So far

Block 1: Fundamentals of agent-based modelling

• Basics of agent-based modelling: the micro-macro gap
• Modelling segregation: Schelling's model
• Modelling cultures

Block 2: Opinion dynamics

• Basics of spreading: Granovetter's threshold model
  • Exercise 1: Schelling's model and Pandas (session 2)
• Today: Opinion dynamics
  • Exercise 2: Threshold models (session 1)
• Modelling hyperpolarization and cognitive balance: Guest lecture by Simon Schweighofer (Webex)
  • Exercise 2: Threshold models (session 2) Deadline: 20.04.2022

Overview

1. The voter model

2. Bounded confidence

3. Modeling polarization in the digital society

The voter model

1. The voter model

2. Bounded confidence

3. Modeling polarization in the digital society

Opinion dynamics outcomes

• Consensus:
  • Distribution of opinions in which the vast majority of agents agree on the same opinion

• Chaos:
  • Distribution of opinions in a population in which no apparent agreement group or structure can be identified (fully random)

• Polarization:
  • Bimodal distribution of opinions in a population towards the extremes of an opinion spectrum

• Fragmentation:
  • Multimodal distribution of opinions in a population into groups of high internal agreement and high external disagreement (multipolarization)

Opinion dynamics modelling principles

• Agent-based modelling approach:
  • internal state include (at least) the agent's opinion
  • interaction rules between agents are specified
  • these interactions cause emergent system properties

• No central authority:
  • collective opinions, consensus, polarisation, cultural traits emerge without central coordination results depend on control parameters

• Reactive rather than reflective agents:
  • no advanced cognitive process, only adaptation to environment (neighbourhood) by following simple rules

Voter models of opinion dynamics

• Rate to change opinion depends on other agents:
  • neighbors (networks, spatial models)
  • randomly chosen agents (also called mean-field interaction)

• Principle: frequency dependent dynamics. Opinion change based on:

$$P(1-\theta_{i}|\theta_{i})=\kappa(f)\; f_{i}^{1-\theta_{i}}$$

• $\theta_{i}$: opinion of agent $i$
• $0\leq f_{i}^{1-\theta_{i}} \leq 1$: frequency of agents with opposite opinions in "neighborhood" of agent $i$
• $\kappa(f)$: response function to frequency of other opinions
• Analysis question: How does the outcome depend on $\kappa(f)$?

Linear voter model

• Dynamics: $P(1-\theta|\theta)=f^{1-\theta}$
• Stochastic simulation agents update at random on a grid
• Initially $f=0.5$, random distribution
• Results:
  • coordination of decisions on medium time scales
  • outcome: consensus as an equilibrium
  • How long does it take to reach consensus?
  • Simulation at: $t=10^{1}$, $10^{2}$, $10^{3}$, $10^{4}$

Observations on linear voter model

• In the time limit, always only one opinion exists
  • Consensus always appears
  • There are two "absorbing" states: all agents are either 0 or 1
  • The probability to reach an all 0 or all 1 consensus equals initial frequency $f(0)$

• Model limitations/drawbacks
  • very limited social/biological interpretation (remember Social Impact Theory)
  • what about coexistence of opinions? The reality is not always consensus

• Some interesting features for analysis:
  • Time to reach consensus (TTC)
  • Intermediate dynamics or dependence on grid topology/network

Nonlinear voter model functions

• Nonlinear response examples for $\kappa(f) f_{i}^{1-\theta_{i}}$

Coexistence of opinions?

