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.

Version

v1 · 2026-09-28 · History

Domain-specific #

10932

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomains

Probability Theory, Markov Chains → Mathematics

Core Idea

Nearly completely decomposable Markov chain is treated here as the recurring crossdomainmodelsstructuresrepresentations identity summarized by this source-grounded definition: 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. 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.

Scope of Application

• 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.

• 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.

• A Markov chain with transition matrix. \frac{1}{2} & \frac{1}{2} & 0 & 0 \.

• A Markov chain with transition matrix. 0 & 0 & \frac{1}{2} & \frac{1}{2} \.

• A Markov chain with transition matrix. \frac{1}{2} & 0 & \frac{1}{2} & 0 \.

Clarity

A clear use of Nearly completely decomposable Markov chain names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is 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.

Manages Complexity

Nearly completely decomposable Markov chain compresses multiple crossdomainmodelsstructuresrepresentations details into a stable diagnostic relation. The source shows both the central mechanism—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.

Abstract Reasoning

1. Type the carrier. Identify the crossdomainmodelsstructuresrepresentations entities to which the claim applies.
2. State the relation. Use the source-grounded identity: 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.
3. Check operation and conditions. \frac{1}{2} & \frac{1}{2} & 0 & 0 \.
4. Demand recognition evidence. 0 & 0 & \frac{1}{2} & \frac{1}{2} \.
5. Test variation.

Knowledge Transfer

Within the home domain. Knowledge about Nearly completely decomposable Markov chain transfers literally when a new case preserves the same carrier type, relation, and recognition test. 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. 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.

Relationships to Other Abstractions

Current abstraction Nearly completely decomposable Markov chain Domain-specific

Parents (1) — more general patterns this builds on

• Nearly completely decomposable Markov chain is a kind of Markov Process Prime

  A nearly decomposable Markov chain is a Markov process with fast within-block and slow between-block transition structure.

Hierarchy paths (4) — routes to 4 parentless roots

• Nearly completely decomposable Markov chain → Markov Process → Stochastic Process

Show alternative paths (3)

Neighborhood in Abstraction Space

Nearly completely decomposable Markov chain sits in a sparse region of the domain-specific corpus (61^st percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Markov Chains & Probabilistic Computation (6 abstractions)

Nearest neighbors

• Metropolis Algorithm — 0.88
• Telescoping Markov chain — 0.85
• Big O in probability notation — 0.85
• Pentadiagonal Matrix — 0.84
• Doubly stochastic matrix — 0.84

Computed from structural-signature embeddings · 2026-10-08