Nearly completely decomposable Markov chain

In probability theory, a nearly completely decomposable (NCD) Markov chain is a Markov chain where the state space can be partitioned in such a way that movement within a partition occurs much more frequently than movement between partitions.^[1] Particularly efficient algorithms exist to compute the stationary distribution of Markov chains with this property.^[2]

Ando and Fisher define a completely decomposable matrix as one where "an identical rearrangement of rows and columns leaves a set of square submatrices on the principal diagonal and zeros everywhere else." A nearly completely decomposable matrix is one where an identical rearrangement of rows and columns leaves a set of square submatrices on the principal diagonal and small nonzeros everywhere else.^[3][4]

A Markov chain with transition matrix

${\displaystyle P={\begin{pmatrix}{\frac {1}{2}}&{\frac {1}{2}}&0&0\\{\frac {1}{2}}&{\frac {1}{2}}&0&0\\0&0&{\frac {1}{2}}&{\frac {1}{2}}\\0&0&{\frac {1}{2}}&{\frac {1}{2}}\\\end{pmatrix}}+\epsilon {\begin{pmatrix}-{\frac {1}{2}}&0&{\frac {1}{2}}&0\\0&-{\frac {1}{2}}&0&{\frac {1}{2}}\\{\frac {1}{2}}&0&-{\frac {1}{2}}&0\\0&{\frac {1}{2}}&0&-{\frac {1}{2}}\\\end{pmatrix}}}$

is nearly completely decomposable if ε is small (say 0.1).^[5]

Special-purpose iterative algorithms have been designed for NCD Markov chains^[2] though the multi–level algorithm, a general purpose algorithm,^[6] has been shown experimentally to be competitive and in some cases significantly faster.^[7]

• Lumpability