Average consensus on networks with transmission noise or quantization

Carli, Ruggero; Fagnani, Fabio; Frasca, Paolo; Taylor, Tom; Zampieri, Sandro

doi:10.23919/ecc.2007.7068829

Cited by 64 publications

(62 citation statements)

References 12 publications

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“…Since giving a complete overview would be impossible here, we just mention some papers whose approaches are particularly close to ours. First of all, in [10] the authors consider a discrete-time version of (2): their analysis is limited to observing that the algorithm may not converge to consensus and then abandoned. The poor perfomance of the dynamics in terms of approaching consensus explains the scarcity 1 of known results about (2).…”

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confidence: 99%

Consensus and Disagreement: The Role of Quantized Behaviors in Opinion Dynamics

Ceragioli¹,

Frasca²

2018

SIAM J. Control Optim.

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Abstract. This paper deals with continuous-time opinion dynamics that feature the interplay of continuous opinions and discrete behaviours. In our model, the opinion of one individual is only influenced by the behaviours of fellow individuals. The key technical difficulty in the study of these dynamics is that the right-hand sides of the equations are discontinuous and thus their solutions must be intended in some generalized sense: in our analysis, we consider both Carathéodory and Krasovskii solutions. We first prove existence and completeness of Carathéodory solutions from every initial condition and we highlight a pathological behavior of Carathéodory solutions, which can converge to points that are not (Carathéodory) equilibria. Notably, such points can be arbitrarily far from consensus and indeed simulations show that convergence to non-consensus configurations is common. In order to cope with these pathological attractors, we study Krasovskii solutions. We give an estimate of the asymptotic distance of all Krasovskii solutions from consensus and we prove its tightness by an example of equilibrium such that this distance is quadratic in the number of agents. This fact implies that quantization can drastically destroy consensus. However, consensus is guaranteed in some special cases, for instance when the communication among the individuals is described by either a complete or a complete bipartite graph.

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confidence: 99%

Consensus and Disagreement: The Role of Quantized Behaviors in Opinion Dynamics

Ceragioli¹,

Frasca²

2018

SIAM J. Control Optim.

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“…Related are also gossip and broadcast-gossip schemes giving rise to a random consensus over a given connected bi-directional communication network [37,38,39,40]. On a broader basis, this work is related to the literature on the consensus over networks with noisy links [41,42,43,44] and the deterministic consensus in decentralized systems models [1,45,2,46,47,21,48,24,49], including the effects of quantization and delay [50,51,52,53,54,40].…”

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confidence: 99%

Product of Random Stochastic Matrices and Distributed Averaging

Touri

2012

Springer Theses

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This thesis is mainly concerned with the study of product of random stochastic matrices and random weighted averaging dynamics. It will be shown that a generalization of a fundamental result in the theory of ergodic Markov chains not only holds for inhomogeneous chains of stochastic matrices, but also remains true for random stochastic matrices. To do this, the concept of infinite flow property will be introduced for a deterministic chain of stochastic matrices and it will be proven that it is necessary for ergodicity of any stochastic chain. This result will further be extended to ergodic classes, through the development of the concept of the infinite flow graph and ℓ 1 -approximation technique.For the converse implications, the product of stochastic matrices will be studied in the more general setting of random adapted stochastic chains. Using a result of A. Kolmogorov, it will be shown that any averaging dynamics admits infinitely many comparison functions including a quadratic one. By identifying the decrease of the quadratic comparison function along the trajectories of the dynamics, it will be proven that under general assumptions on a random chain, the chain is infinite flow stable, i.e. the product of random stochastic matrices is convergent almost surely and, also, the limiting matrices admit certain structures that can be deduced from the infinite flow graph of the chain. It will be shown that a general class of stochastic chains, the balanced chains with feedback property, satisfy the conditions of this result.Some implications of the developed results for products of independent random stochastic matrices will be provided. Furthermore, it will be proven that under general conditions, an independent random chain and its expected chain exhibit the same ergodic behavior. It will be proven that an extension of a well-known result in the theory of homogeneous Markov chains holds for a sequence of inhomogeneous stochastic matrices. Then, link-failure models for averaging dynamics will be introduced and it will be shown that under general conditions, link failure does not affect the limiting behavior of averaging dynamics. Then, the application of the developed methods to the study of Hegselmann-Krause model will be considered. Using the developed results, an upper bound O(m 4 ) will be established for the Hegselmann-Krause dynamics, which is an improvement to the previously known bound ii O(m 5 ). As a final application for the developed tools, an alternative proof for the second Borel-Cantelli lemma will be provided. Motivated by the infinite flow property, a stronger one, the absolute infinite flow property will be introduced. It will be shown that this stronger property is in fact necessary for ergodicity of any stochastic chain. Moreover, the equivalency of the absolute infinite flow property with ergodicity of doubly stochastic chains will be proven. These results will be driven by introduction and study of the rotational transformation of a stochastic chain.Finally, motivated by the study of ...

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“…Closely related is also the work in [10] and [7], [6], which study the effects of quantization on the performance of averaging algorithms. Our work is also related to the utility maximization framework for resource allocation in networks (see [11], [12], [22]).…”

Section: Introductionmentioning

confidence: 99%

Distributed subgradient methods and quantization effects

Nedić

Оlshevsky

Ozdaglar

et al. 2008

2008 47th IEEE Conference on Decision and Control

178

136

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Abstract-We consider a convex unconstrained optimization problem that arises in a network of agents whose goal is to cooperatively optimize the sum of the individual agent objective functions through local computations and communications. For this problem, we use averaging algorithms to develop distributed subgradient methods that can operate over a timevarying topology. Our focus is on the convergence rate of these methods and the degradation in performance when only quantized information is available. Based on our recent results on the convergence time of distributed averaging algorithms, we derive improved upper bounds on the convergence rate of the unquantized subgradient method. We then propose a distributed subgradient method under the additional constraint that agents can only store and communicate quantized information, and we provide bounds on its convergence rate that highlight the dependence on the number of quantization levels.

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Average consensus on networks with transmission noise or quantization

Cited by 64 publications

References 12 publications

Consensus and Disagreement: The Role of Quantized Behaviors in Opinion Dynamics

Consensus and Disagreement: The Role of Quantized Behaviors in Opinion Dynamics

Product of Random Stochastic Matrices and Distributed Averaging

Distributed subgradient methods and quantization effects

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