# Contrarian Voter Model under the Influence of an Oscillating Propaganda: Consensus, Bimodal Behavior and Stochastic Resonance

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. The Model

#### 2.2. Topology of Interactions

**Complete graph (CG):**This is the simplest topology in which every agent is connected to every other agent in the population (all-to-all interactions).**Lattices:**Agents are located at the sites of a lattice with periodic boundary conditions, and interact with their first nearest-neighbors (NNs). The number of neighbors depends on the lattice: one-dimensional ($1D$) lattice or ring (two NNs), two-dimensional ($2D$) square lattice (four NNs), $2D$ triangular lattice (six NNs) and $2D$ hexagonal lattice (three NNs).

#### 2.3. Magnetization and Signal-to-Noise Ratio

## 3. Results

#### 3.1. Evolution of the Magnetization

#### 3.2. Transition Temperature

#### 3.3. Signal-to-Noise Ratio

#### 3.4. Amplitude and Lag of Mean Opinion Oscillations

#### 3.5. Mean-Field Approach

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Solution of the Rate Equation for m

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**Figure 1.**(

**a**) Time evolution of the magnetization m in a single realization of the dynamics for a system of $N=1000$ agents on a complete graph, subject to a periodic field $H\left(t\right)={H}_{0}sin\left(\omega t\right)$ of amplitude ${H}_{0}=0.1$ and period $\tau =512$ ($\omega =2\pi /\tau $), and an external noise associated to a temperature T. Each curve corresponds to a different temperature, as indicated in the legend. Four behaviors are observed: $-1$ consensus for $T=0$, bimodal behavior for $T=0.25<{T}_{c}$, oscillations for $T=1.0>{T}_{c}$, and full disorder for $T=100\gg 1$. Here, ${T}_{c}\simeq 0.2895$ is the transition temperature. The field $H\left(t\right)$ is also plotted for reference. (

**b**) Evolution of m in a single realization for the same parameters as in panel (

**a**), on a $1D$-lattice of $N=8192$ sites (top-left), a $2D$ square lattice of size $N=64\times 64$ (top-right), a $2D$ triangular lattice of size $N=64\times 64$ (bottom-left), and a $2D$ hexagonal lattice of size $N=64\times 64$ (bottom-right).

**Figure 2.**(

**a**) Histogram of m for the same system and parameters as those in Figure 1, and for the temperatures indicated in the legends. At the transition point ${T}_{c}\simeq 0.2895$, the distribution $P\left(m\right)$ is uniform (top-right). (

**b**) Normalized histograms of the residence time ${t}_{r}$ for the same system and parameters of panel. (

**a**) Each plot corresponds to a different temperature T, as indicated in the legends. The bottom-right plot is in linear-log scale, while the other plots are in double logarithmic scale. The dashed line in the top-left plot has slope $-3/2$. Each histogram was obtained by running a single realization up to a time ${10}^{8}$.

**Figure 3.**Signal-to-noise ratio $SNR$ vs. temperature T on the five interaction topologies indicated in the legend. The system sizes are $N=1024$ for the $1D$ lattice and $CG$, and $N=32\times 32$ for the $2D$ lattices (hexagonal, square and triangular). The amplitude of the external field is ${H}_{0}=0.1$. Each panel corresponds to a different period $\tau =128$, 256, 512 and 1024.

**Figure 4.**Top panels: signal-to-noise ratio $SNR$ as a function of the amplitude of the external field ${H}_{0}$, for temperatures $T=0.2,0.3,0.5,1.3$ and $2.3$. Bottom panels: $SNR$ vs. T for ${H}_{0}=0.6,0.7,0.8,0.9$ and $1.0$. The period of the field is $\tau =128$. Left, central and right panels correspond to simulations on a complete graph (CG) of $N=1024$ nodes, a $2D$ square lattice of $N=32\times 32$ sites and a $1D$ lattice of $N=1024$ sites, respectively.

**Figure 5.**Total response $\mathcal{R}$ vs. field amplitude ${H}_{0}$, for the topologies indicated in the legend. The period of the field is $\tau =128$, and the system sizes are $N=1024$ for $CG$ and $1D$ lattice, and $N=32\times 32$ for $2D$ lattices. Inset: $\mathcal{R}$ vs. ${H}_{0}$ on a log-log scale. The dashed line has slope 2.

**Figure 6.**(

**a**) Time evolution of the average magnetization $\langle m\rangle $ for temperatures $T=0.2$ (squares), $T=0.5$ (circles), $T=1.0$ (diamonds), $T=2.0$ (up triangles) and $T=5.0$ (down triangles), on a CG of size $N=100$, with a field of amplitude ${H}_{0}=0.1$ and period $\tau =512$. The average was done over ${10}^{5}$ independent realizations of the dynamics. Solid lines are the analytical approximation from Equation (8). (

**b**) Lag $\mathcal{L}$ respect to the period $\tau $ vs. temperature T for field periods $\tau =256$ (squares), $\tau =512$ (circles) and $\tau =1024$ (diamonds), and amplitude ${H}_{0}=0.1$, for the same topology of panel (

**a**). Solid lines are the approximation from Equation (15). Inset: Amplitude A vs. T for the same parameter values as in the main panel. Solid lines are the analytical approximation from Equation (13).

**Figure 7.**(

**a**) Time evolution of the average magnetization $\langle m\rangle $ over ${10}^{5}$ realizations of the dynamics on a CG of size $N=100$, with temperature $T=0.5$, under a field of period $\tau =512$ and amplitudes ${H}_{0}=0.02$ (squares), ${H}_{0}=0.1$ (circles) and ${H}_{0}=0.5$ (diamonds). The solid pink line corresponds to the numerical integration of Equations (A1) and (A2), while the other solid lines are the analytical approximation from Equation (8). (

**b**) Normalized lag $\mathcal{L}/\tau $ vs. temperature T for a field of period $\tau =512$ and amplitudes ${H}_{0}$ indicated in the legend. The solid line is the approximation from Equation (15). Inset: A vs. T for the same parameter values as in the main panel. Solid lines are the approximation from Equation (13).

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**MDPI and ACS Style**

Gimenez, M.C.; Reinaudi, L.; Vazquez, F.
Contrarian Voter Model under the Influence of an Oscillating Propaganda: Consensus, Bimodal Behavior and Stochastic Resonance. *Entropy* **2022**, *24*, 1140.
https://doi.org/10.3390/e24081140

**AMA Style**

Gimenez MC, Reinaudi L, Vazquez F.
Contrarian Voter Model under the Influence of an Oscillating Propaganda: Consensus, Bimodal Behavior and Stochastic Resonance. *Entropy*. 2022; 24(8):1140.
https://doi.org/10.3390/e24081140

**Chicago/Turabian Style**

Gimenez, Maria Cecilia, Luis Reinaudi, and Federico Vazquez.
2022. "Contrarian Voter Model under the Influence of an Oscillating Propaganda: Consensus, Bimodal Behavior and Stochastic Resonance" *Entropy* 24, no. 8: 1140.
https://doi.org/10.3390/e24081140