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title: Network Effects And Cascading Behaviour
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In this section, we study how a infection propagates through a network. We will look into two classed of model, namely decision based models and probabilistic models. But first lets look at some terminology used throughout the post.

**Terminology**
1. Cascade: Propagation tree created by spreading contagion
2. Contagion: What is spreading in the network, e.g., diseases, tweet, etc.
3. Infection: Adoption/activation of a node
4. Main players: Infected/active nodes, early adopters

# Decision Based Models
In decision based models, every nodes independently decides whether to adopt the contagion or not depending upon its neighbors. The decision is modelled as a two-player coordination game between user and its neighbor and related payoffs. Hence a node with degree $$k$$ plays $$k$$ such games to decide its payoff and correspondingly its behavior.

## Single Contagion Model
There are two contagions $$A$$ and $$B$$ in the network and initially every node has behavior $$B$$. Every node can have only one behavior out of the two. The payoff matrix is given as:

|   | A | B |
|---|---|---|
| A | a | 0 |
| B | 0 | b |

Lets analyze a node with d neighbors, and let p be the fraction of nodes who have adopted $$A$$. Hence the payoff for $$A$$ is $$apd$$ and payoff for $$B$$ is $$b(1-p)d$$. Hence the node adopts behavior $$A$$ if 
$$apd > b(1-p)d \implies p > \frac{b}{a+b} = q$$(threshold)

### Case Study: [Modelling Protest Recruitment on social networks](https://arxiv.org/abs/1111.5595)
Key Insights:
- Uniform activation threhold for users, with two peaks
- Most cascades are short
- Successful cascades are started by central users

#### Note: 
**k-core decomposition**: biggest connected subgraph where every node has at least degree k (iteratively remove nodes with degree less than k)

### Multiple Contagion Model
There are two contagions $$A$$ and $$B$$ in the network and initially every node has behavior $$B$$. In this case a node can have both behavior $$A$$ and $$B$$ at a total cost of $$c$$ (over all interactions). The payoff matrix is given as:

|   | A | B | AB |
|---|---|---|----|
| A | a | 0 | a  |
| B | 0 | b | b  |
| AB| a | b | max(a,b)|

### Example: Infinite Line graph 
**Case 1**:**A-w-B** 
![decision_case_1](../assets/img/decision_model_1.png?style=centerme)

Payoffs for $$w$$: $$A: a$$, $$B: 1$$, $$AB: a+1-c$$

![decision_case_2](../assets/img/decision_model_2.png?style=centerme)

**Case 1**: **AB-w-B**
![decision_case_3](../assets/img/decision_model_3.png?style=centerme)

Payoffs for $$w$$: $$A: a$$, $$B: 1$$, $$AB: max(a, 1) + 1 -c$$

![decision_case_4](../assets/img/decision_model_4.png?style=centerme)