Markov Chain Generator

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K is the average of the expected hitting times from state i to every other state, weighted by each state's stationary mass.

Display:

K = _

Kemeny's constant is the average of the expected hitting times from a given starting state i to every other state j, weighted by each state's stationary mass. Remarkably, for an irreducible chain, this value is the same for every choice of i.

h_x is the weighted average of neighboring values outside A and B

Display:

h_i = P_i(tau_A < tau_B) = _

The splitting probability is the chance that the chain reaches A before B when starting from i. Paths that never reach either set do not count toward this event, so P_i(tau_A < tau_B) + P_i(tau_B < tau_A) can be less than 1.

Display:

E_i[tau_A] = _

The expected hitting time is the average number of steps needed to reach any state in A when starting from i.

Number of steps: n =

State Order: Display:

Computing the n-step distribution is equivalent to calculating Pⁿ and extracting only the row corresponding to the selected starting state. You can perform and inspect the full matrix calculation in the Matrix modal. Nonzero entries show which states can be reached in exactly n steps; as n changes, recurring gaps can reveal periodic behavior.

E_i[tau_i^+] = _

The expected return time to state i is the reciprocal of its stationary probability: E_i[tau_i^+] = 1 / pi_i. If you think about it, over the long run the chain spends a proportion pi_i of its time at state i, so successive visits are separated by 1 / pi_i steps on average.

Display:

E_i[tau_{S^c}] = _

The escape time is the number of steps needed to first leave S. Equivalently, it is the hitting time of S complement. A starting state already outside S has escape time 0.

Tolerance: epsilon =

t_mix(epsilon) = _

Mixing time is the first step n when the distribution from every starting state is within epsilon of the stationary distribution in total variation distance. In other words, the chain forgets where it started.

Phi is the minimum stationary boundary flow divided by stationary mass.

Display:

Phi = _

Conductance ranges from 0 to 1 and measures the smallest stationary flow leaving a set of states relative to that set’s stationary mass. A smaller value identifies a tighter bottleneck. When a representative minimizing cut is shown, the outlined region is S and everything outside it is S complement.

Graph Settings

Colors

Chain Generator

Kemeny's Constant

Kemeny's Constant

Entropy Rate

Splitting Probability

Splitting Probability

Hitting Time

Hitting Time

n-Step Distribution

n-Step Distribution

Return Time

Return Time

Escape Time

Escape Time

Mixing Time

Mixing Time

Cover Time

Cover Time

Spectral Gap

Conductance

Conductance