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
- Degree centrality: C_D(v) = deg(v) / (N-1). O(V+E).
- 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).
- Closeness centrality: C_C(v) = (N-1) / Σd(v,u). Inverse of average shortest path. O(V×(V+E)).
- 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 --- EExpected 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