Completion Problems and Sparsity for Kemeny's Constant

TL;DR

This paper investigates the problem of completing partially specified stochastic matrices to minimize Kemeny's constant, demonstrating the existence of sparse optimal completions and deriving minimum values in specific cases.

Contribution

It introduces the concept of sparse completions for stochastic matrices to minimize Kemeny's constant and provides explicit solutions in special cases.

Findings

Existence of sparse minimizing completions for well-defined problems

Explicit minimum Kemeny's constant when the diagonal is specified

Explicit minimum Kemeny's constant when all specified entries are in one row

Abstract

For a partially specified stochastic matrix, we consider the problem of completing it so as to minimize Kemeny's constant. We prove that for any partially specified stochastic matrix for which the problem is well-defined, there is a minimizing completion that is as sparse as possible. We also find the minimum value of Kemeny's constant in two special cases: when the diagonal has been specified, and when all specified entries lie in a common row.