Opinion Dynamics

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# Opinion Dynamics
### David Garcia, Petar Jerčić, Jana Lasser <br><br> <em>TU Graz</em>
### Computational Modelling of Social Systems

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<div class="my-footer"><span>David Garcia - Computational Modelling of Social Systems</span></div>

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# 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**

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# Overview

## 1. The voter model

## 2. Bounded confidence

## 3. Modeling polarization in the digital society

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# The voter model

## *1. The voter model*

## 2. Bounded confidence

## 3. Modeling polarization in the digital society

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# 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)

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# 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

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# 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 `$\kappa(f)$`?

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# 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: `$t=10^{1}$`, `$10^{2}$`, `$10^{3}$`, `$10^{4}$`
<img src="Figures/voter.svg" width="900" style="display: block; margin: auto;" />

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# 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

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# Nonlinear voter model functions

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

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# 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: `$t=10^{1}$`, `$10^{2}$`, `$10^{3}$`, `$10^{4}$`
  
<img src="Figures/voternl.svg" width="900" style="display: block; margin: auto;" />
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# 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)?
  
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# Effect of memory in TTC

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

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# Bounded confidence

## 1. The voter model

## *2. Bounded confidence*

## 3. Modeling polarization in the digital society

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## Discrete versus continuous opinion models

.pull-left[
- **Voter models:**
  - agents are characterized by a discrete opinion (e.g. binary)
  - adopt other opinions according to their frequency in the agent's neighborhood

- **Bounded confidence models:**
  - *continuous opinions* `$x_i$` (e.g. real number between 0 and 1)
  - agent interactions are randomised and conditional on the difference of their opinions
]

.pull-right[

<img src="Figures/likert.png" width="525" style="display: block; margin: auto;" />
]

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## 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

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# Simulation examples

.pull-left[
<img src="Figures/basic1.png" width="400" style="display: block; margin: auto;" />
`$\epsilon=0.5, \zeta=0.5, N=2000$`
 ]

.pull-right[
<img src="Figures/basic2.png" width="400" style="display: block; margin: auto;" />
`$\epsilon=0.2, \zeta=0.5, N=1000$`
 ]

**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)](https://www.worldscientific.com/doi/abs/10.1142/S0219525900000078)

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# Final opinions vs initial opinions

.pull-left[
<img src="Figures/basic3.svg" width="500" style="display: block; margin: auto;" />
*(Deffuant et al. (2000))*
]

.pull-right[
- qualitative dynamics mostly depend on the threshold `$\epsilon$`:
  - controls the number of peaks of the final distribution of opinions
  - The final expected number of groups is `$\frac{1}{2\epsilon}$`

- `$\zeta$` and `$N$` only influence convergence time and width of the
  distribution of final opinions

- Sometimes you can get "wings" of agents at the edges or stuck between major groups
]

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# Asymmetric confidence model

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

<img src="Figures/Hegselmann-1.svg" width="700" style="display: block; margin: auto;" />
Confidence can extend further on one side than on the other
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# Opinion-independent asymmetry

**Collective opinion drifts in the direction favoured by the asymmetry**

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# One-sided splits

<img src="Figures/Hegselmann-13.png" width="700" style="display: block; margin: auto;" />
**Asymmetric bounded confidence generates temporary fragmentation!**

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# Modeling polarization in the digital society

## 1. The voter model

## 2. Bounded confidence

## *3. Modeling polarization in the digital society*

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## 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)](https://iopscience.iop.org/article/10.1088/1742-5468/2010/06/P06005)

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# 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 `$(-1,1)$`

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# The Information Accumulation Model

- 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 `$(-1,1)$`

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# Interaction in echo chambers

.pull-left[
<img src="Figures/IASCommunities.png" width="450" style="display: block; margin: auto;" />
]

.pull-right[

- Neighborhood `$\Gamma_i$` contains:
  - `$m_O$` connections to neighbors in same community
  - `$m_X$` connections to neighbors in the other community
- Weak inter-community interaction (filter bubble effects):
  `$$m_O>m_X$$`
- Intra-community diffusivity: `$\Omega_O = m_o \omega$`
- Inter-community diffusivity: `$\Omega_x = m_x \omega$`

]

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# 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*}`

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# 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

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# 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: 
`$$\Omega_X^T < \Omega_0+\frac{1}{2}\Delta-\frac{1}{2}\sqrt{\Delta^2+8\Omega_0\Delta}$$`

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# Opinion Attractors ( `$\Delta=0.2$` )

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

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# 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
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# 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
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# 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
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# Tipping diffusivity ratio

.pull-left[

**Tipping diffusivity ratio:**
  Fraction of inter- and intra-diffusivity above which two polarized
  communities would reach a consensus

`$$\Phi^T=\frac{\Omega_X^T}{\Omega_0}$$`
Green arrow: fostering consensus by increasing inter-community diffusivity (creating links between groups)

]

.pull-right[

]

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# Connectivity can increase polarization

.pull-left[

**More links, less consensus effect:**
For low `$\frac{\Omega_X}{\Omega_0}$` there is a level of total diffusivity that creates
polarization

Red arrow: echo chamber effect is constant (fixed `$\frac{\Omega_X}{\Omega_0}$` ), increase in total connectivity (thus increase in `$\Omega_0$` ):
**Polarization appears!**

Only for low `$\frac{\Omega_X}{\Omega_0}$`: weakening echo chamber effect fosters consensus above a threshold

]

.pull-right[

]

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# 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

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# 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?