Centrality

Centrality

PageRank, betweenness, closeness — who matters in this graph?

Why it matters

Centrality scores answer 'which node has outsized influence?' — the metric is the question: PageRank for endorsement, betweenness for bottleneck, closeness for reach.

Going deeper

Use-case → metric, with the trap each one has:

Question	Best metric	Trap
'Which page should rank top in search?'	PageRank	Gameable via link farms
'Which router would, if removed, partition the network?'	Betweenness	O(V³) naïve; sample on large graphs
'Who can reach every customer in fewest steps?'	Closeness	Disconnected components inflate scores
'Who has the most direct connections?'	Degree	Doesn't see the quality of connections

Most production deployments compute two or three centralities and let the downstream model or analyst pick. Centrality without a clear question is just another column on the dashboard.

Each centrality metric is a different question you'd ask about a city's residents:

• Degree — 'who has the most friends on Facebook?' Cheap, often misleading.
• PageRank — 'who do influential people recommend?' The mayor is influential not because they know everyone but because the people who know them are themselves influential. The metric recurses on its own definition.
• Betweenness — 'whose absence would split the city in two?' The bridge keeper. Often a broker rather than a celebrity.
• Closeness — 'who can deliver a message to everyone fastest?' The dispatcher.

Choose the metric whose question matches your business question. A fraud team asking 'who's the choke point in this money-laundering ring?' wants Betweenness, not PageRank.