On the structure of stochastic matrices with a subdominant eigenvalue near 1

Hartfiel, D. J.; Meyer, Carl D.

doi:10.1016/s0024-3795(97)00333-9

Cited by 51 publications

(30 citation statements)

References 4 publications

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“…We further find the 1/κ 6 (T ) = 0.0882 < 0.089 = min λ∈σ (T ),λ =1 |1 − λ| and so, in this example, κ 6 (T ) furnishes a close estimate to an upper bound on the measure for near-uncoupling introduced by Hartfiel and Meyer [11].…”

Section: Proof Of Theorem 33supporting

confidence: 57%

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On optimal condition numbers for Markov chains

2008

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Let T andT = T − E be arbitrary nonnegative, irreducible, stochastic matrices corresponding to two ergodic Markov chains on n states. A function κ is called a condition number for Markov chains with respect to the (α, β)-norm pair if π −π α ≤ κ(T ) E β . Here π andπ are the stationary distribution vectors of the two chains, respectively. Various condition numbers, particularly with respect to the (1, ∞) and (∞, ∞)-norm pairs have been suggested in the literature. They were ranked according to their size by Cho and Meyer in a paper from 2001. In this paper we first of all show that what we call the generalized ergodicity coefficient τ p (A # ) = sup y t e=0 y t A # p y 1 , where e is the n-vector of all 1's and A # is the group generalized inverse of A = I − T , is the smallest condition number of Markov chains with respect to the ( p, ∞)-norm pair. We use this result to identify the smallest condition number of Markov chains among the (∞, ∞) and (1, ∞)-norm pairs. These are, respectively, κ 3 and κ 6 in the Cho-Meyer list of 8 condition numbers. Kirkland has studied κ 3 (T ). He has shown that κ 3 (T ) ≥ n−1 2n and he has characterized transition matrices for which equality holds. We prove here again that 2κ 3 (T ) ≤ κ(6) which appears in the Cho-Meyer paper and we characterize the transition matrices T for which κ 6 (T ) = n−1 n . There is actually only one such matrix: T = (J n − I )/(n − 1), where J n is the n × n matrix of all 1's.

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Section: Proof Of Theorem 33supporting

confidence: 57%

“…In [11], Hartfiel and Meyer showed that the closer this quantity is to 0, the more nearly uncoupled the chain becomes. We further mention that in Kirkland and Neumann [18], conditions are studied on T under which equality holds throughout (4.5).…”

Section: Proof Of Theorem 33mentioning

confidence: 99%

On optimal condition numbers for Markov chains

2008

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“…That is to say, if ( ) → 0, then the second largest eigenvalue modulus is nearly 1. The inverse of Proposition 3 has been proved by [22]. If second largest eigenvalue is sufficiently close to 1, then is nearly uncoupled.…”

Section: Proposition 1 Consensus Will Be Reached Ultimatelymentioning

confidence: 95%

The Impact of Community Structure on the Convergence Time of Opinion Dynamics

Sun

Liu

2017

Discrete Dynamics in Nature and Society

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“…We may use a stochastic matrix P to describe the states of such a chain. In [3], Hartfiel and Meyer defined the uncoupling measure of P as following:…”

Section: Applications To Markov Chainsmentioning

confidence: 99%