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@ -4,40 +4,56 @@ title: Network Effects And Cascading Behaviour
header-includes:
- \usepackage{amsmath}
---
The phenomenon of spreading through networks and cascading behaviors is prevalent in a wide range of real networks. Examples include contagion of diseases, cascading failure of technologies, diffusion of fake news, and viral marketing. Formally, an “infection” event can spread contagion along main players (active/infected nodes) which constitute a propagation tree, known as a cascade.
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
The phenomenon of spreading through networks and cascading behaviors is prevalent in a wide range of real networks. Examples include contagion of diseases, cascading failure of technologies, diffusion of fake news, and viral marketing. Formally, an **“infection” event** can spread **contagion** through **main players** (active/infected nodes) which constitute a propagation tree, known as a **cascade**. We will examine two model classes of diffusion:
- Decision-based: each node decides whether to activate based on its neighbors' decisions. Deterministic rule, nodes are active players, and suited for modeling adoption
- Probabilistic: infected nodes "push" contagion to uninfected nodes with some probability. Can involve randomness, nodes are passive, and suited for modeling epidemic spreading
# 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.
# Decision Based Diffusion
## Game Theoretic Model of Cascades: single behavior adoption
The key intuition behind the game theoretic model is that a node will gain more payoff if its neighbors adopt the same behavior as it. An example is competing technological products: if your friends have the same type DVD players and discs (e.g. Blu-ray vs. HD DVD), then you can enjoy sharing DVDs with them.
## 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:
Every node independently decides whether to adopt the contagion depending upon its neighbors. The decision is modelled as a two-player game between a node and a given neighbor. Hence a node with degree $$k$$ plays $$k$$ such games to evaluate its payoff and correspondingly its behavior. The total payoff is the sum of node payoffs over all games.
If there are two behavior $$A$$ and $$B$$ in the network and each node can adopt a single behavior, the payoff matrix for the two-player game is as follows:
| | 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)
where rows correspond to node $$v$$'s behavior, columns correspond to node $$w$$'s behavior, and entries represent each node's payoff.
Let's analyze a node with $$d$$ neighbors, and let $$p$$ be the fraction of nodes which have adopted $$A$$. The payoff for choosing $$A$$ is $$apd$$ and the payoff for choosing $$B$$ is $$b(1-p)d$$. Hence the node adopts behavior $$A$$ if the following is met:
$$apd > b(1-p)d \implies p > \frac{b}{a+b}$$
We define $$q = \frac{b}{a+b}$$ to be the **threshold** fraction of a node's neighbors required for the node to choose $$A$$ i.e. requires $$p > q$$.
### Example:
Scenario:
- Graph where all nodes start with $$B$$
- Small set $$S$$ of early adopters of $$A$$. Hardwire this set such that these nodes will persistently use $$A$$ regardless of payoff
-Set $$a=b-\epsilon$$ and $$q = 0.5$$ for a small constant $$\epsilon>0$$. Interpretation: I adopt $$A$$ if more than 50% of my neighbors adopt $$A$$.
### 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
Undirected network of Twitter users. 70 identified hashtags associated with 2011 Spain anti-austerity protests.
For each user (node):
- User activation time = moment when user starts tweeting protest messages
- $$k_{in}$$ = total number of neighbors at user activation time
- $$k_{a}$$ = number of activate neighbors at user activation time
- Activation threshold $$\frac{k_{a}}{k_{in}}$$ = fraction of neighbors that are active at user activation time
#### Note:
**k-core decomposition**: biggest connected subgraph where every node has at least degree k (iteratively remove nodes with degree less than k)
Key Insights:
- The distribution of activation threshold had two local peaks: i) at $$\frac{k_{a}}{k_{in}} \approx 0$$, indicating many self-active users who join with without social pressure ii) at $$\frac{k_{a}}{k_{in}} \approx 0.5$$ indicating many users join once half their neighbors have. Remainder of distribution mostly uniform.
- A "burst" of neighbors joining the movement has greater impact on users with high threshold and lesser impact on users with low threshold
- Most cascades were small
- Larger cascades were started by users with higher $$k$$-core number i.e. more central. The $$k$$-core is defined as the largest connected subgraph where every node has at least degree $$k$$ and can be evaluated by iteratively removing 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:
## Extending Game Theoretic Model: multi-behavior adoption
A node can adopt both behaviors and become $$AB$$ by paying a cost $$c$$. The resulting payoff matrix (without cost $$c$$ applied) is as follows:
| | A | B | AB |
|---|---|---|----|
@ -45,7 +61,9 @@ There are two contagions $$A$$ and $$B$$ in the network and initially every node
| B | 0 | b | b |
| AB| a | b | max(a,b)|
### Example: Infinite Line graph
### Example: Infinite path graph
Let us examine an infinite path graph where everyone begins with behavior/product $$B$$ except for three nodes of the following cases. Let us also set $$b=1$$.
The graphs show how different regions of $$(a,c)$$ values impact the decision-based diffusion:
- B->A (Direct Conquest): If $$a$$ and $$c$$ are both high, then the cost of being compatible to both products is not worth it and conversion to $$A$$ is direct
- B->AB->A (Infiltration): If $$a$$ is high and $$c$$ is lower, users first transition to $$AB$$ before fully committing to $$A$$
jthluke
4 years ago
Ben Hannel
