Noisy voter model for the parliamentary presence

In a previous post we have defined the parliamentary presence and shown that the data from Brazilian and Lithuanian parliaments exhibits anomalous diffusion. Actually we have analyzed Lithuanian data [1], while Brazilian data was considered by Vieira and others [2]. This time let me constructed a voter model to replicate the observations.

The model

Let us assume that binary states in the voter model represent willingness to attend. Let state \( 1 \) indicate willingness to attend, while state \( 0 \) indicate willingness to skip. Let the agents rethink their willingness after each daily session. Let the transitions between the states be described by a typical voter model transition probabilities:

\begin{equation} p^{(i)}_{1 \rightarrow 0} = h \left[ \varepsilon_0 + \frac{X_0}{N} \right] , \end{equation}

\begin{equation} p^{(i)}_{0 \rightarrow 1} = h \left[ \varepsilon_1 + \frac{X_1}{N} \right] = h \left[ \varepsilon_1 + \left( 1 - \frac{X_0}{N} \right) \right] . \end{equation}

In the above \( h \) sets the timescale on which the changes of states occur, while \( \varepsilon_i \) control independence of changes.

Via numerical analysis of the model with \( \varepsilon_0 = \varepsilon_1 = \varepsilon \), we have found that it exhibits scaling behavior in the standard deviation series, \( \sigma_t \). The scaling law being given by:

\begin{equation} \sigma_t = \frac{\theta_0 t}{\sqrt{\theta_1 + S t}} . \end{equation}

In the above \( \theta_i \) are independent of the model parameters, while \( S = h ( 1 + 2 \varepsilon ) \). As this law was discovered numerically we can provide only estimates of \( \theta_i \): \( \theta_0 \approx 0.66 \) and \( \theta_1 \approx 1.4 \).

Scaling law implies that there should be just two regimes - ballistic regime for small \( S \) and normal diffusion for larger \( S \). Larger \( S \) may imply one of two things: changes are happening faster and changes are more independent.

Scaling law is consistent with the empirical observations made in [2], but is not fully consistent with what we have observed for Lithuanian parliament. We could claim that Lithuanian parliament is in the cross-over region and due to lack of the data our observation of the cross-over is imperfect, but there is another possible explanation, which we will consider in the next post.

The interactive app

Feel free to explore the model in the interactive app below. The upper plot shows how the attendance of the individual agents, \( N = 100 \), evolves within the window of 250 days, \( N_s = 250 \). Agent index varies on the y-axis, while time is on the x-axis. The lower left plot shows the cumulative presence series of three agents. The lower right plot shows the scaling law and standard deviation series for the data within the window of 250 days. Try varying model parameters to see how the law and the dynamics of the model change.

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Acknowledgment. This post was written while reviewing literature relevant to the planned activities in postdoctoral fellowship ''Physical modeling of order-book and opinion dynamics'' (09.3.3-LMT-K-712-02-0026) project. The fellowship is funded by the European Social Fund under the No 09.3.3-LMT-K-712 ''Development of Competences of Scientists, other Researchers and Students through Practical Research Activities'' measure.