chain_structure: Qualitative structure of a discrete-time Markov chain

In Nestimate: Network Estimation, Bootstrap, and Higher-Order Analysis

Qualitative structure of a discrete-time Markov chain

Description

Computes properties that depend only on the transition matrix support, not on any starting distribution: state classification, communicating classes, periods, irreducibility / aperiodicity / regularity / reversibility, hitting probabilities, and absorption analysis when absorbing states exist.

Usage

chain_structure(x, normalize = TRUE, tol = 1e-10)

Arguments

x	A netobject, cograph_network, tna model, transition matrix, or sequence data.frame (passed through build_network() with method = "relative").
normalize	Logical. If TRUE (default), rows of the transition matrix are renormalized to sum to 1 before analysis (see passage_time() for the same convention).
tol	Numerical tolerance for the reversibility check (detailed balance) and for treating near-zero entries as zero when building the support graph (which drives classification, communicating_classes, period, and hitting_probabilities). It does not govern the absorbing-state test: a state is absorbing only when P[i, i] equals 1 to an internal fixed tolerance of .Machine$double.eps^0.5, independent of tol (so raising tol to ignore tiny transition probabilities never reclassifies a near-deterministic state as absorbing). Default 1e-10.

Details

Built specifically as a diagnostic to run before trusting the output of passage_time() or markov_stability(). Both implicitly assume a regular chain (irreducible + aperiodic) so that the stationary distribution is unique and meaningful. Use is_regular to check.

The fundamental-matrix absorption math follows Kemeny & Snell (1976); the hitting-probability linear system follows Norris (1997).

Value

A chain_structure object: a list with elements

states

Character vector of state names.

classification

Named character vector. One of "absorbing", "recurrent", "transient" per state.

communicating_classes

List of state-name vectors. Each sublist is a strongly connected component of the support graph.

recurrent_classes

Subset of communicating_classes that are closed (no transitions leaving the class).

transient_classes

Subset that are not closed.

absorbing_states

Character vector of states with P[i, i] = 1 (tested exactly, to within .Machine$double.eps^0.5; the user-facing tol does not relax this).

period

Named integer vector. Period of each recurrent state; NA for transient states.

is_irreducible

Logical. TRUE iff there is exactly one communicating class.

is_aperiodic

Logical. TRUE iff every recurrent state has period 1.

is_regular

Logical. is_irreducible && is_aperiodic.

is_reversible

Logical or NA. TRUE iff the chain satisfies detailed balance against its stationary distribution. NA for non-irreducible chains (no unique stationary).

hitting_probabilities

⁠n x n⁠ matrix. ⁠[i, j] = P(ever reach j starting from i)⁠, computed over the same tol-thresholded support graph that drives classification so the two are mutually consistent (a state classified "absorbing"/closed never shows hitting probability to states outside its class).

absorption_probabilities

⁠n_transient x n_absorbing⁠ matrix or NULL if no transient -> absorbing pathway exists. ⁠[i, j] = P(eventual absorption in j | start in i)⁠.

mean_absorption_time

Named numeric vector or NULL. Expected number of steps until absorption from each transient state.

P

The (possibly normalized) transition matrix used.

References

Kemeny, J. G. and Snell, J. L. (1976). Finite Markov Chains. Springer-Verlag.

Norris, J. R. (1997). Markov Chains. Cambridge University Press.

See Also

passage_time(), markov_stability(), build_network()

Examples

net <- build_network(as.data.frame(trajectories), method = "relative")
cs  <- chain_structure(net)
print(cs)

summary(cs)

Nestimate documentation built on July 11, 2026, 1:09 a.m.