D. Aldous and J. Fill. Reversible Markov chains and random walks on graphs, 2002. In preparation. Home/Search Document Not in Database Summary Related Articles Check |
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Rapidly Mixing Random Walks And Bounds On Characters Of The.. - Lulov, Pak (2002) (Correct)
....converges to a uniform distribution on G, and the mixing time mix can be de ned (see section 1) Intuitively, the mixing time is the rst time when the probability distribution of the random walk is close to uniform (in 1 distance. A general bound ( mix c (G;R) jRj log jGj (see e.g. [AF, x15]) is often used to bound the mixing time, where (G;R) is the diameter of the corresponding Cayley graph, and c is a universal constant. This bound can be sharpened when R is symmetric, i.e. transitive under the action of Aut(G) mix c (G;R) log jGj: For example, when R is a conjugacy ....
.... (G;R) is the diameter of the corresponding Cayley graph, and c is a universal constant. This bound can be sharpened when R is symmetric, i.e. transitive under the action of Aut(G) mix c (G;R) log jGj: For example, when R is a conjugacy class, the above formula applies (see e.g. [AF]) As was pointed out in [DS] the diameter ) in this case is easy to bound from a ratio of character values. The problem with bound ( is the factor log jGj, which is not tight in many special cases. Some examples show that it cannot be removed completely, and the best one could hope for is ....
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D. Aldous, J. Fill, Reversible Markov Chains and Random Walks on Graphs, monograph in preparation, 1996.
The Markov Chain Simulation Method for Generating.. - Gkantsidis, Mihail.. (2003) (1 citation) (Correct)
....example, the diameter appears to deviate consistently from its mean (see Figure 1) Therefore, a better heuristic is to estimate the sample average (y 0 y T . y kT ) k 1) wherey iT is the metric under consideration at time iT . This method of sample averages has been first considered in [5]. In addition, we will consider two (or more) separate runs of the Markov chain, where the initial points of each run are qualitatively di#erent. For example, we may consider a dense core and a sparse core starting point, as mentioned in Section 2. Now, we may consider the case where the ....
D. Aldous and J. Fill. Reversible markov chains and random walks on graphs. Monograph, http://statwww. berkeley.edu/users/aldous/book.html, 2002.
Distributed Construction of Random Expander Networks - Law, Siu (2003) (6 citations) (Correct)
....Propp and Wilson [15]introduced faster algorithms based on their coupling from the past (CFTP) procedure. They gave an algorithm COVER CFTP which generates a state distributed with stationary # in expected time at most 15 times the cover time. The expected cover time for is around n log n [16] with high probability. Thus the running time of COVER CFTP on an is O(n log n) V. APPROXIMATE SAMPLING All the sampling algorithms in Section IV require O(n) messages. We believe that it is unlikely that perfect sampling with o(n) message complexity is possible for graphs other than the ....
D. J. Aldous and J. A. Fill, "Reversible Markov chains and random walks on graphs," monograph in preparation.
Mixing Properties of the Swendsen-Wang Process on the.. - Colin Cooper Martin (2000) (2 citations) (Correct)
....given that x is the initial state. The variation distance at time t with respect to the initial state x is then defined as ,xx(t) We define the function d(t) maxE A(t) and the mixing time r( by r( min t: d(t) In particular we let r = r(e 1) By a useful property of d(t) given in [1] d(8 t) 2d)d(t) By iterating this inequality we see that for any e e 1, r(e) r(e 1)eexp(21og i e) For our purposes, the Swendsen Wang process is rapidly mixing, if the mixing time rsw = rsw(G, is bounded by a polynomial in n, the number of vertices of G. 3.1 Coupling We prove our ....
