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Digraph-based conditioning for Markov chains. (English) Zbl 1056.15022
For an irreducible stochastic matrix $T$, the author considers a certain condition number $c(T)$, which measures the stability of the corresponding stationary distribution when $T$ is perturbed, and then characterizes the strongly connected directed graphs $D$ such that $c(T)$ is bounded as $T$ ranges over $\Cal S_D$, the set of stochastic matrices whose directed graph is contained in $D$. For those digraphs $D$ for which $c(T)$ is bounded, the maximum value of $\{c(T)\mid T$ ranges over $\Cal S_D\}$ is determined.

MSC:
15B51Stochastic matrices
15A18Eigenvalues, singular values, and eigenvectors
60J10Markov chains (discrete-time Markov processes on discrete state spaces)
05C20Directed graphs (digraphs), tournaments
15A12Conditioning of matrices
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References:
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