# Anomalous diffusion in non-linear transformation of voter model

Lets continue the topic of anomalous diffusion by considering my recent manuscript together with R. Kazakevicius [1]. In the said manuscript we have considered two different non-linear transformations of the noisy voter model. Here we present you with non-linear transformation of the observable.

## Model

Here we consider the standard noisy voter model (which is equivalent to Kirman's ant colony model we have considered previously on Physics of Risk) and generalize observable transformation we have used earlier in the financial market scenario (also see [2]):

\begin{equation} y = \left( \frac{x}{1-x} \right)^{\frac{1}{\alpha}} . \end{equation}

By relying on similarity with the Bessel process we were able to derive expressions for the temporal evolution of the moments of \( y \). The most important thing these expressions contain is the power-law dependence of the moments on time:

\begin{equation} \langle y^k \rangle \sim t^{\frac{k}{\alpha}} . \end{equation}

Thus the mean will scale as power-law with exponent \( \frac{1}{\alpha} \), while the variance will scale with exponent \( \frac{2}{\alpha} \). Note that these theoretical results work well only for very small and very large \( y \) (if \( x \) is either close to \( 0 \) or to \( 1 \)).

## Downscaling the model

For some of the parameter sets we need to use extremely many agents. Otherwise we do not observe anomalous diffusion (due to discretization effects). Unfortunately, the time complexity of the model is a square of number of agents, so increasing number of agents quickly becomes unfeasible. Therefore we have implemented automatic downscaling of the model.

Namely, as soon as number of agents in the counted state, \( X \), reaches critical amount, both the total number of agents, \( N \), and \( X \) are divided by 10. This procedure is performed as long as \( N > 100 \). This number is not special, we just find it to be sufficient for the simulation to be presented on the web (our simulations in [1] stop as soon as \( N \leq 10^3 \)).

We can use such downscaling, because our \( \varepsilon \) parameters will always be larger than \( 2 \) (otherwise variance is not defined) and the model in the steady state will spend most of the time far from the edges. Also we care more about small initial times and not the times when the model has switched to steady state behavior (this happens close to \( t = 1 \)).

## Interactive app

In the interactive app below you can see two plots: left one is for the temporal evolution of mean, right one is for the temporal evolution of variance (not standard deviation we have used in the previous posts on anomalous diffusion). In those figures you'll see two main curves - red dots show the simulation results (calculated over \( 100 \) sample series), while dark gray lines show the theoretical power-law dependence (these are meant just to guide your eye). If \( \alpha \neq 2 \), then variance plot also shows a third curve, light gray line, which compares the simulation results against what is expect from the normal diffusion.

Observe that for \( \alpha < 2 \), super-diffusion is observed, while for \( \alpha > 2 \) sub-diffusion is present. Observe that for \( x(0) \) close to \( 1 \), the localization phenomenon might be observed (initially variance growing, but then becoming smaller until reaching the steady state value). It might not be always observable due to simulation constraints we have imposed for web presentation.

## References

- R. Kazakevicius, A. Kononovicius. Anomalous diffusion in nonlinear transformations of the noisy voter model. Physical Review E 103: 032154 (2021). doi: 10.1103/PhysRevE.103.032154. arXiv:2011.02927 [cond-mat.stat-mech].
- A. Kononovicius, V. Gontis. Agent based reasoning for the non-linear stochastic models of long-range memory. Physica A 391: 1309-1314 (2012). doi: 10.1016/j.physa.2011.08.061. arXiv: 1106.2685 [q-fin.ST].