Analyzing poll-delayed voter model as AR(2) process

In a recent post, we have discussed how to obtain the stationary variance of the AR(2) process using Yule-Walker equations. While intuitively, it is trivial to see that the poll-delayed voter model is an AR(2) process, showing this formally is a bit more involved. Let us delve into this question further.

Macroscopic simulation method

In [1], we have used multiple different numerical simulation methods to simulate the poll-delayed voter model. Of all the methods we have used, the macroscopic simulation method was instrumental due to its speed and simplicity. It has also inspired analytical derivations presented in the paper.

The core idea of the macroscopic simulation method is that the individual agent transition rates remain constant during a single polling period. Such an approach allows us to derive the probability for an agent starting in any particular state to end up in state \( 1 \) after a single polling period passes [1]:

\begin{equation} P_{1}\left(s|P_{1}\left(0\right)\right) = P_{1}\left(\infty\right) + \left[P_{1}\left(0\right)-P_{1}\left(\infty\right)\right] \exp\left[-\left(\varepsilon_{0}+\varepsilon_{1}+N\right)s\right]. \end{equation}

In the above, \( s \) represents an increment of time (assuming that it does not cross the polling boundary). While \( P_{1}\left(0\right) \) encodes the initial condition as a "probability" to initially observe the agent in state \( 1 \) (as we can assume to know the initial state of the agent, commonly it would be either \( 0 \) or \( 1 \)). Likewise, \( P_{1}\left(\infty\right) \) is the stationary probability of observing an agent in state \( 1 \):

\begin{equation} P_{1}\left(\infty\right) = \frac{\varepsilon_{1}^{\left(k\right)}}{\varepsilon_{0}^{\left(k\right)}+\varepsilon_{1}^{\left(k\right)}} = \frac{\varepsilon_{1}+A_{k-1}}{\varepsilon_{0}+\varepsilon_{1}+N}. \end{equation}

Armed with these expression we can determine the system state of the poll-delayed voter model by evaluating:

\begin{equation} X\left(t+s\right) = B_{1\rightarrow 1}\left[X\left(t\right),P_{1}\left(s|1\right)\right] + B_{0\rightarrow 1}\left[N-X\left(t\right),P_{1}\left(s|0\right)\right].\label{eq:main-macro} \end{equation}

In the above \( B_{i\rightarrow j} \) stands for a sample from a binomial distribution with respective values of parameters \( N \) (number of trials) and \( p \) (probability of "success"). The index here emphasizes that a particular sample encodes a specific transition.

So, this method allows us to simulate all relevant transitions by generating just two binomial random values. Generating just two binomial random values is often far more efficient than generating lots of exponential random values when using a variation of the Gillespie algorithm. However, if the polling period is short, and few transitions occur within a single polling period, then Gillespie's algorithm would be superior performance-wise.

Obtaining stationary mean

From \eqref{eq:main-macro} one can easily obtain an expression for a conditional expected outcome:

\begin{equation} \left\langle A_{k+1}|A_{k},A_{k-1}\right\rangle = \varphi_{1}A_{k} + \left(1-\varphi_{1}\right)\varphi_{2} \left(\varepsilon_{1}+A_{k-1}\right). \end{equation}

In the above, we have introduced two new symbols (to shorten the expressions obtained): \( \varphi_{1}=\exp\left[-\left(\varepsilon_{0}+\varepsilon_{1}+N\right)\tau\right] \) and \( \varphi_{2}=\frac{N}{\varepsilon_{0}+\varepsilon_{1}+N} \).

Averaging this expression over the stationary distribution, we obtain the stationary mean:

\begin{equation} \left\langle A_{\infty}\right\rangle = \frac{\varphi_{2}\varepsilon_{1}}{1-\varphi_{2}} = \frac{N\varepsilon_{1}}{\varepsilon_{0}+\varepsilon_{1}}. \end{equation}

Obtaining zero-centered AR(2) process

Currently, \( A_k \) takes values from \( 0 \) to \( N \). To make the process zero-centered (we need this condition to hold for us to be able to use the previously obtained results), let us introduce a new state variable by subtracting the stationary mean from the \( A_k \):

\begin{equation} \tilde{A}_{k}=A_{k}-\left\langle A_{\infty}\right\rangle . \end{equation}

Conditional expectation is then given by:

\begin{equation} \left\langle\tilde{A}_{k+1}|\tilde{A}_{k},\tilde{A}_{k-1}\right\rangle = \varphi_{1}\tilde{A}_{k} + \left(1-\varphi_{1}\right)\varphi_{2}\tilde{A}_{k-1}. \end{equation}

Based on the conditional expectation given above, let us approximate the process by:

\begin{equation} \tilde{A}_{k+1} = \varphi_{1}\tilde{A}_{k} + \left(1-\varphi_{1}\right)\varphi_{2}\tilde{A}_{k-1} + \xi_{k+1}.\label{eq:ar2-process} \end{equation}

In the above, \( \xi_{k+1} \) would encode random deviations from the expectation. It does not need to follow any particular distribution as long as the values are uncorrelated and have finite variance.

