# Analytical and Numerical Treatment of Continuous Ageing in the Voter Model

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Voter Model with Age-Dependent Switching Rates

#### 2.2. Modified Thinning Algorithm for Simulation

- Set $t=0$. Initialise all ages ${\tau}_{i}\left(0\right)=0$ and draw the states ${s}_{i}\left(0\right)$ from the desired initial distribution.
- Assume that the simulation has reached time t. Set ${R}_{\mathrm{max}}=aN+2{p}_{\mathrm{max}}{n}^{-}\left(t\right){n}^{+}\left(t\right)/N$.
- Draw a uniform random number $\mathit{u}$ from the interval $(0,1]$ and calculate the time interval to the next event $\Delta =-ln\phantom{\rule{0.166667em}{0ex}}\mathit{u}/{R}_{\mathrm{max}}$. Update time and the ages of all voters such that $t\to t+\Delta $, and ${\tau}_{i}\to {\tau}_{i}+\Delta $ for $i=1,\cdots ,N$.
- Choose the type of event to occur (all rates are evaluated at the updated time):
- (i)
- With probability ${R}_{0}\left(t\right)/{R}_{\mathrm{max}}$, nothing happens.
- (ii)
- With probability ${r}_{i}\left(t\right)/{R}_{\mathrm{max}}$, voter i switches opinion, ${s}_{i}\left(t\right)\to -{s}_{i}\left(t\right)$; set ${\tau}_{i}\left(t\right)=0$; update ${n}^{\pm}\left(t\right)$.

- Go to item 2.

## 3. Results

#### 3.1. Deterministic Approximation for the Model without Spontaneous Opinion Changes ($a=0$)

#### 3.2. Ordering Dynamics with Continuous Ageing

#### 3.2.1. Linearisation Close to the Absorbing State

#### 3.2.2. Power-Law Ageing and the Approach to Consensus

**-****Fractional differential equation**

**-****Asymptotic behaviour**

#### 3.2.3. Ageing Profile with Non-Zero Asymptotic Value

#### 3.2.4. Exponential Ageing and the Frozen State

#### 3.3. Model with Spontaneous Opinion Changes ($a>0$)

#### 3.3.1. Continuous Phase Transition

#### 3.3.2. Fluctuations about the Steady State

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Derivation of Equation (17)

## Appendix B. An Alternative to the Linearised Equation (17)

## References

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**Figure 1.**The ageing-induced power-law approach to consensus. Individual-based simulations of the ageing voter model were performed using the method given in Section 2.2 for the case where ${p}_{\tau}=\gamma /(t+{t}_{0})$. The solid red line is the mean $\langle {x}^{+}\left(t\right)\rangle $ averaged over 1000 trials and the dotted black line is ${x}^{+}\left(t\right)={t}^{-\gamma}$. The remaining system parameters were $\gamma =0.8$, $N=100$, ${t}_{0}=0.8$.

**Figure 2.**The location of the pole ${u}^{*}$ of the right-hand side of Equation (30), given by the solution of $\widehat{\Psi}\left({u}^{*}\right)=\frac{1}{{p}_{\infty}}$, as a function of $\gamma $ for ${t}_{0}=2.0$, ${p}_{\infty}=0.5$. Note that the sign of ${u}^{*}$ matches that of $\gamma $ (see the text for the discussion of this result).

**Figure 3.**We plot (solid line) the evolution of the average density $\langle {x}^{*}\left(t\right)\rangle $ of agents holding the $+1$ opinion as a function of time t using the functional form for the activation rate given by Equation (28) with ${p}_{\infty}=0.5,\phantom{\rule{0.166667em}{0ex}}{t}_{0}=0.8,\phantom{\rule{0.166667em}{0ex}}\gamma =0.1$. The results have been averaged over 100 realisations of the stochastic dynamics starting at ${x}^{*}\left(0\right)=0.2$. The dashed line is the theoretical prediction of an asymptotic exponential decay ${x}^{+}\left(t\right)\sim {e}^{{u}^{*}t}$ with a value of ${u}^{*}=-0.0517$ given by the solution of $\widehat{\Psi}\left({u}^{*}\right)=\frac{1}{{p}_{\infty}}$, the pole of the right-hand side of Equation (30).

**Figure 4.**The final frozen state in the case of exponentially decaying transition rate. Results of individual-based simulations of the ageing voter model were performed using the Lewis and Shedler thinning method (see Section 2.2) for the case where ${p}_{\tau}={p}_{0}{e}^{-\tau /{t}_{0}}$. The red points are the simulation results for the mean $\langle {x}^{+}\left(t\right)\rangle $ averaged over 100 trials and the dotted black line is the series in Equation (36) truncated after 100 terms. The remaining system parameters were ${t}_{0}=1$, $N=1000$.

**Figure 5.**Ageing-modified noise-induced phase transition. Here, ${t}_{0}=1$, $\gamma =2$, $N=100$. Coloured crosses represent the modal value of the magnetisation $m=|{x}^{+}-{x}^{-}|=|2{x}^{\pm}-1|$ obtained from simulations using the method detailed in Section 2.2. The dashed line is the solution of Equation (45).

**Figure 6.**The stationary distribution of the concentration ${x}^{+}$ for various values of the noise strength ((

**a**) $a=1$, (

**b**) $a=0.2$, (

**c**) $a=0.126$). The age-dependent rate in Equation (18) was used with fixed parameters ${t}_{0}=1$, $\gamma =2$ and $N=100$. Coloured crosses are the results of simulations using the Lewis and Shedler thinning method discussed in Section 2.2 and the dashed line is given by Equation (50).

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**MDPI and ACS Style**

Baron, J.W.; Peralta, A.F.; Galla, T.; Toral, R.
Analytical and Numerical Treatment of Continuous Ageing in the Voter Model. *Entropy* **2022**, *24*, 1331.
https://doi.org/10.3390/e24101331

**AMA Style**

Baron JW, Peralta AF, Galla T, Toral R.
Analytical and Numerical Treatment of Continuous Ageing in the Voter Model. *Entropy*. 2022; 24(10):1331.
https://doi.org/10.3390/e24101331

**Chicago/Turabian Style**

Baron, Joseph W., Antonio F. Peralta, Tobias Galla, and Raúl Toral.
2022. "Analytical and Numerical Treatment of Continuous Ageing in the Voter Model" *Entropy* 24, no. 10: 1331.
https://doi.org/10.3390/e24101331