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51
Fast linear iterations for distributed averaging.
 Systems & Control Letters,
, 2004
"... Abstract We consider the problem of finding a linear iteration that yields distributed averaging consensus over a network, i.e., that asymptotically computes the average of some initial values given at the nodes. When the iteration is assumed symmetric, the problem of finding the fastest converging ..."
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Cited by 433 (12 self)
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Abstract We consider the problem of finding a linear iteration that yields distributed averaging consensus over a network, i.e., that asymptotically computes the average of some initial values given at the nodes. When the iteration is assumed symmetric, the problem of finding the fastest converging linear iteration can be cast as a semidefinite program, and therefore efficiently and globally solved. These optimal linear iterations are often substantially faster than several common heuristics that are based on the Laplacian of the associated graph. We show how problem structure can be exploited to speed up interiorpoint methods for solving the fastest distributed linear iteration problem, for networks with up to a thousand or so edges. We also describe a simple subgradient method that handles far larger problems, with up to one hundred thousand edges. We give several extensions and variations on the basic problem.
Distributed average consensus with leastmeansquare deviation
 Journal of Parallel and Distributed Computing
, 2005
"... 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 averag ..."
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Cited by 205 (4 self)
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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 meansquare deviation of the individual variables from their average, which converges to a steadystate value. We consider the problem of finding the (symmetric) edge weights that result in the least meansquare 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 meansquare deviations obtained by this method and several other weight design methods.
Fastest mixing markov chain on a graph
 SIAM REVIEW
, 2003
"... We consider a symmetric random walk on a connected graph, where each edge is labeled with the probability of transition between the two adjacent vertices. The associated Markov chain has a uniform equilibrium distribution; the rate of convergence to this distribution, i.e., the mixing rate of the ..."
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Cited by 155 (15 self)
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We consider a symmetric random walk on a connected graph, where each edge is labeled with the probability of transition between the two adjacent vertices. The associated Markov chain has a uniform equilibrium distribution; the rate of convergence to this distribution, i.e., the mixing rate of the Markov chain, is determined by the second largest (in magnitude) eigenvalue of the transition matrix. In this paper we address the problem of assigning probabilities to the edges of the graph in such a way as to minimize the second largest magnitude eigenvalue, i.e., the problem of finding the fastest mixing Markov chain on the graph. We show that this problem can be formulated as a convex optimization problem, which can in turn be expressed as a semidefinite program (SDP). This allows us to easily compute the (globally) fastest mixing Markov chain for any graph with a modest number of edges (say, 1000) using standard numerical methods for SDPs. Larger problems can be solved by
Consensus propagation
 IEEE Transactions on Information Theory
"... Abstract — 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 a ..."
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Cited by 89 (5 self)
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Abstract — 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 singlyconnected graphs, there are very few classes of relevant problems for which belief propagation is known to converge. Index Terms — belief propagation, distributed averaging, distributed consensus, distributed signal processing, Gaussian Markov random fields, messagepassing algorithms, maxproduct algorithm, minsum algorithm, sumproduct algorithm. I.
Communication Constraints in the Average Consensus Problem
, 2007
"... The interrelationship between control and communication theory is becoming of fundamental importance in many distributed control systems, such as the coordination of a team of autonomous agents. In such a problem, communication constraints impose limits on the achievable control performance. We cons ..."
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Cited by 81 (19 self)
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The interrelationship between control and communication theory is becoming of fundamental importance in many distributed control systems, such as the coordination of a team of autonomous agents. In such a problem, communication constraints impose limits on the achievable control performance. We consider as instance of coordination the consensus problem. The aim of the paper is to characterize the relationship between the amount of information exchanged by the agents and the rate of convergence to the consensus. We show that timeinvariant communication networks with circulant symmetries yield slow convergence if the amount of information exchanged by the agents does not scale well with their number. On the other hand, we show that randomly timevarying communication networks allow very fast convergence rates. We also show that, by adding logarithmic quantized data links to timeinvariant networks with symmetries, control performance significantly improves with little growth of the required communication effort.
Minimizing effective resistance of a graph
 SIAM Review
, 2005
"... Abstract. The effective resistance between two nodes of a weighted graph is the electrical resistance seen between the nodes of a resistor network with branch conductances given by the edge weights. The effective resistance comes up in many applications and fields in addition to electrical network a ..."
