Course Summary and Q&A Session

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.title[
# Course Summary and Q&A Session
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.author[
### David Garcia, Petar Jerčić, Jana Lasser <br><br> <em>TU Graz</em>
]
.date[
### 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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# Course overview

- **Block 1: Fundamentals of Agent-Based Modelling**
  - Micro-macro gap
  - Segregation and culture
  
  
- **Block 2: Opinion dynamics**
  - Granovetter's spreading model
  - Voter models and bounded confidence

- **Block 3: Network formation**
  - Random graphs and phase transitions
  - Small Worlds and Scale-free networks

- **Block 4: Processes on networks**
  - Epidemic models: SIR
  - Epidemics in practice: SEIRX

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# The macro-micro gap

*Causal Mechanisms in the Social Sciences. Peter Hedström and Petri Ylikoski. Annual Review of Sociology, 2010.*
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## Interdisciplinarity in complex social behavior
<img src="Figures/Interdisciplinarity.svg" width="1200" style="display: block; margin: auto;" />

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# Kalick & Hamilton dating model

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<img src="Figures/preference.png" width="400" style="display: block; margin: auto;" />
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- % matched couples for the case of preferring attractive partners

- Correlation starts low but raises pretty fast up to about 0.55, closer to empirical values than the matching attractiveness case

- **Main result:** attractiveness matching is not necessary for observed correlations

]

*The matching hypothesis reexamined. Michael Kalick and Thomas Hamilton. Journal of Personality and Social Psychology, 1986.*

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# Schelling's segregation model

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<img src="Figures/RND.png" width="350" style="display: block; margin: auto;" />
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- Agents are aware of the fraction of similar agents in their neighborhood: `$f$`

- Agents are satisfied with `$f \geq F$`, otherwise they relocate to a position in which they are satisfied
  - `$F$` measures intolerance

- Outcome: Segregation measured with Moran's I

]

*Dynamic Models of Segregation. Thomas Schelling. Journal of Mathematical Sociology, 1971*
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## Segregation versus tolerance in Schelling's model

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<img src="Figures/IvsF.png" width="750" style="display: block; margin: auto;" />
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- 3x3 neighborhood (up to 8 neighbors), torus edges

- Boxplots of I after convergence in several simulations

- Moran's I stays low for low F values

- Sharp increase above two neighbors for F

- Substantial segregation for F>0.33

]
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# Axelrod's culture model

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1. Choose a cell (agent) uniformly at random to be the active agent

2. Choose at random one of its neighbors

3. With probability equal to their cultural similarity:
  
  - Active agent copies a random feature of its neighbor in which they differed

- Key parameters: size, `$F$` and `$k$`

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.pull-right[
<img src="Figures/AxelrodSim.png" width="420" style="display: block; margin: auto;" />
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# The role of grid size in Axelrod's model

<img src="Figures/Size.png" width="700" style="display: block; margin: auto;" />
*The Dissemination of Culture: A Model with Local Convergence and Global Polarization. Robert Axelrod, Journal of Conflict Resolution 41(20), 1997*

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# Cultural affinity eurovision model

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## `$FoF$` distribution in affinity model

*Measuring cultural dynamics through the Eurovision song contest. David Garcia and Dorian Tanase. Advances in Complex Systems, 16 (2013)*
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# Granovetter's threshold model

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Net benefit =  benefit - costs

- Threshold to join: Net benefit is >0

- benefits increase and costs decrease with more people in the action (monotonic net benefit)

- weaker assumption: there is only one crossing of zero in the function of net benefit vs people in action

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.pull-right[
<img src="Figures/Benefit.png" width="800" style="display: block; margin: auto;" />
Example of net benefit function from Granovetter (1978)
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*Threshold Models of Collective Behavior. Mark Granovetter. American Journal of Sociology (1978)*
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# `$r_e$` versus `$\sigma$` in Granovetter's model

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<img src="Figures/STD.png" width="500" style="display: block; margin: auto;" />
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- Assumption: Thresholds follow normal distribution with `$\mu$` and `$\sigma$`
- `$r_e$`: equilibrium number of active agents (simulation ended)
- `$\sigma$`: standard deviation of distribution of thresholds
- Number of agents is constant: 100
- `$\mu$` is constant: 25

- Sharp increase in `$r_e$` at a critical `$\sigma$` value: phase transition
- Diversity-induced collective behavior
]

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## Opinion dynamics: bounded confidence model

- 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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# Final opinions vs initial opinions

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<img src="Figures/basic3.svg" width="500" style="display: block; margin: auto;" />

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- 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
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*Mixing beliefs among interacting agents. Guillaume Deffuant, David Neau, Frederic Amblard and Gerard Weisbuch. Advances in Complex Systems (2000)*
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# Hyperpolarization

<img src="Figures/Hyperpolarization.png" width="950" style="display: block; margin: auto;" />
<center>
*Hyperpolarization: Opinion extremeness x Opinion constraint*

