State Space Orderings for Gauss-Seidel in Markov Chains Revisited

States of a Markov chain may be reordered to reduce the magnitude of the subdominant eigenvalue of the Gauss--Seidel (GS) iteration matrix. Orderings that maximize the elemental mass or the number of nonzero elements in the dominant term of the GS splitting (that is, the term approximating the coefficient matrix) do not necessarily converge faster. An ordering of a Markov chain that satisfies Property-R is semiconvergent. On the other hand, there are semiconvergent state space orderings that do not satisfy Property-R. For a given ordering, a simple approach for checking Property-R is shown. Moreover, a version of the Cuthill--McKee algorithm may be used to order the states of a Markov chain so that Property-R is satisfied. The computational complexity of the ordering algorithm is less than that of a single GS iteration. In doing all this, the aim is to gain insight into (faster) converging orderings.