Summary. This paper provides a theoretical framework for analysis of consensus algorithms for multi-agent networked systems with an emphasis on the role of directed information flow, robustness to changes in network topology due to link/node failures, time-delays, and performance guarantees. An overview of basic concepts of information consensus in networks and methods of convergence and performance analysis for the algorithms are provided. Our analysis framework is based on tools from matrix theory, algebraic graph theory, and control theory. We discuss the connections between consensus problems in networked dynamic systems and diverse applications including synchronization of coupled oscillators, flocking, formation control, fast consensus in small-world networks, Markov processes and gossip-based algorithms, load balancing in networks, rendezvous in space, distributed sensor fusion in sensor networks, and belief propagation. We establish direct connections between spectral and structural properties of complex networks and the speed of information diffusion of consensus algorithms. A brief introduction is provided on networked systems with nonlocal information flow that are considerably faster than distributed systems with latticetype nearest neighbor interactions. Simulation results are presented that demonstrate the role of small-world effects on the speed of consensus algorithms and cooperative control of multi-vehicle formations.

We consider a stochastic model for distributed average consensus, which arises in applications such as load balancing for parallel processors, distributed coordination of mobile autonomous agents, and network synchronization. In this model, each node updates its local variable with a weighted average of its neighbors ’ values, and each new value is corrupted by an additive noise with zero mean. The quality of consensus can be measured by the total mean-square deviation of the individual variables from their average, which converges to a steady-state value. We consider the problem of finding the (symmetric) edge weights that result in the least mean-square deviation in steady state. We show that this problem can be cast as a convex optimization problem, so the global solution can be found efficiently. We describe some computational methods for solving this problem, and compare the weights and the mean-square deviations obtained by this method and several other weight design methods.

We propose consensus propagation, an asynchronous distributed protocol for averaging numbers across a network. We establish convergence, characterize the convergence rate for regular graphs, and demonstrate that the protocol exhibits better scaling properties than pairwise averaging, an alternative that has received much recent attention. Consensus propagation can be viewed as a special case of belief propagation, and our results contribute to the belief propagation literature. In particular, beyond singly-connected graphs, there are very few classes of relevant problems for which belief propagation is known to converge.

We consider distributed iterative algorithms for the averaging problem over timevarying topologies. Our focus is on the convergence time of such algorithms when complete (unquantized) information is available, and on the degradation of performance when only quantized information is available. We study a large and natural class of averaging algorithms, which includes the vast majority of algorithms proposed to date, and provide tight polynomial bounds on their convergence time. We also describe an algorithm within this class whose convergence time is the best among currently available averaging algorithms for time-varying topologies. We then propose and analyze distributed averaging algorithms under the additional constraint that agents can only store and communicate quantized information, so that they can only converge to the average of the initial values of the agents within some error. We establish bounds on the error and tight bounds on the convergence time, as a function of the number of quantization levels.

We study the distributed averaging problem on arbitrary connected graphs, with the additional constraint that the value at each node is an integer. This discretized distributed averaging problem models several problems of interest, such as averaging in a network with finite capacity channels and load balancing in a processor network. We describe simple randomized distributed algorithms which achieve consensus to the extent that the discrete nature of the problem permits. We give bounds on the convergence time of these algorithms for fully connected networks and linear networks.

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.

Abstract—This paper proposes a design methodology to stabilize relative equilibria in a model of identical, steered particles moving in the plane at unit speed. Relative equilibria either correspond to parallel motion of all particles with fixed relative spacing or to circular motion of all particles around the same circle. Particles exchange relative information according to a communication graph that can be undirected or directed and time-invariant or timevarying. The emphasis of this paper is to show how previous results assuming all-to-all communication can be extended to a general communication framework. Index Terms—Cooperative control, geometric control, multiagent systems, stabilization. I.

In this paper, we study the behavior of a network of N agents, each evolving on the circle. We propose a novel algorithm that achieves synchronization or balancing in phase models under mild connectedness assumptions on the (possibly time-varying and unidirectional) communication graphs. The global convergence analysis on the N-torus is a distinctive feature of the present work with respect to previous results that have focused on convergence in the Euclidean space.

Abstract — A decentralized, adaptive control law is presented to drive a network of mobile robots to a near-optimal sensing configuration. The control law is adaptive in that it integrates sensor measurements to provide a converging estimate of the distribution of sensory information in the environment. It is decentralized in that it requires only information local to each robot. A Lyapunov-type proof is used to show that the control law causes the network to converge to a near-optimal sensing configuration, and the controller is demonstrated in numerical simulations. This technique suggests a broader application of adaptive control methodologies to decentralized control problems in unknown dynamical environments. I.

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.