Social Network Analysis — Graph Metrics, Centrality, Community Detection

🎯 Learning Objectives

• Calculate degree, betweenness, and closeness centrality
• Compute clustering coefficient for nodes and networks
• Explain PageRank algorithm and its application
• Detect communities using modularity

1. Graph Fundamentals

1.1 Intuition

1.2 Graph Types

1.3 Basic Metrics

• Nodes (N) = 5
• Edges (E) = 6
• Density = 2E / (N(N-1)) = 12 / 20 = 0.6

2. Centrality Measures

2.1 Degree Centrality

2.2 Betweenness Centrality

• Shortest paths A↔E: A-C-E (passes C). A-B-E? A-C-E is shorter.
• Shortest paths B↔E: B-C-E (passes C)
• Shortest paths D↔E: D-C-E (passes C)
• C lies on every path between nodes on different sides High betweenness = gatekeeper / bridge.

2.3 Closeness Centrality

3. Clustering Coefficient

3.1 Local Clustering Coefficient

• Edges between B-C: yes, C-D: yes, B-D: no
• ${k \choose 2} = {3 \choose 2} = 3$
• Edges among neighbors = 2
• $C(A) = 2/3 = 0.67$ For node E (neighbor: C):
• $C(E) = 0$ (only one neighbor, no pairs to evaluate)

3.2 Global Clustering Coefficient

4. PageRank Algorithm

4.1 Intuition

4.2 Algorithm

python
Run Inline
def pagerank(graph, damping=0.85, iterations=100):
    N = len(graph)
    PR = {node: 1.0/N for node in graph}
    for _ in range(iterations):
        new_PR = {}
        for node in graph:
            # Sum of PR(incoming neighbors) / out-degree
            rank_sum = 0
            for neighbor in graph:
                if node in graph[neighbor]:  # neighbor links to node
                    rank_sum += PR[neighbor] / len(graph[neighbor])
            # Damping factor + teleportation
            new_PR[node] = (1 - damping) / N + damping * rank_sum
        PR = new_PR
    return PR

4.3 Worked Example

pseudo
PR(A) = 0.15/3 + 0.85 × (PR(C)/1) = 0.05 + 0.85 × 0.33 = 0.331
PR(B) = 0.05 + 0.85 × (PR(A)/2) = 0.05 + 0.85 × 0.165 = 0.190
PR(C) = 0.05 + 0.85 × (PR(A)/2 + PR(B)/1) = 0.05 + 0.85 × (0.165 + 0.190) = 0.352

5. Community Detection

5.1 Modularity

• Q > 0.3 → significant community structure
• Q = 0 → random connections

5.2 Louvain Algorithm

6. 📝 Practice Questions

Q1: Calculate degree centrality for each node in: A-B-C-D (path graph of 4 nodes).

Answer: A=1, B=2, C=2, D=1. Normalized (÷3): A=0.33, B=0.67, C=0.67, D=0.33. Q2: Why does betweenness centrality identify "bridges" in a network?

Answer: Betweenness counts how many shortest paths pass through a node. Nodes that connect different communities lie on many shortest paths between members of different communities. Removing such a bridge would disconnect the graph. Q3: Compute PageRank for a 2-node graph where A→B and B→A (d=0.85).

Answer: PR(A) = 0.05 + 0.85×PR(B)/1 PR(B) = 0.05 + 0.85×PR(A)/1

Solving: PR(A) = PR(B). 2×PR = 0.05 + 0.85·PR → PR = 0.05/0.15 = 0.33 each. Q4: What does a clustering coefficient of 1 mean for a node?

Answer: A clustering coefficient of 1 means all the node's neighbors are connected to each other — they form a complete graph (clique). Every pair of friends is also friends. Q5: How is the small-world phenomenon measured?

Answer: Small-world networks have high clustering (like regular lattices) and short average path lengths (like random graphs). Measured by comparing clustering coefficient and path length to equivalent random graphs.

7. 🔗 Cross-References

• Week 7 - Information Diffusion: Central nodes are influential spreaders
• Week 8 - Privacy: Anonymization preserves network structure
• BSCS4021 (Advanced Algorithms): PageRank, graph algorithms Join Discord NextData Collection