To completely determine the corresponding AR(2) process, we need to derive the variance of the random deviation term. Notably, random deviation comes from the binomial random sampling, thus:

\begin{equation} \mathrm{Var}\left[\xi_{k}|\tilde{A}_{k},\tilde{A}_{k-1}\right] = \mathrm{Var}\left[B_{1\rightarrow1}|\tilde{A}_{k},\tilde{A}_{k-1}\right] + \mathrm{Var}\left[B_{0\rightarrow1}|\tilde{A}_{k},\tilde{A}_{k-1}\right] \end{equation}

Averaging over the stationary poll outcome distribution yields:

\begin{equation} \mathrm{Var}\left[\xi_{\infty}\right] = \left\langle\mathrm{Var}\left[\xi_{k}|\tilde{A}_{k},\tilde{A}_{k-1}\right]\right\rangle = \psi_{0}+\left(\psi_{12}\rho_{1}+\psi_{22}\right)\mathrm{Var}\left[A_{\infty}\right]. \end{equation}

In the above, \( \psi_{ij} \) and \( \rho_1 \) symbols are introduced for notational convenience. \( \rho_1 \) symbol also has a meaning of the stationary lag-one auto-correlation. These symbols are defined as follows:

\begin{equation} \psi_{0} = \frac{N\varepsilon_{0}\varepsilon_{1}\left(1-\varphi_{1}^{2}\right)}{\left(\varepsilon_{0}+\varepsilon_{1}\right)^{2}}, \end{equation}

\begin{equation} \psi_{12} = -\frac{2\varphi_{1}\left(1-\varphi_{1}\right)}{\varepsilon_{0}+\varepsilon_{1}+N}, \end{equation}

\begin{equation} \rho_{1}=\frac{\varphi_{1}}{1-\left(1-\varphi_{1}\right)\varphi_{2}}. \end{equation}

\begin{equation} \psi_{22} = -\frac{N\left(1-\varphi_{1}\right)^{2}}{\left(\varepsilon_{0}+\varepsilon_{1}+N\right)^{2}}. \end{equation}

By using the result from the previous post, and properly rearranging it, we get the expression for the stationary poll outcome variance:

\begin{equation} \mathrm{Var}\left[A_{\infty}\right] = \frac{\psi_{0}}{1-\left(\varphi_{1}+\psi_{12}\right)\rho_{1}-\left(1-\varphi_{1}\right)\varphi_{2}\rho_{2}-\psi_{22}}. \end{equation}

In the above, \( \rho_2 \) is another convenience symbol (it also has a meaning of the stationary lag-two auto-correlation):

\begin{equation} \rho_{2} = \left(1-\varphi_{1}\right)\varphi_{2} + \frac{\varphi_{1}^{2}}{1-\left(1-\varphi_{1}\right)\varphi_{2}}. \end{equation}

Use the interactive figure below to explore how the relationship between stationary variance and polling period evolves with changes in model parameters.

Observe that altering the \( \varepsilon \) parameters results in relatively modest shifts - primarily raising or lowering the variance curve (though small changes in the vicinity of the minimum can be observed). In contrast, varying \( N \) has a bigger effect on the variance curve. Also note, that variance for the small \( \tau \) values is always roughly twice as large as variance for the large \( \tau \).

Uncovering a neat scaling law

The idea behind these derivations was that the stationary distribution of the poll-delayed voter model should be well approximated by Beta or Beta-binomial distributions. Given that these distributions have two free (shape) parameters, then knowing the expressions for the stationary mean and variance allows us to provide a reasonable approximation.

Detailed numerical exploration of the stationary distributions in [1] confirms this intuition. The dependence is quite neat, as apparently we can separate the base value and the scaling law for any set of the model parameters. I.e., the dependence between the shape parameters (estimated from multiple numerical simulations) and the polling period is neatly determined by a universal scaling law, \( L\left(\tau\right) \):

\begin{equation} \frac{\hat{\alpha}}{\varepsilon_1} = \frac{\hat{\beta}}{\varepsilon_0} = L\left(\tau\right) \end{equation}

The scaling is given by [1]:

\begin{equation} L\left(\tau\right)=\frac{\varepsilon_{0}\varepsilon_{1}N^{2}-\left(\varepsilon_{0}+\varepsilon_{1}\right)^{2}\mathrm{Var}\left(\tau\right)}{\left(\varepsilon_{0}+\varepsilon_{1}\right)^{3}\mathrm{Var}\left(\tau\right)-\varepsilon_{0}\varepsilon_{1}\left(\varepsilon_{0}+\varepsilon_{1}\right)N}. \end{equation}

In the above, \( \mathrm{Var}\left(\tau\right) \) stands for the dependence between the stationary variance and the polling period.

Use the interactive figure below to explore how the scaling law responds to the changes in the model parameters. Notice that modifying the \( \varepsilon \) parameters has a minimal impact on the overall shape of the scaling curve. In contrast, changes in \( N \) significantly influence the scaling behavior - the location and height of the maximum shift noticeably with varying \( N \). Additionally, observe that for short polling periods, or small \( \tau \), the scaling law is close to \( 1 \) (this means that for short polling periods the model is effectively equivalent to the standard noisy voter model), while for long polling periods, or large \( \tau \), it tends toward \( 2 \).

References

• A. Kononovicius, R. Astrauskas, M. Radavičius, F. Ivanauskas. Delayed interactions in the noisy voter model through the periodic polling mechanism. Physica A 652: 130062 (2024). doi: 10.1016/j.physa.2024.130062. arXiv:2403.10277 [physics.soc-ph].