Centrality Metrics Explained

Centrality metrics measure how "important" a node is in a graph. But "important" can mean different things. This article explains each metric and when to use it.

Overview

Metric	Question It Answers	Good For
Degree	How many connections?	Finding popular nodes
Betweenness	How many paths go through?	Finding bridges/bottlenecks
Closeness	How quickly can it reach others?	Finding efficient spreaders
PageRank	How important are its connections?	Finding true influence
Eigenvector	How connected to well-connected?	Finding inner circle

Degree Centrality

Question: How many direct connections does this node have?

Calculation: Count of edges connected to the node, normalized by the maximum possible.

Interpretation:

• High degree = many direct connections
• In social networks: popular accounts
• In transport: hub stations
• In dependencies: commonly used packages

For Directed Graphs:

• In-degree: How many edges point TO this node (followers, dependents)
• Out-degree: How many edges point FROM this node (following, dependencies)

grph centrality graph.gexf --type degree
grph degree graph.gexf --direction in   # For directed: incoming
grph degree graph.gexf --direction out  # For directed: outgoing

When to use: Quick measure of popularity or connectivity.

Limitation: Doesn't consider the quality of connections. A node with 100 low-quality connections scores higher than one with 10 high-quality connections.

Betweenness Centrality

Question: How often does this node appear on the shortest path between other nodes?

Calculation: For each pair of nodes, find all shortest paths. Count what fraction pass through this node.

Interpretation:

• High betweenness = bridge position
• Controls information flow between parts of the network
• Critical for network connectivity
• Removing it would increase average path length

Examples:

• Transport: Stations where many routes converge
• Social: People who connect different groups
• Infrastructure: Routers handling traffic between networks

grph centrality graph.gexf --type betweenness

When to use: Finding bottlenecks, single points of failure, or critical connectors.

Limitation: Computationally expensive for large graphs.

Closeness Centrality

Question: How quickly can this node reach all other nodes?

Calculation: Inverse of the average shortest path length to all other nodes.

Interpretation:

• High closeness = central position
• Can spread information quickly
• Low average distance to everyone else

Examples:

• Epidemiology: Who can spread disease most efficiently?
• Information: Who can reach everyone with fewest hops?
• Logistics: Best location for a distribution center

grph centrality graph.gexf --type closeness

When to use: Finding efficient starting points for spreading information or influence.

Limitation: Only works well for connected graphs. Disconnected graphs have infinite distances.

PageRank

Question: How important is this node, considering the importance of nodes that connect to it?

Calculation: Iterative algorithm originally designed by Google. A node is important if important nodes link to it.

Interpretation:

• High PageRank = endorsed by important nodes
• Quality over quantity
• Being followed by an influencer matters more than being followed by many nobodies

Examples:

• Web: Pages linked by authoritative sites rank higher
• Social: Users followed by influencers are more influential
• Citations: Papers cited by landmark papers are more significant

grph centrality graph.gexf --type pagerank

When to use: Finding true influence or authority, especially in directed networks.

Key insight: Unlike degree, PageRank considers WHO links to you, not just how many.

Eigenvector Centrality

Question: How well-connected is this node to other well-connected nodes?

Calculation: Based on the graph's adjacency matrix eigenvalues. A node is important if its neighbors are important.

Interpretation:

• Similar to PageRank but symmetric
• Measures being "in the club"
• High score means connected to the well-connected

Examples:

• Social: Part of an influential inner circle
• Collaboration: Works with productive researchers
• Business: Connected to powerful organizations

grph centrality graph.gexf --type eigenvector

When to use: Finding nodes embedded in influential clusters.

Limitation: Can be unstable for directed graphs. Use PageRank for directed networks.

Choosing the Right Metric

For Social Networks

1. PageRank - True influence (who has influential followers?)
2. In-degree - Raw popularity (who has the most followers?)
3. Betweenness - Connectors (who bridges different communities?)

For Infrastructure/Transport

1. Betweenness - Critical nodes (which stations/routers are bottlenecks?)
2. Degree - Hub nodes (which have most connections?)
3. Closeness - Efficiency (which can reach everywhere quickly?)

For Dependencies

1. In-degree - Most depended upon (what do most packages use?)
2. Betweenness - Critical path (what's in most dependency chains?)
3. PageRank - Core importance (what do the important packages depend on?)

Comparing Metrics

Different metrics highlight different things:

# Run multiple metrics on the same graph
grph centrality graph.gexf --type degree --top 5
grph centrality graph.gexf --type betweenness --top 5
grph centrality graph.gexf --type pagerank --top 5

When the same nodes appear across metrics, they're genuinely central. When different nodes appear, each metric is capturing a different type of importance.

Summary

Want to find...	Use...
Most connected	Degree
Bottlenecks	Betweenness
Best spreaders	Closeness
True influencers	PageRank
Inner circle	Eigenvector