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):

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

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