Boundary conditions and zealotry in the noisy voter model

When writing our recent article [1], we (I and Rytis) had argued about the relationship between having stable voter base (zealots) and imposing boundary conditions on the noisy voter model. This post explores the subtle differences between these notions.

The noisy voter model

Let us assume that we have \( N \) agents. Each of the agents can be in state \( 0 \) or in state \( 1 \). In general, the labeling of the states doesn't matter, here I have chosen such labels to emphasize that we observe the number of agents in state \( 1 \), let us denote it by \( X \).

Whenever a change in \( X \) occurs, it will either increase or decrease by a single unit. I.e., one agent will either move from state \( 1 \) to state \( 0 \) (let us call this transition "death"), or the agent will move \( 0 \rightarrow 1 \) (let us call this transition "birth"). We have chosen these strange names for the rates to emphasize that we can use continuous time birth-death process formalism.

Let the agents change their state either independently (with rate \( \varepsilon_0 \) or \( \varepsilon_1 \)), or due to peer interaction (let the interaction rate be our unit of time). Then the birth rate of the model is given by,

\begin{equation} \lambda(X) = \left( N-X \right) \left(\varepsilon_1 + X \right) . \end{equation}

While the death rate of the model

\begin{equation} \mu(X) = X \left(\varepsilon_1 + \left[N-X\right] \right) . \end{equation}

The rates take this form in the "standard" case.

They will retain their form when the boundary conditions are applied, because the boundary conditions would be applied only after the transition would occur. If the transition would drive \( X \) outside the desired (valid) value range, the transition would be canceled.

Let us now introduce zealots into the model. Let \( Z_0 \) stand for the number of zealots in state \( 0 \). Likewise, let \( Z_1 \) stand for the number of zealots in state \( 1 \). Let \( X \) stand for the number of zealots and normal agents in state \( 1 \), then the rates would be modified as follows,

\begin{equation} \lambda(X) = \left( N-X-Z_0 \right) \left(\varepsilon_1 + X \right) , \end{equation}

\begin{equation} \mu(X) = \left(X-Z_1\right) \left(\varepsilon_1 + \left[N-X\right] \right) . \end{equation}

There are lots of terms, and while they are simple... It might be hard to see through them. To gain an intuition of what happens, look at the interactive figure below.

Note that to make the models comparable, I have introduced a restriction. I have assumed that the number of zealots in each state would imply the boundary conditions (i.e., \( X_{\text{min}} \) and \( X_{\text{max}} \)) as well. Namely, in both models \( X \) can't become smaller than \( Z_1 \) (i.e., than \( X_{\text{min}} \)), and it can't become larger than \( N - Z_0 \) (i.e., than \( X_{\text{max}} \)).

Interactive figure for the rates

Red curve in the interactive figure below represents the model with zealots, while the blue curve shows how the rates change in the model with boundary conditions.

Observe that the red curve (model with zealots) always goes to zero in one end. Birth rate declines to zero as \( X \) approaches \( N - Z_0 \), because in state \( 0 \) there would be no more agents who can change their state. Death rate declines to zero as \( X \) approaches \( Z_1 \), because at this point there are zero agents in state \( 1 \) who are not zealots. The blue curve (model with boundary conditions) does not do that - the curve remains the same, only some transition are effectively prevented.

So, how would individual agent logic look like in the model with boundary conditions? Each agent would have to monitor the global system state. If the global state drifted too far in one direction, the agent would then adjust their public opinion. Imagine you support party A, but you also value healthy competition in politics. If party A gained so much power that it could suppress that competition, you would feel compelled to publicly support party B.

Interactive exploration of time series and opinion distribution

App below simulates the model. The plot on the left shows fragment of the simulated time series, while the plot on the right shows the probability density function of opinions.

Observe that the opinion distribution in the two variants of the noisy voter model changes differently as we change the size of stable support. If the support is provided by the zealots (red curve), the opinion distribution narrows (grows close to the delta function). If the support is provided by the boundary conditions, the opinion distribution retains familiar form of \( \mathrm{BetaBinomial}(\varepsilon_1,\varepsilon_0,N) \) (stationary distribution of the unmodified noisy voter model), with a caveat that it is truncated to the range between \( X_{\text{min}} \) and \( X_{\text{max}} \).

References

• R. Kazakevicius, A. Kononovicius. Mean first passage time of the symmetric noisy voter model with arbitrary initial and boundary conditions. Chaos, Solitons and Fractals 203: 117649 (2026). doi: 10.1016/j.chaos.2025.117649. arXiv:2512.02519 [cond-mat.stat-mech].