Modelling small worlds

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# Modelling small worlds
### 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
  - Opinion dynamics
  - Modelling hyperpolarization and cognitive balance

- **Block 3: Fundamentals of agent-based modelling**
  - Basic network models
  - **Today: Modelling small worlds**
  - Scale-free networks
  - Growth processes

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# Overview

## 1. The small world phenomenon

## 2. The Watts-Strogatz model

## 3. Social small world model

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# The small world phenomenon

## *1. The small world phenomenon*

## 2. The Watts-Strogatz model

## 3. Social small world model

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# It's a small world!

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# Six degrees of separation

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<img src="Figures/SixDegrees.jpg" width="330" style="display: block; margin: auto;" />
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<img src="Figures/Linked.jpg" width="300" style="display: block; margin: auto;" />
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# The Bacon number

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## Milgram's small world experiment

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- 160 people in Omaha try to reach one person in Boston by mail to their acquaintances

- 44 letters reached the target with six steps on average

- Short path length as evidence of small-world

- However, some letters did not agree, are those paths infinite?
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<img src="Figures/map_us.png" width="500" style="display: block; margin: auto;" />
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# Refresher: Network distance

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The **distance** between nodes `$v$` and `$w$` is denoted as `$dist(v,w)$` and measures the minimum length among all the paths connecting `$v$` and `$w$`.

If there is no path between `$v$` and `$w$`, the distance between them is defined as `$dist(v,w) := \infty$`.

Example: `$dist(b, e) = 2$`

In directed networks, it might happen that `$dist(v,w) \neq dist(w,v)$`. 
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<img src="Figures/networkUndirected.png" width="400" style="display: block; margin: auto;" />
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# Average path length

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$$ \langle l \rangle = \frac{1}{N(N-1)}\sum_{u,v} dist(u,v) $$

- Global metric for the whole network

- It makes sense when network is connected, otherwise `$\langle l \rangle= \infty$`

- In the example, `$\langle l \rangle=1.7$`

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<img src="Figures/networkUndirected.png" width="400" style="display: block; margin: auto;" />
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# Refresher: Clustering coefficient

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**Local clustering coefficient**:

`$$C_i = \frac{2*t(i)}{d_i *(d_i−1)}$$`

- `$d_{out}(i)$` is the degree of `$i$` (>1) 
- `$t(i)$` is the number of pairs of neighbors of `$i$` that are connected

**Average clustering coefficient**:

`$$C = \frac{1}{N} \sum_i C_i$$`

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.pull-right[
<img src="Figures/closedTriad.png" width="400" style="display: block; margin: auto;" />
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# The Watts-Strogatz model

## 1. The small world phenomenon

## *2. The Watts-Strogatz model*

## 3. Social small world model

*Collective dynamics of ‘small-world’ networks. Duncan J. Watts & Steven H. Strogatz. Nature (1998)*

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# Clustering versus small distances

- Triangles reduce distances: all nodes in a triangle are at distance one

- Clustering and short path lengths appear to be opposing properties

- Social networks have high clustering (lots of triangles), can they also have short paths? Is the six degrees observation a robust one?

** Research question: can a model produce networks with both high clustering and low average path distance?**

Additional conditions:
- Network is large ( `$N \gg 1$` )
- Network is sparse, like a social network ( `$\langle k \rangle \ll N$` )

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# The Watts-Strogatz model

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- Start with a fixed ring where `$N$` nodes are connected to `$k$` neighbors in the ring

(example: `$N=20$`, `$k=4$`)

- For each edge:
  
  - With probability `$p$`: rewire the edge uniformly at random
  
(two versions: only one endpoint or both are rewired)
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<img src="Figures/WS-1.png" width="300" style="display: block; margin: auto;" />
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# From order to randomness

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# Properties of the lattice ( `$p=0$` )

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**Average clustering coefficient:**

$$C(0) \sim 3/4   $$
(tunable value of `$C$` based on `$k$`)

