Perturbation bounds for the stationary probabilities of a finite Markov chain

Haviv, Moshe; Heyden, Ludo Van der

doi:10.2307/1427341

Cited by 60 publications

(36 citation statements)

References 7 publications

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“…is a poor estimate of the sensitivity of the ij th entry for this problem. Some of these condition numbers are based on ergodicity coefficients [20,21], some on mean first passage times [2], and some on generalized inverses of the characteristic matrix I − F [5,7,9,16,19]. We prove for the hilly landscape transition matrix F that κ i (F ) increases exponentially with L for all i.…”

Section: 3mentioning

confidence: 99%

Sharp Entrywise Perturbation Bounds for Markov Chains

Thiede¹,

Koten²,

Weare

2015

SIAM J. Matrix Anal. & Appl.

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For many Markov chains of practical interest, the invariant distribution is extremely sensitive to perturbations of some entries of the transition matrix, but insensitive to others; we give an example of such a chain, motivated by a problem in computational statistical physics. We have derived perturbation bounds on the relative error of the invariant distribution that reveal these variations in sensitivity. Our bounds are sharp, we do not impose any structural assumptions on the transition matrix or on the perturbation, and computing the bounds has the same complexity as computing the invariant distribution or computing other bounds in the literature. Moreover, our bounds have a simple interpretation in terms of hitting times, which can be used to draw intuitive but rigorous conclusions about the sensitivity of a chain to various types of perturbations.

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Section: 3mentioning

confidence: 99%

Sharp Entrywise Perturbation Bounds for Markov Chains

Thiede¹,

Koten²,

Weare

2015

SIAM J. Matrix Anal. & Appl.

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“…The theorem is based on a result due to Schweitzer (1968) (see also Haviv and Van Der Heyden, 1984).…”

Section: Theorem 3 Consider a Regular Markov Chain With N States Andmentioning

confidence: 99%

Spread of (mis)information in social networks

Acemoğlu

Ozdaglar

ParandehGheibi

2010

Games and Economic Behavior

363

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We provide a model to investigate the tension between information aggregation and spread of misinformation. Individuals meet pairwise and exchange information, which is modeled as both individuals adopting the average of their pre-meeting beliefs. "Forceful" agents influence the beliefs of (some of) the other individuals they meet, but do not change their own opinions. We characterize how the presence of forceful agents interferes with information aggregation. Under the assumption that even forceful agents obtain some information from others, we first show that all beliefs converge to a stochastic consensus. Our main results quantify the extent of misinformation by providing bounds or exact results on the gap between the consensus value and the benchmark without forceful agents (where there is efficient information aggregation). The worst outcomes obtain when there are several forceful agents who update their beliefs only on the basis of information from individuals that have been influenced by them.

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“…The proof uses results from Markov Chain Theory, which enable us to decompose the mean interaction matrixW in (24) into a component given by the social network matrix T , which is doubly stochastic, and an influence matrix D, which is the source of deviation of Ex from θ (see, in particular, [57,83]). …”

Section: Theoremmentioning

confidence: 99%