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39
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 82 (20 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.
Symmetry analysis of reversible markov chains
 INTERNET MATHEMATICS
, 2005
"... We show how to use subgroups of the symmetry group of a reversible Markov chain to give useful bounds on eigenvalues and their multiplicity. We supplement classical representation theoretic tools involving a group commuting with a selfadjoint operator with criteria for an eigenvector to descend to ..."
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Cited by 55 (15 self)
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We show how to use subgroups of the symmetry group of a reversible Markov chain to give useful bounds on eigenvalues and their multiplicity. We supplement classical representation theoretic tools involving a group commuting with a selfadjoint operator with criteria for an eigenvector to descend to an orbit graph. As examples, we show that the Metropolis construction can dominate a maxdegree construction by an arbitrary amount and that, in turn, the fastest mixing Markov chain can dominate the Metropolis construction by an arbitrary amount.
A SuperClass Walk on UpperTriangular Matrices
, 2003
"... Let G be the group of n x n uppertriangular matrices with elements in a finite field and ones on the diagonal. This paper applies the character theory of Andre, Carter and Yan to analyze a natural random walk based on adding or subtracting a random row from the row above. ..."
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Cited by 26 (3 self)
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Let G be the group of n x n uppertriangular matrices with elements in a finite field and ones on the diagonal. This paper applies the character theory of Andre, Carter and Yan to analyze a natural random walk based on adding or subtracting a random row from the row above.
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.
Mixing times for random kcycles and coalescencefragmentation chains
, 1961
"... Dedicated to the memory of ODED SCHRAMM Let Sn be the permutation group on n elements, and consider a random walk on Sn whose step distribution is uniform on kcycles. We prove a wellknown conjecture that the mixing time of this process is (1/k)n log n, with threshold of width linear in n. Our proo ..."
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Cited by 8 (1 self)
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Dedicated to the memory of ODED SCHRAMM Let Sn be the permutation group on n elements, and consider a random walk on Sn whose step distribution is uniform on kcycles. We prove a wellknown conjecture that the mixing time of this process is (1/k)n log n, with threshold of width linear in n. Our proofs are elementary and purely probabilistic, and do not appeal to the representation theory of Sn.
ABRUPT CONVERGENCE AND ESCAPE BEHAVIOR FOR BIRTH AND DEATH CHAINS
, 2009
"... We link two phenomena concerning the asymptotical behavior of stochastic processes: (i) abrupt convergence or cutoff phenomenon, and (ii) the escape behavior usually associated to exit from metastability. The former is characterized by convergence at asymptotically deterministic times, while the c ..."
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Cited by 8 (2 self)
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We link two phenomena concerning the asymptotical behavior of stochastic processes: (i) abrupt convergence or cutoff phenomenon, and (ii) the escape behavior usually associated to exit from metastability. The former is characterized by convergence at asymptotically deterministic times, while the convergence times for the latter are exponentially distributed. We compare and study both phenomena for discretetime birthanddeath chains on Z with drift towards zero. In particular, this includes energydriven evolutions with energy functions in the form of a single well. Under suitable drift hypotheses, we show that there is both an abrupt convergence towards zero and escape behavior in the other direction. Furthermore, as the evolutions are reversible, the law of the final escape trajectory coincides with the time reverse of the law of cutoff paths. Thus, for evolutions defined by onedimensional energy wells with sufficiently steep walls, cutoff and escape behavior are related by time inversion.
Convergence rates of random walk on irreducible representations of finite groups
 J. Theoret. Probab
"... Abstract. Random walk on the set of irreducible representations of a finite group is investigated. For the symmetric and general linear groups, a sharp convergence rate bound is obtained and a cutoff phenomenon is proved. As a related result, an asymptotic description of Plancherel measure of the fi ..."
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Cited by 6 (3 self)
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Abstract. Random walk on the set of irreducible representations of a finite group is investigated. For the symmetric and general linear groups, a sharp convergence rate bound is obtained and a cutoff phenomenon is proved. As a related result, an asymptotic description of Plancherel measure of the finite general linear groups is given. 1.
The looperased random walk and the uniform spanning tree on the fourdimensional discrete torus
, 2008
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A survey of results on random random walks on finite groups
 Probab. Surv
, 2005
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