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. Particularly efficient algorithms exist to compute the stationary distribution of Markov chains with this property.
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Definition
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.
Example
A Markov chain with transition matrix
is nearly completely decomposable if ε is small (say 0.1).
Stationary distribution algorithms
Special-purpose iterative algorithms have been designed for NCD Markov chains though the multi–level algorithm, a general purpose algorithm, has been shown experimentally to be competitive and in some cases significantly faster.