....in the discussions below. Let e = i p where p = c n and let a = kin. 2 n e Cna(l a) a(l a)w O[w) I( A O = an As c Co we have (Aa) a) alog a (1 a)log(1 a) ca(1 c 0. Lemma 5 The extrema of (a) are given by the solutions of ace ac = 1 a)c e (1 a)c. The maximum in [0, 1] occurs at the unique value of a given below. i) If c 2 then a = ii) If c 2 then a(c) is the unique solution in (0, 1) of c(a) 1 log 1 . a 1 2a a (4) I) a) log a log(1 a) c(1 2a) The values of a given in the statement of the Lemma satisfy q) a) 0. q) a) 0 ....
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D. Aldous and J. Fill. Reversible Markov Chains and Random Walks on Graphs. In preparation.
Faster Mixing via average Conductance - Lovász, Kannan (8 citations) (Correct)
.... stopping rule) it can be shown that for any 0, if we stop the Markov chain after c(log( H steps (where the constant c is an absolute constant independent of the chain) then the total variation distance of the ensuing distribution to the steady state distribution is at most (see [AF, LW1]) Thus, if we upper bound H, this yields an upper bound on the number of steps needed to get close to the steady state. The following theorem (in somewhat di erent terms) was proved by Jerrum and Sinclair [JS] Theorem 2.1 The mixing time H of any Markov chain can be bounded by H 32 log(1= 0 ....
D. J. Aldous and J. Fill (1998) : Reversible Markov Chains and Random Walks on Graphs (book), to appear. URL for draft at http://www.stat.Berkeley.edu/users/aldous/book.html
Exact Mixing in an Unknown Markov Chain - Lovasz, Winkler (1995) (6 citations) (Correct)
....we shall need that N ij # (1 (1 n) # j for all i and j. We can achieve this by choosing t = #nh#. This is good enough if we are only interested in polynomiality, but the time bounds we get this way are too pessimistic on two counts. We could apply the multiplicativity property in Aldous [4] to show that the factor n could be replaced by log n, and results from [5] to show that h can be replaced by the mixing time T mix . More exactly, let M = #log n# and s =8#T mix #,andletZ be the sum of M independent random variables Y 1 , Y M , each distributed uniformly in 0, s ....
D.J. Aldous, Reversible Markov Chains and Random Walks on Graphs (book), to appear.
Mixing Properties of the Swendsen-Wang Process on the.. - Cooper, Dyer, Frieze.. (2000) (2 citations) (Correct)
....state. The variation distance at time t with respect to the initial state x is then de ned as x (t) P t;x ; We de ne the function d(t) max x2 x (t) and the mixing time ( by ( minft : d(t) g: In particular we let = e 1 ) By a useful property of d(t) given in [1] d(s t) 2d(s)d(t) By iterating this inequality we see that for any e 1 , e 1 ) exp(2 log 1= For our purposes, the Swendsen Wang process is rapidly mixing, if the mixing time SW = SW (G; is bounded by a polynomial in n, the number of vertices of G. 4 3.1 ....
D. Aldous and J. Fill. Reversible Markov Chains and Random Walks on Graphs. In preparation.
How to Combine Fast Heuristic Markov Chain Monte Carlo with.. - Bandyopadhyay (2001) Self-citation (Aldous) (Correct)
....some small part of the state space which is almost disconnected from the remainder. Rigorous estimates typically require an a priori bound on some notion of the chain s mixing time (e.g. the relaxation time defined at (3) and while there is now substantial theoretical literature on mixing times [1, 3, 5] it deals with settings more tractable than most statistical applications. This paper investigates the interface between rigor and heuristics in a particular (perhaps artificial) context. Suppose we have a guess for the mixing time of the chain, based on simulation diagnostics or heuristic ....
D.J. Aldous and J.A. Fill. Reversible Markov chains and random walks on graphs. Book in preparation, 2001.
Semisupervised Alignment of Manifolds - Jihun Ham Systems (2004) (2 citations) (Correct)
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D. Aldous and J. Fill. Reversible Markov chains and random walks on graphs, 2002. In preparation.
