512 research outputs found
Hitting times for random walks with restarts
The time it takes a random walker in a lattice to reach the origin from
another vertex , has infinite mean. If the walker can restart the walk at
at will, then the minimum expected hitting time (minimized over
restarting strategies) is finite; it was called the ``grade'' of by
Dumitriu, Tetali and Winkler. They showed that, in a more general setting, the
grade (a variant of the ``Gittins index'') plays a crucial role in control
problems involving several Markov chains. Here we establish several conjectures
of Dumitriu et al on the asymptotics of the grade in Euclidean lattices. In
particular, we show that in the planar square lattice, is asymptotic
to as . The proof hinges on the local variance
of the potential kernel being almost constant on the level sets of . We
also show how the same method yields precise second order asymptotics for
hitting times of a random walk (without restarts) in a lattice disk.Comment: 12 page
Hitting Times in Markov Chains with Restart and their Application to Network Centrality
Motivated by applications in telecommunications, computer scienceand physics,
we consider a discrete-time Markov process withrestart. At each step the
process eitherwith a positive probability restarts from a given distribution,
orwith the complementary probability continues according to a Markovtransition
kernel. The main contribution of the present work is thatwe obtain an explicit
expression for the expectation of the hittingtime (to a given target set) of
the process with restart.The formula is convenient when considering the problem
of optimizationof the expected hitting time with respect to the restart
probability.We illustrate our results with two examplesin uncountable and
countable state spaces andwith an application to network centrality
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
Restart expedites quantum walk hitting times
Classical first-passage times under restart are used in a wide variety of
models, yet the quantum version of the problem still misses key concepts. We
study the quantum hitting time with restart using a monitored quantum walk. The
restart strategy eliminates the problem of dark states, i.e. cases where the
particle evades detection, while maintaining the ballistic propagation which is
important for fast search. We find profound effects of quantum oscillations on
the restart problem, namely a type of instability of the mean detection time,
and optimal restart times that form staircases, with sudden drops as the rate
of sampling is modified. In the absence of restart and in the Zeno limit, the
detection of the walker is not possible and we examine how restart overcomes
this well-known problem, showing that the optimal restart time becomes
insensitive to the sampling period.Comment: 13 pages, 11 figure
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