6,187 research outputs found
Robustness of large-scale stochastic matrices to localized perturbations
Upper bounds are derived on the total variation distance between the
invariant distributions of two stochastic matrices differing on a subset W of
rows. Such bounds depend on three parameters: the mixing time and the minimal
expected hitting time on W for the Markov chain associated to one of the
matrices; and the escape time from W for the Markov chain associated to the
other matrix. These results, obtained through coupling techniques, prove
particularly useful in scenarios where W is a small subset of the state space,
even if the difference between the two matrices is not small in any norm.
Several applications to large-scale network problems are discussed, including
robustness of Google's PageRank algorithm, distributed averaging and consensus
algorithms, and interacting particle systems.Comment: 12 pages, 4 figure
Theoretical analysis of a Stochastic Approximation approach for computing Quasi-Stationary distributions
This paper studies a method, which has been proposed in the Physics
literature by [8, 7, 10], for estimating the quasi-stationary distribution. In
contrast to existing methods in eigenvector estimation, the method eliminates
the need for explicit transition matrix manipulation to extract the principal
eigenvector. Our paper analyzes the algorithm by casting it as a stochastic
approximation algorithm (Robbins-Monro) [23, 16]. In doing so, we prove its
convergence and obtain its rate of convergence. Based on this insight, we also
give an example where the rate of convergence is very slow. This problem can be
alleviated by using an improved version of the algorithm that is given in this
paper. Numerical experiments are described that demonstrate the effectiveness
of this improved method
General solution of the Poisson equation for Quasi-Birth-and-Death processes
We consider the Poisson equation , where
is the transition matrix of a Quasi-Birth-and-Death (QBD) process with
infinitely many levels, is a given infinite dimensional vector and is the unknown. Our main result is to provide the general solution of this
equation. To this purpose we use the block tridiagonal and block Toeplitz
structure of the matrix to obtain a set of matrix difference equations,
which are solved by constructing suitable resolvent triples
Bounding inferences for large-scale continuous-time Markov chains : a new approach based on lumping and imprecise Markov chains
If the state space of a homogeneous continuous-time Markov chain is too large, making inferences becomes computationally infeasible. Fortunately, the state space of such a chain is usually too detailed for the inferences we are interested in, in the sense that a less detailed—smaller—state space suffices to unambiguously formalise the inference. However, in general this so-called lumped state space inhibits computing exact inferences because the corresponding dynamics are unknown and/or intractable to obtain. We address this issue by considering an imprecise continuous-time Markov chain. In this way, we are able to provide guaranteed lower and upper bounds for the inferences of interest, without suffering from the curse of dimensionality
Markov-modulated Brownian motion with two reflecting barriers
We consider a Markov-modulated Brownian motion reflected to stay in a strip
[0,B]. The stationary distribution of this process is known to have a simple
form under some assumptions. We provide a short probabilistic argument leading
to this result and explaining its simplicity. Moreover, this argument allows
for generalizations including the distribution of the reflected process at an
independent exponentially distributed epoch. Our second contribution concerns
transient behavior of the reflected system. We identify the joint law of the
processes t,X(t),J(t) at inverse local times.Comment: 13 pages, 1 figur
Opinion influence and evolution in social networks: a Markovian agents model
In this paper, the effect on collective opinions of filtering algorithms
managed by social network platforms is modeled and investigated. A stochastic
multi-agent model for opinion dynamics is proposed, that accounts for a
centralized tuning of the strength of interaction between individuals. The
evolution of each individual opinion is described by a Markov chain, whose
transition rates are affected by the opinions of the neighbors through
influence parameters. The properties of this model are studied in a general
setting as well as in interesting special cases. A general result is that the
overall model of the social network behaves like a high-dimensional Markov
chain, which is viable to Monte Carlo simulation. Under the assumption of
identical agents and unbiased influence, it is shown that the influence
intensity affects the variance, but not the expectation, of the number of
individuals sharing a certain opinion. Moreover, a detailed analysis is carried
out for the so-called Peer Assembly, which describes the evolution of binary
opinions in a completely connected graph of identical agents. It is shown that
the Peer Assembly can be lumped into a birth-death chain that can be given a
complete analytical characterization. Both analytical results and simulation
experiments are used to highlight the emergence of particular collective
behaviours, e.g. consensus and herding, depending on the centralized tuning of
the influence parameters.Comment: Revised version (May 2018
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