Fastest mixing Markov chain on a path - Stephen Boyd Persi (2004) (Correct)
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D. Aldous and J. Fill. Reversible Markov Chains and Random Walks on Graphs. stat-www.berkeley.edu/users/aldous/RWG/book.html, 2003. Forthcoming book.
Random Walks in Peer-to-Peer Networks - Gkantsidis, Mihail, Saberi (2004) (13 citations) (Correct)
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D. Aldous and J. Fill, "Reversible markov chains and random walks on graphs," Monograph, http://statwww. berkeley.edu/users/aldous/book.html, 2002.
Random Walks in Peer-to-Peer Networks - Gkantsidis, Mihail, Saberi (2004) (13 citations) (Correct)
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D. Aldous and J. Fill, "Reversible markov chains and random walks on graphs," Monograph, http://statwww. berkeley.edu/users/aldous/book.html, 2002.
Random Walk for Self-Stabilizing Group Communication in.. - Dolev, Schiller, Welch (2002) (12 citations) (Correct)
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D. Aldous and J. A. Fill, Reversible Markov Chains and Random Walks on Graphs (book draft), October 1999. http://www.stat.berkeley.edu/aldous/book.html
A Kernel View Of The Dimensionality Reduction Of Manifolds - Ham, Lee, Mika, Schölkopf (2003) (5 citations) (Correct)
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D. Aldous and J. Fill. Reversible markov chains and random walks on graphs. In preparation.
Symmetry Analysis of Reversible Markov Chains - Boyd, Diaconis, Parrilo, Xiao (2003) (Correct)
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D. Aldous and J. Fill. Reversible Markov Chains and Random Walks on Graphs. http://stat-www.berkeley.edu/users/aldous/RWG/book.html, 2003. Forthcoming book.
Fastest mixing Markov chain on a path - Stephen Boyd Persi (2004) (Correct)
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D. Aldous and J. Fill. Reversible Markov Chains and Random Walks on Graphs. stat-www.berkeley.edu/users/aldous/RWG/book.html, 2003. Forthcoming book.
Stability and Capacity of Wireless Networks with Probabilistic .. - Mergen, Tong (2003) (Correct)
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D. Aldous and J. Fill, "Reversible Markov Chains and Random Walks on Graphs," manuscript.
Random Walks in Peer-to-Peer Networks - Christos Gkantsidis Milena (2004) (13 citations) (Correct)
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D. Aldous and J. Fill, "Reversible markov chains and random walks on graphs," Monograph, http://stat-www.berkeley.edu/users/aldous/book.html, 2002.
Cover Time and Favourite Points for Planar Random Walks - Dembo (2001) (Correct)
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Aldous (D.) and Fill (J.). { Reversible Markov chains and random walks on graphs. { Monograph in preparation.
On Kazhdan Constants and Mixing of Random Walks - Pak, Zuk (Correct)
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D. Aldous, J. Fill, Reversible Markov Chains and Random Walks on Graphs, monograph in preparation, 1996.
New Bounds for Randomized Busing - Steven Seiden Peter (Correct)
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D. J. Aldous, J. A. Fill, Reversible Markov Chains and Random Walks on Graphs. In preparation. 10
Unknown - (Correct)
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Aldous, D.: Reversible Markov chains and random walks on graphs, 1994, Preprint.
Extension of the PAC Framework to Finite and Countable Markov.. - Gamarnik (2002) (5 citations) (Correct)
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D. Aldous and J. Fill. Reversible Markov Chains and Random Walks on Graphs. In preparation.
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D. Aldous and J. A. Fill (2002), Reversible Markov chains and random walks on graphs, book in preparation.
Distributed Construction of Random Expander Graphs (Extended.. - Law, al. (2003) (7 citations) (Correct)
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Aldous, D. J., and Fill, J. A. Reversible Markov chains and random walks on graphs. Monograph in preparation.
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