58,719 research outputs found
Path Coupling Using Stopping Times and Counting Independent Sets and Colourings in Hypergraphs
We give a new method for analysing the mixing time of a Markov chain using
path coupling with stopping times. We apply this approach to two hypergraph
problems. We show that the Glauber dynamics for independent sets in a
hypergraph mixes rapidly as long as the maximum degree Delta of a vertex and
the minimum size m of an edge satisfy m>= 2Delta+1. We also show that the
Glauber dynamics for proper q-colourings of a hypergraph mixes rapidly if m>= 4
and q > Delta, and if m=3 and q>=1.65Delta. We give related results on the
hardness of exact and approximate counting for both problems.Comment: Simpler proof of main theorem. Improved bound on mixing time. 19
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A characterization of L<inf>2</inf> mixing and hypercontractivity via hitting times and maximal inequalities
There are several works characterizing the total-variation mixing time of a
reversible Markov chain in term of natural probabilistic concepts such as
stopping times and hitting times. In contrast, there is no known analog for the
mixing time, (while there are sophisticated analytic tools
to bound , in general they do not determine up to a constant
factor and they lack a probabilistic interpretation). In this work we show that
can be characterized up to a constant factor using hitting times
distributions. We also derive a new extremal characterization of the
Log-Sobolev constant, , as a weighted version of the spectral
gap. This characterization yields a probabilistic interpretation of
in terms of a hitting time version of hypercontractivity. As
applications of our results, we show that (1) for every reversible Markov
chain, is robust under addition of self-loops with bounded weights,
and (2) for weighted nearest neighbor random walks on trees, is
robust under bounded perturbations of the edge weights
Mixing and relaxation time for Random Walk on Wreath Product Graphs
Suppose that G and H are finite, connected graphs, G regular, X is a lazy
random walk on G and Z is a reversible ergodic Markov chain on H. The
generalized lamplighter chain X* associated with X and Z is the random walk on
the wreath product H\wr G, the graph whose vertices consist of pairs (f,x)
where f=(f_v)_{v\in V(G)} is a labeling of the vertices of G by elements of H
and x is a vertex in G. In each step, X* moves from a configuration (f,x) by
updating x to y using the transition rule of X and then independently updating
both f_x and f_y according to the transition probabilities on H; f_z for z
different of x,y remains unchanged. We estimate the mixing time of X* in terms
of the parameters of H and G. Further, we show that the relaxation time of X*
is the same order as the maximal expected hitting time of G plus |G| times the
relaxation time of the chain on H.Comment: 30 pages, 1 figur
Metric Construction, Stopping Times and Path Coupling
In this paper we examine the importance of the choice of metric in path
coupling, and the relationship of this to \emph{stopping time analysis}. We
give strong evidence that stopping time analysis is no more powerful than
standard path coupling. In particular, we prove a stronger theorem for path
coupling with stopping times, using a metric which allows us to restrict
analysis to standard one-step path coupling. This approach provides insight for
the design of non-standard metrics giving improvements in the analysis of
specific problems.
We give illustrative applications to hypergraph independent sets and SAT
instances, hypergraph colourings and colourings of bipartite graphs.Comment: 21 pages, revised version includes statement and proof of general
stopping times theorem (section 2.2), and additonal remarks in section
Age Optimal Information Gathering and Dissemination on Graphs
We consider the problem of timely exchange of updates between a central
station and a set of ground terminals , via a mobile agent that traverses
across the ground terminals along a mobility graph . We design the
trajectory of the mobile agent to minimize peak and average age of information
(AoI), two newly proposed metrics for measuring timeliness of information. We
consider randomized trajectories, in which the mobile agent travels from
terminal to terminal with probability . For the information
gathering problem, we show that a randomized trajectory is peak age optimal and
factor- average age optimal, where is the mixing
time of the randomized trajectory on the mobility graph . We also show that
the average age minimization problem is NP-hard. For the information
dissemination problem, we prove that the same randomized trajectory is
factor- peak and average age optimal. Moreover, we propose an
age-based trajectory, which utilizes information about current age at
terminals, and show that it is factor- average age optimal in a symmetric
setting
Estimating graph parameters with random walks
An algorithm observes the trajectories of random walks over an unknown graph
, starting from the same vertex , as well as the degrees along the
trajectories. For all finite connected graphs, one can estimate the number of
edges up to a bounded factor in
steps, where
is the relaxation time of the lazy random walk on and
is the minimum degree in . Alternatively, can be estimated in
, where is
the number of vertices and is the uniform mixing time on
. The number of vertices can then be estimated up to a bounded factor in
an additional steps. Our
algorithms are based on counting the number of intersections of random walk
paths , i.e. the number of pairs such that . This
improves on previous estimates which only consider collisions (i.e., times
with ). We also show that the complexity of our algorithms is optimal,
even when restricting to graphs with a prescribed relaxation time. Finally, we
show that, given either or the mixing time of , we can compute the
"other parameter" with a self-stopping algorithm
Mixing times of random walks on dynamic configuration models
The mixing time of a random walk, with or without backtracking, on a random
graph generated according to the configuration model on vertices, is known
to be of order . In this paper we investigate what happens when the
random graph becomes {\em dynamic}, namely, at each unit of time a fraction
of the edges is randomly rewired. Under mild conditions on the
degree sequence, guaranteeing that the graph is locally tree-like, we show that
for every the -mixing time of random walk
without backtracking grows like
as , provided
that . The latter condition
corresponds to a regime of fast enough graph dynamics. Our proof is based on a
randomised stopping time argument, in combination with coupling techniques and
combinatorial estimates. The stopping time of interest is the first time that
the walk moves along an edge that was rewired before, which turns out to be
close to a strong stationary time.Comment: 23 pages, 6 figures. Previous version contained a mistake in one of
the proofs. In this version we look at nonbacktracking random walk instead of
simple random wal
The Best Mixing Time for Random Walks on Trees
We characterize the extremal structures for mixing walks on trees that start
from the most advantageous vertex. Let be a tree with stationary
distribution . For a vertex , let denote the expected
length of an optimal stopping rule from to . The \emph{best mixing
time} for is . We show that among all trees with
, the best mixing time is minimized uniquely by the star. For even ,
the best mixing time is maximized by the uniquely path. Surprising, for odd
, the best mixing time is maximized uniquely by a path of length with
a single leaf adjacent to one central vertex.Comment: 25 pages, 7 figures, 3 table
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