** Average path length:**

$$L(0) =  \frac{N}{2k} $$
(very high path length, grows linearly with network size)

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.pull-right[
<img src="Figures/WS-1.png" width="300" style="display: block; margin: auto;" />
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# C(p) and L(p) versus C(0) and L(0)

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# Empirical versus random networks

Empirical network analysis of `$L$` and `$C$` compared to random networks with the same number of nodes and edges (null model, `$G(n,m)$` )

| Network | `$L_{actual}$` | `$L_{random}$` | `$C_{actual}$` | `$C_{random}$` |
| ----------- | ----------- | ----------- | ----------- | ----------- |
| Film actors | 3.65 | 2.99 | 0.79 | 0.00027 |
| Power grid | 18.7 | 12.4 | 0.080 | 0.005 |
| C. elegans | 2.65 | 2.25 | 0.28 | 0.05 |

**Evidence of small world networks: empirical networks have average path lenghts close to random networks but much higher average clustering**

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#  Social small world model

## 1. The small world phenomenon

## 2. The Watts-Strogatz model

## *3. Social small world model*

*Networks, Dynamics, and the Small‐World Phenomenon. Duncan J. Watts. American Journal of Sociology (1999)*

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# Social networks aren't lattices

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<img src="Figures/Karate.jpg" width="750" style="display: block; margin: auto;" />
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.pull-right[
<img src="Figures/Politnetz.png" width="750" style="display: block; margin: auto;" />
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# Extreme case: the caveman graph

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# Modeling social small worlds

- Start with a ring of n nodes

- For each pair of nodes (in random order):
      - Calculate number of shared friends `$m_{i,j}$`
      - Calculate propensity to connect `$R_{i,j}$` based on `$m_{i,j}$`
      - Connect them with probability `$R_{i,j}$`
      
      
`$R_{i,j}$` has a non-zero but low base probability `$p$` of any two nodes connecting regardless of their number of shared friends

The dependence of `$R_{i,j}$` on `$m_{i,j}$` interpolates between a regular (caveman network) and a random network. The reality should be somewhere in between
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# Modeling propensity to triadic closure

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# Formalizing propensity to triadic closure

- `$R_{i,j}$`: propensity to connect of agents `$i$` and `$j$`
- `$m_{i,j}$`: mutual friends between `$i$` and `$j$`
- `$p$`: base probability to connect when no common friends
- `$k$`: average degree of network
- `$\alpha$`: exponent that defines curvature of `$R_{i,j}$` versus `$m_{i,j}$` 
  - `$0 \le \alpha \le \infty$`
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# `$\alpha$` in propensity

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# Avg path length vs `$\alpha$`

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# Clustering coefficient vs `$\alpha$`

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# "Small-worldness" vs `$\alpha$`

<img src="Figures/CavemanResult.png" width="670" style="display: block; margin: auto;" />
**For an intermediate range of values of `$\alpha$`, average path length is low while the clustering coefficient is high.**

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## Summary

-  The small world phenomenon
  - Anecdotes and literature suggest that the distance between two random people in the world is not so large
  - Refresher: average path length and average clustering coefficient

- The Watts-Strogatz model
  - From a regular lattice to a random network rewiring with probability `$p$`
  - For a wide range of middle values of `$p$`, clustering is high and average path lengths are low
  - Empirical networks show this when compared to random networks

- Social small world model
  - Cavemen rather than lattices as regular network
  - `$\alpha$` modulates shape of propensity to connect vs friendships share
  - Shows small-world behavior for an intermediate range of `$\alpha$`

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# Quiz

- What fraction of letters in Milgram's experiment reached their destination?

- What is the average path length of a network with an isolated node?

- What is the average clustering coefficient of a tree?

- What is the null model to compare small-world measurements?

- What is a better model of a social network, a lattice or a cavemen network?

- if `$\alpha$` is zero, do you connect to the friends of your friends very often?