• Coexistence can happen for some nonlinear $\kappa(f)$
• Absorbing states can be destabilized with small random component in the linear case
  • small pertubation for $f^{1-\theta}=1$ ( for example $\epsilon=10^{-4}$)
  • coordination of decisions on long time scales
  • asymptotically: coexistence but non-equilibrium
  • Simulation at: $t=10^{1}$, $10^{2}$, $10^{3}$, $10^{4}$

Including memory effects

• $\nu_{i}(\tau_{i})$: reluctance of agent $i$ to change opinion $\theta_{i}$
  • persistence time $\tau_{i}$ (opinion was not changed) $\to$ "history"
  • reflects local experience with agents in neighborhood

$$\frac{d \nu}{d \tau}=\mu\, \nu(1-\nu) \quad \to \quad \nu_{i}= \frac{1}{1+ e^{-\mu \tau_{i}}}$$

• Decision dynamics:

$$w(\theta_{i}^{\prime}|\theta_{i})=\left[1- \nu_{i}(\tau_{i})\right] f_{i}^{\theta_{i}^{\prime}}$$

• $\mu>0$: slowing down of opinion dynamics
• What is the role of $\mu$ in time to consensus (TTC)?

Effect of memory in TTC

Heterogeneity of agents important when memory is present:
local groups of "confident" agents convince an indifferent neighborhood

Bounded confidence

1. The voter model

2. Bounded confidence

3. Modeling polarization in the digital society

Discrete versus continuous opinion models

The baseline model: pairwise bounded confidence

• Consider a population of $N$ agents $i$ with continuous opinions $x_i$
• At each time step any two randomly chosen agents meet
• Re-adjust opinion if absolute opinion difference is smaller than a threshold $\epsilon$
  • In other words: agents $i$ and $j$ with opinions $x_i$ and $x_j$ interact if: $$|x_i-x_j|<\epsilon$$

• New opinions are adjusted according to $$x_i(t+1)=x_i(t)+ \zeta \cdot (x_j(t)-x_{i}(t))$$ $$x_j(t+1)=x_j(t)+ \zeta \cdot (x_i(t)-x_j(t))$$

• $\zeta$ is the convergence parameter: measures speed of opinions approaching

Simulation examples

Consensus, polarization, and fragmentation are possible outcomes
Mixing beliefs among interacting agents. Guillaume Deffuant, David Neau, Frederic Amblard and Gerard Weisbuch. Advances in Complex Systems (2000)

Final opinions vs initial opinions

Asymmetric confidence model

• symmetric: $|x_i - x_j| < \epsilon$
• asymmetric: $-\epsilon_l < x_i - x_j < \epsilon_r$

Confidence can extend further on one side than on the other

Opinion-independent asymmetry

Collective opinion drifts in the direction favoured by the asymmetry

One-sided splits

Asymmetric bounded confidence generates temporary fragmentation!

Modeling polarization in the digital society

1. The voter model

2. Bounded confidence

3. Modeling polarization in the digital society

The question of connectivity in the digital society

Could more links create polarization? (see Axelrod's conjecture)

Tipping diffusivity in information accumulation systems: more links, less consensus. Jae K Shin and Jan Lorenz Journal of Statistical Mechanics: Theory and Experiment (2010)

The Information Accumulation Model

• n agents with continuous opinions $y_i$ and the following opinion dynamics: $$y_i^{t+1} = (1-\Delta) y_i^ t + \sum_{j \in \Gamma_i} \omega y_j^t (1- |y_i^t|)$$
• $\Delta$: measures how much agents opinions relax over time
  • In absence of interaction $y \to 0$

• Diffusivity $\omega$: coupling of opinions between agents
  • Agents approach the opinions of others in their neighborhood
• Saturation $1-|y_i^t|$: limits opinions to interval $(-1,1)$

The Information Accumulation Model

• n agents with continuous opinions $y_i$ and the following opinion dynamics: $$y_i^{t+1} = (1-\Delta) y_i^ t + \sum_{j \in \Gamma_i} \omega y_j^t (1- |y_i^t|)$$
• $\Delta$: measures how much agents opinions relax over time
  • In absence of interaction $y \to 0$

• Diffusivity $\omega$: coupling of opinions between agents
  • Agents approach the opinions of others in their neighborhood

