scholarly journals Mixing time for the random walk on the range of the random walk on tori

2016 ◽  
Vol 21 (0) ◽  
Author(s):  
Jiří Černý ◽  
Artem Sapozhnikov
Keyword(s):  
Random Walk ◽  
Mixing Time

scholarly journals Uniform mixing time for random walk on lamplighter graphs

2014 ◽  
Vol 50 (4) ◽  
pp. 1140-1160 ◽  
Author(s):  
Júlia Komjáthy ◽  
Jason Miller ◽  
Yuval Peres
Keyword(s):  
Random Walk ◽  
Mixing Time

scholarly journals Assessing significance in a Markov chain without mixing

2017 ◽  
Vol 114 (11) ◽  
pp. 2860-2864 ◽  
Author(s):  
Maria Chikina ◽  
Alan Frieze ◽  
Wesley Pegden
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Stationary Distribution ◽  
Potential Application ◽  
Null Hypothesis ◽  
Value Function ◽  
Mixing Time ◽  
Statistical Test ◽  
Reversible Markov Chain ◽  
A Value

We present a statistical test to detect that a presented state of a reversible Markov chain was not chosen from a stationary distribution. In particular, given a value function for the states of the Markov chain, we would like to show rigorously that the presented state is an outlier with respect to the values, by establishing a p value under the null hypothesis that it was chosen from a stationary distribution of the chain. A simple heuristic used in practice is to sample ranks of states from long random trajectories on the Markov chain and compare these with the rank of the presented state; if the presented state is a 0.1% outlier compared with the sampled ranks (its rank is in the bottom 0.1% of sampled ranks), then this observation should correspond to a p value of 0.001. This significance is not rigorous, however, without good bounds on the mixing time of the Markov chain. Our test is the following: Given the presented state in the Markov chain, take a random walk from the presented state for any number of steps. We prove that observing that the presented state is an ε-outlier on the walk is significant at p=2ε under the null hypothesis that the state was chosen from a stationary distribution. We assume nothing about the Markov chain beyond reversibility and show that significance at p≈ε is best possible in general. We illustrate the use of our test with a potential application to the rigorous detection of gerrymandering in Congressional districting.

scholarly journals A non-local random walk on the hypercube

2017 ◽  
Vol 49 (4) ◽  
pp. 1288-1299
Author(s):  
Evita Nestoridi
Keyword(s):  
Random Walk ◽  
Mixing Time ◽  
Non Local

Abstract In this paper we study the random walk on the hypercube (ℤ / 2ℤ)n which at each step flips k randomly chosen coordinates. We prove that the mixing time for this walk is of the order (n / k)logn. We also prove that if k = o(n) then the walk exhibits cutoff at (n / 2k)logn with window n / 2k.

scholarly journals Random walks on generating sets for finite groups

10.37236/1322 ◽  
1996 ◽  
Vol 4 (2) ◽  
Author(s):  
F. R. K. Chung ◽  
R. L. Graham
Keyword(s):  
Finite Group ◽  
Random Walk ◽  
Random Walks ◽  
Finite Groups ◽  
Mixing Time ◽  
Cartesian Product ◽  
Generating Sets ◽  
Random Elements ◽  
A Finite Group

We analyze a certain random walk on the cartesian product $G^n$ of a finite group $G$ which is often used for generating random elements from $G$. In particular, we show that the mixing time of the walk is at most $c_r n^2 \log n$ where the constant $c_r$ depends only on the order $r$ of $G$.

scholarly journals The Mixing Time for a Random Walk on the Symmetric Group Generated by Random Involutions

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Megan Bernstein
Keyword(s):  
Information Theory ◽  
Random Walk ◽  
Symmetric Group ◽  
Random Walks ◽  
Binomial Distribution ◽  
Unitary Group ◽  
Mixing Time ◽  
Sufficient Time ◽  
International Audience ◽  
A New Technique

International audience The involution walk is a random walk on the symmetric group generated by involutions with a number of 2-cycles sampled from the binomial distribution with parameter p. This is a parallelization of the lazy transposition walk onthesymmetricgroup.Theinvolutionwalkisshowninthispapertomixfor1 ≤p≤1fixed,nsufficientlylarge 2 in between log1/p(n) steps and log2/(1+p)(n) steps. The paper introduces a new technique for finding eigenvalues of random walks on the symmetric group generated by many conjugacy classes using the character polynomial for the characters of the representations of the symmetric group. This is especially efficient at calculating the large eigenvalues. The smaller eigenvalues are handled by developing monotonicity relations that also give after sufficient time the likelihood order, the order from most likely to least likely state. The walk was introduced to study a conjecture about a random walk on the unitary group from the information theory of back holes.

scholarly journals Speed up random walk by leveraging community affiliation information

2019 ◽  
Vol 2 (1) ◽  
pp. 51-65
Author(s):  
Naian Yin ◽  
Yachao Lu ◽  
Nan Zhang
Keyword(s):  
Random Walk ◽  
Network Topology ◽  
Mixing Time ◽  
Third Party ◽  
Online Networks ◽  
Web Interfaces ◽  
Related Data ◽  
Speed Up ◽  
Selection Algorithms ◽  
Digital Assets

Abstract Large online networks are most massive and opulent data sources these days. The inherent growing demands of analyses related data fetching conflict greatly with network providers’ efforts to protect their digital assets as well as users’ increasing awareness of privacy. Restrictions on web interfaces of online networks prevent third party researchers from gathering sufficient data and further global images of these networks are also hidden. Under such circumstances, only techniques like random walk approaches that can run under local neighborhood access will be adopted to fulfill large online network sampling tasks. Meanwhile, the presence of highly clustered community like structure in large networks leads to random walk’s poor conductance, causing intolerable and hard-to-foresee long mixing time before useful samples can be collected. With lack of techniques incorporate online network topology features being the context, in this paper we focus on taking use of community affiliation information that possibly comes with metadata when querying objects in online networks, and proposed a speeded version of random walk by raising the probability of inter-community edges being selected. Assuming the community structure is well established as promised, the community speeded random walk expects better conductance and faster convergence. Our method forces the sampler to travel rapidly among different communities that conquers the bottlenecks and thus the samples being collected are of higher quality. We also consider the scenario when community affiliation is not directly available, where we apply feature selection algorithms to select features as community.

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