Detailed balance

One of the things I wondered about the previous summer was the difference between stationarity (balance of all inflows and outflows from the state) and detailed balance (balance of flows between two particular states). For example, when looking for the stationary distribution of the noisy voter model it is sufficient to solve equations for the detailed balance condition. But why? Do all stationary stochastic models satisfy detailed balance condition?

Is detailed balance ubiquitous and universal?

The answer to the question is no. Not all stochastic models (e.g., Markov chains) that converge to a stationary distribution satisfy detailed balance condition. Simplest example of this is a Markov chain with the following right stochastic transition matrix:

\begin{equation} \mathbf{T} = \begin{pmatrix} 0.8 & 0.2 & 0 \\ 0 & 0.8 & 0.2 \\ 0.2 & 0 & 0.8 \end{pmatrix} . \end{equation}

This matrix is actually doubly stochastic, but our intended meaning is that \( T_{i,j} \) encode probability of \( i \rightarrow j \) transitions. So we refer to this matrix as right stochastic matrix.

Stationary distribution of this Markov chain can be found by solving the eigenproblem, but the solution should be evident from the fact that all our states are identical. So the stationary distribution is \( \vec{p} = \left( 1/3, 1/3, 1/3 \right) \), but the detailed balance does not hold. For example, transition \( 0 \rightarrow 2 \) has probability of zero, while the opposite transition, \( 2 \rightarrow 0 \), has a non-zero probability. So,

\begin{equation} p_0 T_{0,2} \neq p_2 T_{2,0} . \end{equation}

Why does detailed balance hold for the noisy voter model?

For the two state noisy voter model all possible transitions between the observed system states (corresponding to the number of agents in the "active" state \( X \)) lie on a single line. Unlike in the example considered in this post, there are no loops, which would allow to have persisting directed flows.

Namely, in the noisy voter model it is only possible to increment or decrement the number of "active" agents by one at a time, i.e., only \( X \rightarrow X \pm 1 \) transitions are allowed. So, if there would be a directed flow, then the system would eventually get stuck in the state \( X=0 \) (no "active" agents) or state \( X=N \) (all "active" agents). After reaching those states there would be no flows between any of the states, the system would freeze and then it would make little sense to talk about the detailed balance condition.

In other words, in any one-step process, such as the noisy voter model, it is impossible to have a directed flow, breaking of the detailed balance condition, and a stationary distribution at the same time. Hence, for these processes stationary distribution can be found by solving for the detailed balance condition.

Interactive app

This interactive app allows you to visually see the state occupancy distribution (the darker the color the more particles are in that state), and how many transitions of each type occur in the model (once again the darker the arrow the more transitions in that direction occur). The sole parameter of the app can be used to change the transition matrix. It is given by:

\begin{equation} \mathbf{T} = \begin{pmatrix} 0.8 & 0.2\cdot\frac{1-\varepsilon}{2} & 0.2\cdot\frac{1+\varepsilon}{2} \\ 0.2\cdot\frac{1+\varepsilon}{2} & 0.8 & 0.2\cdot\frac{1-\varepsilon}{2} \\ 0.2\cdot\frac{1-\varepsilon}{2} & 0.2\cdot\frac{1+\varepsilon}{2} & 0.8 \end{pmatrix} . \end{equation}

So \( \varepsilon \), in certain sense, controls the degree to which the detailed balance condition is violated. When \( \varepsilon = 1 \) the difference between the respective transition probabilities is largest (probability of one of the transitions is zero). With \( \varepsilon = 0 \) there is no difference between the transition probabilities, and therefore the detailed balance condition is satisfied.

So, experiment by changing \( \varepsilon \) value and observe which transitions occur most often. The transition matrix implies that for \( \varepsilon > 0 \) the transition shown on the inner ring (those going anti-clockwise) should be more frequent. Also observe that no matter the value of \( \varepsilon \) the stationary distribution remains the same.