• Saturation $1-|y_i^t|$: limits opinions to interval $(-1,1)$

Interaction in echo chambers

Community dynamics

Agent dynamics simplifies to community-level variables: $$\begin{eqnarray*} y_{1}^{t+1}&=(1-\Delta) y_{1}^{t}+(\Omega_0 y_{1}^{t}+\Omega_X y_{2}^{t})(1-|y_{1}^{t}|) \nonumber \\ y_{2}^{t+1}&=(1-\Delta) y_{2}^{t}+(\Omega_X y_{1}^{t}+\Omega_0 y_{2}^{t})(1-|y_{2}^{t}|). \end{eqnarray*}$$

• 00: no opinion in both communities
• PP, NN: consensus (both same sign)
• PN, NP: polarization (differn signs)

When $t \to \infty$ in PN mode, opinions follow this system of equations: $$\begin{eqnarray*} Y_1&=(1-\Delta)Y_1+(\Omega_0Y_1+\Omega_XY_2)(1-Y_1) \nonumber \\ Y_2&=(1-\Delta)Y_2+(\Omega_XY_1+\Omega_0Y_2)(1+Y_2). \end{eqnarray*}$$

Solutions of model dynamics (I)

1. Trivial: 00 mode:

   • Exists for any combination of parameter values
   • Only stable if $\Omega_O + \Omega_x < \Delta$ (relaxation $>$ total diffusivity)

2. Consensus: PP and NN modes:

   • $Y_1 = Y_2 = 1-\frac{\Delta}{\Omega_0+\Omega_X}$
   • Exists when $\Omega_0+\Omega_X\geq\Delta$ (total diffusivity $>$ relaxation)
   • Always stable

3. Polarization: PN and NP modes

Solutions of model dynamics (II)

1. Trivial: 00 mode

2. Consensus: PP and NN modes

3. Polarization: PN and NP modes:

   • $-Y_2 = Y_1 = 1-\frac{\Delta}{\Omega_0-\Omega_X}$

   • Exists when $\Omega_0-\Omega_X \geq \Delta$ (difference in diffusivities $>$ relaxation)

   • Polarization can only be stable if: $$\Omega_X^T < \Omega_0+\frac{1}{2}\Delta-\frac{1}{2}\sqrt{\Delta^2+8\Omega_0\Delta}$$

Opinion Attractors ( $\Delta=0.2$ )

• Total diffusivity below $\Delta$ ( $\Omega_O=0.14$, $\Omega_X=0.04$ )
• Trivial 00 solution

Consensus Attractors ( $\Delta=0.2$ )

• $\Omega_0-\Omega_X < \Delta$ ( $\Omega_O=0.24$, $\Omega_X=0.08$ )
• 00 marginally stable, PP and NN stable

Existence of Polarization ( $\Delta=0.2$ )

• $\Omega_0-\Omega_X > \Delta$ ( $\Omega_O=0.34$, $\Omega_X=0.08$ )
• 00 unstable, PN and NP marginally stable, PP and NN stable

Stable Polarization ( $\Delta=0.2$ )

• $\Omega_0-\Omega_X > \Delta$ ( $\Omega_O=0.44$, $\Omega_X=0.06$ )
• 00 unstable, PN , NP, PP, and NN stable

Tipping diffusivity ratio

Connectivity can increase polarization

Summary

• The voter model: binary opinions model
  • Probability to change opinion based on opinions of neighbors
  • Linear voter model: consensus. Nonlinear model can have coexistence
  • Adding reluctance to change can speed up consensus

• Bounded confidence: continuous opinions
  • Interaction only when opinions are close enough
  • Generates consensus, polarization, and fragmentation
  • Asymmetry of thresholds creates one-sided splits

• Information accumulation systems
  • Interaction in echo chambers with relaxation towards zero
  • Polarization depends on intra- and inter-community diffusivity
  • Tipping ratio shows that more links can generate polarization

Quiz

• The memory effect in the voter model makes you more or less susceptible to change your opinion?

• Is the Information Accumulation System a model of binary or continuous opinions?

• If $\epsilon=0.2$, how many opinion groups can you expect in the bounded confidence model?

• In the base version of the bounded confidence model, can the number of groups increase during the simulation?

• In our online society, do you expect $\Omega_X/\Omega_O$ to be above or below 1?