Qualitative structure of a discrete-time Markov chain

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.

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.

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).

References

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

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

Examples

net <- build_network(as.data.frame(trajectories), method = "relative")
cs  <- chain_structure(net)
print(cs)
#> Chain structure  [3 states, 1 communicating classes]
#>   irreducible: TRUE   aperiodic: TRUE   regular: TRUE   reversible: FALSE
#>   recurrent classes: 1   transient classes: 0
#> 
#> Use summary(x) for the per-state table, plot(x) for the heatmap.
# \donttest{
summary(cs)
#> Chain structure summary  [3 states, 1 classes]
#>   irreducible: TRUE   aperiodic: TRUE   regular: TRUE   reversible: FALSE
#> 
#>       state classification period persistence return_probability sojourn_steps
#>      Active      recurrent      1      0.6976                  1          3.31
#>     Average      recurrent      1      0.6099                  1          2.56
#>  Disengaged      recurrent      1      0.4831                  1          1.93
#>  stationary_probability
#>                  0.3719
#>                  0.4431
#>                  0.1850
# }