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## Weighted Balance Theory and hyperpolarization

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- Cognitive balance +   
 evaluative extremeness

- ABM show emergence of hyperpolarization

- Filter bubbles and echo chambers are not a necessary condition

- **Predicts that issues become aligned and polarized over time**
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<img src="Figures/WBT.png" width="450" />
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[A Weighted Balance Model of Opinion Hyperpolarization. Schweighofer, Schweitzer & Garcia, JASSS (2020)](http://jasss.soc.surrey.ac.uk/23/3/5.html)

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## Hyperpolarization in Weighted Balance Theory

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# G(n,p) ensamble

- Probability of such G(n,p) networks `$P(G)$`:
  - `$P(G) = p^m(1-p)^{(\frac{n}{2})-m}$` , for such networks
  - `$P(G)=0$` , for non-simple graphs

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- p = 0.1
<img src="Figures/g_n_p_0_1.png" width="300" style="display: block; margin: auto;" />
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- p = 0.05
<img src="Figures/g_n_p_0_0_5.png" width="300" style="display: block; margin: auto;" />
]

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## Phase transition of largest component size with `$p$`

*Kang, M., & Petrášek, Z. (2014). Random graphs: Theory and applications from nature to society to the brain*

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# Watts & Strogatz Small World model

*Collective dynamics of ‘small-world’ networks. Duncan J. Watts & Steven H. Strogatz. Nature (1998)*
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# Clustering and average path length

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# The scale-free property

Power-law distributions are of the form:
`$$P(x) \propto x^{-\alpha}$$`  
If we multiply the random variable by a constant, the distribution is just multiplied too
`$$P(Cx) = C^{-\alpha} P(x)$$`
**Scale-free property:** The shape of the distribution is the same across different scales of the variable

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# Poisson degree distributions vs data

`$G(n,m)$` produces Poisson degree distributions, not power-laws. Similar problems with the Watts-Strogatz small world model.

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# The Barabási-Albert model

- **Growth:** stating from an empty network, add one node at each iteration

- **Preferential attachment:** when a node is added, connect it to `$m$` neighbors chosen at random in the network. Neighbors to connect are chosen with probability proportional to their degree.

The probability `$\Pi$` that a new vertex will be connected to vertex `$i$` with degree `$k_i$`  is

$$ \Pi(k_i) = \frac{k_i}{\sum_j k_j} $$

*Emergence of Scaling in Random Networks. Albert-László Barabási & Réka Albert. Science (1999)*
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# Degree distribution in the BA model

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<img src="Figures/BAdist.png" width="440" style="display: block; margin: auto;" />
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- Degree distribution of simulations with `$m=5$` for t=150,000 and t=200,000 (circles and squares, indistinguishable)

- Line has slope -2.9 in log-log plot

- The degree distribution in the BA model follows a power-law distributuion with `$\alpha=3$`:

$$ P(x) = \frac{2m^2}{x^3} \propto x^{-3} $$
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# The vertex copying model
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<img src="Figures/VertexCopying.png" width="500" style="display: block; margin: auto;" />
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- Start with one node and grow one node at a time

- When a new node connects, it samples one node at random and connects to it

- Then it copies all edges to neighbors of that node

- Some versions copy only a random fraction R of neighbors
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*Network growth by copying. P. L. Krapivsky and S. Redner. Physical Review E (2005)*

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## Multiplicative growth with heterogeneous ages

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- Multiplicative growth:

`$$X_{t+1} = X_t + \gamma * \epsilon * X_t$$`

- A Poisson birth process produces new elements at a constant rate: Aggregates have heterogeneous ages

- Inset: vertical axis is log-transformed
 ]
 
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*A Brief History of Generative Models for Power Law and Lognormal Distributions. Michael Mitzenmacher. Internet Mathematics (2004)*
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# Compartmental models: SIR

- Individuals are separated into *compartments* by their state of infection
- Individuals are indistinguishable within compartments
- Individuals transition between compartments when their state changes
- Not really Agent-Based Model: population or equation-based model

*Modeling Epidemics With Compartmental Models. Juliana Tolles and ThaiBinh Luong, JAMA Guide to Statistics and Methods (2020)*

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# Epidemic models in practice: SEIRX
<img src="Figures/SEIRX.png" width="700" style="display: block; margin: auto;" />
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## Outbreak distributions in the TU Graz network

<img src="Figures/Outbreaks.png" width="840" style="display: block; margin: auto;" />
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# Wrap-up

1. Fundamentals of Agent-Based Modelling
2. Opinion dynamics
3. Network formation
4. Processes on networks

Final notes and announcements:

- Register your group soon

- Presentation schedule will be published early next week

- Looking forward to your projects!

- **Course evaluation open from 14.06, please help us improve our courses!** Your opinion is very important, especially for younger lecturers