2004
DOI: 10.1137/s0036144503423264
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Fastest Mixing Markov Chain on a Graph

Abstract: 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 minim… Show more

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Cited by 619 publications
(512 citation statements)
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References 36 publications
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“…, A t , the matrix Chebyshev polynomial approximation problem is actually a special case of the MNA problem (1). In contrast to the unconstrained example (2), some other problems may have prescribed linear constraints, for example, the fastest mixing Markov chain (FMMC) problem studied in [2,3]. Let G = (N , E) be an undirected connected graph with n nodes.…”
Section: A(x)mentioning
confidence: 99%
“…, A t , the matrix Chebyshev polynomial approximation problem is actually a special case of the MNA problem (1). In contrast to the unconstrained example (2), some other problems may have prescribed linear constraints, for example, the fastest mixing Markov chain (FMMC) problem studied in [2,3]. Let G = (N , E) be an undirected connected graph with n nodes.…”
Section: A(x)mentioning
confidence: 99%
“…if x(k f ) =m then x(k) =m ∀k > k f . Equation (3) says that the average of all the estimates is conserved at each iteration which guarantees the final consensus value to be the average of the initial measurements. Finally, equation (4) is the contraction condition, i.e.…”
Section: Average Consensus Algorithmsmentioning
confidence: 99%
“…The convergence speed of these algorithms is related to the mixing time of Markovian chains, it is possible to show that the "error" ||x(k) −m|| 2 can be upper bounded by cλ k 2 , where λ 2 is the second largest eigenvalue of the weight matrix W and c is a constant [3], [13]. In [2], [3], the authors address the problem of selecting the optimal weights and formulate a semidefinite program to determine the matrix W with the smallest value of λ 2 . If the network topology is not known a priory (or changes in time), the implementation of this optimization procedure requires a single sensor to discover the topology of the whole network and perform the calculations.…”
Section: Average Consensus Algorithmsmentioning
confidence: 99%
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“…Two simple and popular heuristic methods for designing P that satisfy the above conditions are maximum-degree (MD) and Metropolis-Hastings (MH) algorithm [9] [10].…”
Section: B Random Walks On Graphs and Uniform Node Samplingmentioning
confidence: 99%