I.