Noise and Opinion Dynamics

This repository contains model code for our manuscript:

Steiglechner, P., Keijzer, M.A., Smaldino, P.e., Moser, D., Merico, A. (in press) ‘Noise and opinion dynamics: How ambiguity promotes pro-majority consensus in the presence of confirmation bias’. Journal of the Royal Society Open Science. Preprint available at: https://doi.org/10.31235/osf.io/u2t7r.

The model

We investigate here the impact of four different types of noise on the bounded confidence (BC) model of opinion formation. Without noise, the BC model for an interaction of a receiving agent $i$ and a sending agent $j$ with opinions $x_i$ and $x_j$ reads:

$$ x_{i} \mapsto \begin{cases} x_i + \mu \cdot (m_j - x_{i}) & \text{if } \left| m_j - x_{i} \right| < \epsilon \\ x_i & \text{else} \end{cases} \ \ \text{with the message } m_j = x_{j} $$

where $\epsilon$ is the bounded confidence radius.

The four types of noise include:

Selection/Selectivity noise $\xi_{\rm se}$

$$ x_{i} \mapsto \begin{cases} x_i + \mu \cdot (m_j - x_{i}) & \text{if } \left| m_j - x_{i} \right| < \epsilon + \xi_{\rm se} \\ x_i & \text{else} \end{cases} \ \ \text{with the message } m_j = x_{j} $$

Adaptation noise $\xi_{\rm ad}$

$$ x_{i} \mapsto \begin{cases} x_i + \mu \cdot (m_j - x_{i}) + \xi_{\rm ad} & \text{if } \left| m_j - x_{i} \right| < \epsilon \\ x_i & \text{else} \end{cases} \ \ \text{with the message } m_j = x_{j} $$

Exogenous noise $\xi_{\rm ex}$

$$ x_{i} \mapsto {\rm BC \ model} + \xi_{\rm ex} \ {\rm with \ some \ probability }\omega $$

Ambiguity noise $\xi_{\rm am}$

$$ x_{i} \mapsto \begin{cases} x_i + \mu \cdot (m_j - x_{i}) & \text{if } \left| m_j - x_{i} \right| < \epsilon \\ x_i & \text{else} \end{cases} \ \ \text{with the message } m_j = x_{j} + \xi_{\rm am} $$

Noise is drawn from a zero-mean Gaussian distribution with Gaussian width $\nu$. Noise is truncated such that the opinion $x_i$ or the message $m_j$ is within the bounds of the opinion space $[0, 1]$. Note, for exogenous noise, we choose $\nu=\omega$.

Running simulations

The simulation can simply be run by executing the python model.py. Parameters that can be changed to reproduce the results of the manuscript should be changed in the lines below # CHOOSE PARAMETERS (l. 325)

Test

To analyse the model, the following code should produce a very simplified version of figure 2a in the manuscript.

1. Run the model for "ambiguity noise", with "uniform" and several seeds.
2. Run the analysis (e.g. in a Jupyter notebook)

import xarray as xr
import matplotlib.pyplot as plt
import numpy as np

eps_vals = [0.05, 0.175, 0.3]  # TODO which BC radius values simulated (here for locations in figure 3a) 
seedmax=9  # TODO how many seed values were simulated
initial_condition = "2G-6AM"  # or "uniform"

data = xr.merge([ xr.open_dataset(f"data/model-ambiguityNoise_lowRes_{initial_condition}Initial_eps{eps:.3f}_seeds0-{seedmax}.ncdf", engine="netcdf4") for eps in eps_vals])
   
data.std(dim="id").sel({"t":1e4, "mu":0.5}).mean(dim="seed").x.plot(x="nu",cmap="Reds")
plt.ylim(0.4,0)

Parameter	Description	Example Value
track_times	List of times at which a snapshot of the simulation should be saved; must include 0 (initial time) and the final time	[0,10e5]
mu_arr	List of the $\mu$ to be run	[0.5]
n	Number of agents	100
seeds	List of the seeds to be run	list(range(1000))
resolution	The resolution of the phase space of bias and noise levels (eps, nu)	"low" or "high"
ic	the description of the initial conditions to be used. We tested more options, but these are commented out to increase clarity	"uniform" or "2G-6AM" (which means a superposition of two Gaussian functions to represent climate change opinions in the society according to the six America data, see [Maibach et al. (2011)])
noise_type	the type of noise in the opinion formation process (see above)	"ambiguityNoise"

Relevant Python Libraries and Dependencies

conda env create -f environment.yml

Package	Version
python	3.9.5
scipy	1.6.2
numpy	1.20.2
optional: networkx	2.5.1
matplotlib	3.3.4
xarray	0.18.0
netcdf4	1.5.7