is.reversible: Check whether a Markov chain is reversible

In markovchain: Easy Handling Discrete Time Markov Chains

Check whether a Markov chain is reversible

Description

Checks whether a finite, irreducible discrete-time Markov chain is reversible with respect to its (unique) stationary distribution, i.e. whether it satisfies the detailed balance equations.

Usage

is.reversible(object, tolerance = sqrt(.Machine$double.eps))

## S4 method for signature 'markovchain'
is.reversible(object, tolerance = sqrt(.Machine$double.eps))

Arguments

object	A markovchain object representing a finite, irreducible discrete-time Markov chain.
tolerance	A single finite non-negative number. Detailed balance is accepted as holding when every pair (i,j) satisfies |\pi_i P_{ij} - \pi_j P_{ji}| \le \code{tolerance}. The default is a small numerical-noise tolerance, not a modelling tolerance: it exists to absorb floating-point rounding in the eigendecomposition-based steadyStates computation, not to declare "almost reversible" chains reversible.

Details

A chain with transition matrix P and stationary distribution \pi is reversible if

\pi_i P_{ij} = \pi_j P_{ji} \quad \text{for every } i,j.

Intuitively, if you started the chain from \pi and watched a long run of it, running the recorded sequence of states backwards would look statistically identical to running it forwards: at stationarity, the "flow" of probability from i to j exactly balances the flow from j to i.

Only irreducibility is required, not aperiodicity: detailed balance is a purely algebraic condition on P and \pi and is perfectly well defined for periodic chains too. For example, a simple random walk on any undirected graph (moving to a uniformly random neighbour) is always reversible, whether or not it happens to be periodic.

A 2-state irreducible chain is always reversible: with only two states, the single detailed balance equation \pi_1 P_{12} = \pi_2 P_{21} is just a restatement of the stationarity equation \pi P = \pi, so it holds automatically.

Every reversible chain has a real spectrum (all eigenvalues of P are real), which is why slem and spectralGap are especially easy to interpret for reversible chains: there are no complex-conjugate eigenvalue pairs to reason about.

The implementation calls steadyStates once, then compares the two triangles of the flow matrix \pi_i P_{ij}. Its time complexity is dominated by steadyStates(), plus an additional O(n^2) comparison for a dense n-state transition matrix. It supports both row- and column-stochastic storage.

Value

A single logical value, TRUE or FALSE.

References

Norris, J. R. (1998). Markov Chains. Cambridge University Press.

Levin, D. A. and Peres, Y. (2017). Markov Chains and Mixing Times, 2nd edition. American Mathematical Society.

See Also

steadyStates, is.irreducible, slem, mixingTime

Examples

# A random walk on a triangle is reversible.
statesNames <- c("a", "b", "c")
triangle <- new("markovchain", states = statesNames,
  transitionMatrix = matrix(c(0, 0.5, 0.5,
                              0.5, 0, 0.5,
                              0.5, 0.5, 0), byrow = TRUE, nrow = 3,
                            dimnames = list(statesNames, statesNames)))
is.reversible(triangle)

# A directed cycle (states only move "forward") is not reversible: there
# is a net clockwise flow of probability at stationarity.
cycle3 <- new("markovchain", states = statesNames,
  transitionMatrix = matrix(c(0, 1, 0,
                              0, 0, 1,
                              1, 0, 0), byrow = TRUE, nrow = 3,
                            dimnames = list(statesNames, statesNames)))
is.reversible(cycle3)

markovchain documentation built on Oct. 10, 2026, 9:07 a.m.