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Cited by 65 (4 self)
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Abstract. The effective resistance between two nodes of a weighted graph is the electrical resistance seen between the nodes of a resistor network with branch conductances given by the edge weights. The effective resistance comes up in many applications and fields in addition to electrical network analysis, including, for example, Markov chains and continuoustime averaging networks. In this paper we study the problem of allocating edge weights on a given graph in order to minimize the total effective resistance, i.e., the sum of the resistances between all pairs of nodes. We show that this is a convex optimization problem, and can be solved efficiently either numerically, or, in some cases, analytically. We show that optimal allocation of the edge weights can reduce the total effective resistance of the graph (compared to uniform weights) by a factor that grows unboundedly with the size of the graph. We show that among all graphs with n nodes, the path has the largest value of optimal total effective resistance, and the complete graph the least. 1. Introduction. Let N be a network with n nodes and m edges, i.e., an undirected graph (V, E) with V  = n, E  = m, and nonnegative weights on the edges. We call the weight on edge l its conductance, and denote it by gl. The effective resistance between a pair of nodes i and j, denoted Rij, is the electrical resistance measured across nodes i and j, when the network represents an electrical circuit with each edge (or branch, in the terminology of electrical circuits) a resistor with (electrical) conductance gl. In other
Convex optimization of graph Laplacian eigenvalues
 IN INTERNATIONAL CONGRESS OF MATHEMATICIANS
"... We consider the problem of choosing the edge weights of an undirected graph so as to maximize or minimize some function of the eigenvalues of the associated Laplacian matrix, subject to some constraints on the weights, such as nonnegativity, or a given total value. In many interesting cases this p ..."
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Cited by 51 (0 self)
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We consider the problem of choosing the edge weights of an undirected graph so as to maximize or minimize some function of the eigenvalues of the associated Laplacian matrix, subject to some constraints on the weights, such as nonnegativity, or a given total value. In many interesting cases this problem is convex, i.e., it involves minimizing a convex function (or maximizing a concave function) over a convex set. This allows us to give simple necessary and sufficient optimality conditions, derive interesting dual problems, find analytical solutions in some cases, and efficiently compute numerical solutions in all cases. In this overview we briefly describe some more specific cases of this general problem, which have been addressed in a series of recent papers. • Fastest mixing Markov chain. Find edge transition probabilities that give the fastest mixing (symmetric, discretetime) Markov chain on the graph. • Fastest mixing Markov process. Find the edge transition rates that give the fastest mixing (symmetric, continuoustime) Markov process on the graph. • Absolute algebraic connectivity. Find edge weights that maximize the algebraic
Linearly edgereinforced random walks
, 2006
"... We review results on linearly edgereinforced random walks. On finite graphs, the process has the same distribution as a mixture of reversible Markov chains. This has applications in Bayesian statistics and it has been used in studying the random walk on infinite graphs. On trees, one has a represe ..."
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Cited by 20 (6 self)
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We review results on linearly edgereinforced random walks. On finite graphs, the process has the same distribution as a mixture of reversible Markov chains. This has applications in Bayesian statistics and it has been used in studying the random walk on infinite graphs. On trees, one has a representation as a random walk in an independent random environment. We review recent results for the random walk on ladders: recurrence, a representation as a random walk in a random environment, and estimates for the position of the random walker.
Communication constraints in the state agreement problem
 IN PREPARATION
, 2005
"... The interrelationship between control and communication theory is becoming of fundamental importance in many distributed control systems. Particular examples are systems comprised of multiple agents. When it comes to coordinately control a group of autonomous mobile agents in order to achieve a comm ..."
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Cited by 17 (7 self)
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The interrelationship between control and communication theory is becoming of fundamental importance in many distributed control systems. Particular examples are systems comprised of multiple agents. When it comes to coordinately control a group of autonomous mobile agents in order to achieve a common task, communication constraints impose limits on the achievable control performance. In this paper we consider a widely studied problem in the robotics and control communities, called consensus or state agreement problem. The aim of the paper is to characterize the relationship between the amount of information exchanged by the agents and the rate of convergence to the agreement. Timeinvariant communication networks that exhibit particular symmetries are shown to yield slow convergence if the amount of information exchanged does not scale with the number of agents. On the other hand, we show that, randomly timevarying communication networks allow very fast convergence rates. The last part of the paper is devoted to the study of timeinvariant communication networks with logarithmic quantized data exchange among the agents. It is shown that, by adding quantized data links to the network, the control performance significantly improves with little growth of the required communication effort.