Network Centrality Metrics

Overview

Centrality measures quantify node importance in a network. Four classical metrics: degree (connections), betweenness (bridge role), closeness (proximity), eigenvector (connection quality). Each captures a different aspect of importance. Complexity ranges from O(V+E) for degree to O(V×E) for betweenness.

When to Use

Trigger conditions:

• Identifying key influencers or critical nodes in social/organizational networks
• Analyzing network vulnerabilities (which node failure causes most damage)
• Comparing node importance across different dimensions

When NOT to use:

• For group/community detection (use community detection algorithms)
• For information spread modeling (use epidemic models)

Algorithm

IRON LAW: Different Centrality Metrics Answer DIFFERENT Questions
- Degree: Who has the most connections? (popularity)
- Betweenness: Who bridges communities? (brokerage)
- Closeness: Who can reach everyone fastest? (efficiency)
- Eigenvector: Who is connected to important people? (prestige)
Using the WRONG metric answers the WRONG question. Choose based on
what "important" means in your context.

Phase 1: Input Validation

Build network graph from edge list or adjacency matrix. Determine: directed vs undirected, weighted vs unweighted, connected vs disconnected. Gate: Graph is well-formed, largest connected component identified.

Phase 2: Core Algorithm

1. Degree centrality: C_D(v) = deg(v) / (N-1). O(V+E).
2. Betweenness centrality: C_B(v) = Σ(σ_st(v) / σ_st) for all s,t pairs. Fraction of shortest paths through v. O(V×E).
3. Closeness centrality: C_C(v) = (N-1) / Σd(v,u). Inverse of average shortest path. O(V×(V+E)).
4. Eigenvector centrality: Score proportional to sum of neighbors' scores. Power iteration until convergence. O(k×E).

Phase 3: Verification

Check: centrality values normalized [0,1]. Top nodes by each metric may differ — this is expected and informative. Sanity check top-5 against domain knowledge. Gate: All metrics computed, top nodes make intuitive sense.

Phase 4: Output

Return centrality scores with multi-metric comparison.

Output Format

{
  "centralities": [{"node": "Alice", "degree": 0.85, "betweenness": 0.42, "closeness": 0.71, "eigenvector": 0.90}],
  "metadata": {"nodes": 500, "edges": 2000, "directed": false, "connected_components": 1}
}

Examples

Sample I/O

Input: 5-node undirected graph (bridge topology): edges = {(A,B), (A,C), (B,C), (C,D), (D,E)}

    A --- B
     \  /
      C
      |
      D --- E

Expected centralities (normalized by N-1 = 4):

Node	Degree	Betweenness	Closeness	Eigenvector
A	0.50 (2/4)	0.000	0.571 (4/7)	0.452
B	0.50 (2/4)	0.000	0.571 (4/7)	0.452
C	0.75 (3/4)	0.667	0.800 (4/5)	0.628
D	0.50 (2/4)	0.500	0.667 (4/6)	0.386
E	0.25 (1/4)	0.000	0.500 (4/8)	0.201

Verify: C is the bridge — highest in ALL four metrics. E is the periphery — lowest in all metrics. A and B are symmetric (identical scores). D has nonzero betweenness (bridges C to E) but lower degree than C.

Edge Cases

Input	Expected	Why
Star graph	Center has max all centralities	Hub dominates in all metrics
Disconnected graph	Closeness undefined for disconnected pairs	Use harmonic centrality instead
Directed graph	In-degree ≠ out-degree centrality	Popularity (in) vs activity (out)

Gotchas

• Disconnected graphs: Closeness centrality is undefined when nodes can't reach each other. Use harmonic centrality: C_H(v) = Σ(1/d(v,u)) as an alternative.
• Scale dependence: Raw centrality values depend on network size. Use normalized versions for cross-network comparison.
• Betweenness is expensive: O(V×E) makes it impractical for very large networks (millions of nodes). Use approximation algorithms (random sampling of shortest paths).
• Dynamic networks: Centrality in a snapshot may not reflect influence over time. Temporal centrality metrics exist but are more complex.
• Correlation between metrics: In many real networks, centrality metrics are correlated. But the DIFFERENCES are often the most informative (high degree but low betweenness = local hub, not broker).

References

• For centrality metric comparison framework, see references/metric-comparison.md
• For approximate betweenness algorithms, see references/approximate-betweenness.md