Module 04: Friendship paradox

Learning Objectives

📚 Understand the friendship paradox
🔍 Explore its importance and consequences
📊 How to plot degree distribution? (not as easy as it seems!)
- Histogram
- CDF
- CCDF
- Log-log plot
🧠 Excess degree distribution

Your friends have more friends than you, on average

Let’s approach this mathematically.

Let \(P_0(k)\) be the probability of a node having degree \(k\)
Let \(P_1(k)\) be the probability of a friend having degree \(k\)
How does \(P_1(k)\) look like in terms of \(P_0(k)\) and \(k\)?
Hint:
- Nodes with \(k\) edges appear \(k\) times more likely as a friend of someone than a person with \(1\) edge. Thus, \(P_1(k) \propto \; ????\). To get the actual distribution, you will need \(\langle k \rangle = \sum_{k} k P_0(k)\)
Compute the average degree of friends over edges

\(P_1(k) = \frac{k}{\langle k \rangle} P_0(k)\)
Average degree of friends
- \(\langle k_f \rangle = \sum_{k} k P_1(k) = \sum_{k} \frac{k^2}{\langle k \rangle} P_0(k) = \frac{\langle k^2 \rangle}{\langle k \rangle}\)

Understanding degree distribution is crucial for network analysis. But how do we plot it? 🤔 (it’s not as easy as it seems!)

Definitions CDF: \(F(x) = P(X \leq x)\), CCDF: \(\bar{F}(x) = P(X > x) = 1 - F(x)\)