Fastest Expected Time to Mixing for a Markov
Chain on a Directed Graph.
Linear Algebra and its Applications, 433.
For an irreducible stochastic matrix T, the Kemeny constant K(T)
measures the expected time to mixing of the Markov chain corresponding
to T. Given a strongly connected directed graph D, we consider the set
ΣD of stochastic matrices whose directed graph is subordinate to D, and
compute the minimum value of K, taken over the set ΣD. The matrices
attaining that minimum are also characterised, thus yielding a description
of the transition matrices in ΣD that minimise the expected time to
mixing. We prove that K(T) is bounded from above as T ranges over
the irreducible members of ΣD if and only if D is an intercyclic directed
graph, and in the case that D is intercyclic, we find the maximum value
of K on the set ΣD. Throughout, our results are established using a mix
of analytic and combinatorial techniques.
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