Hub and Spoke Dossier
Degree, closeness, and betweenness are three different questions put to the same graph — which is why they can crown three different winners.
Centrality is the family of measures that rank the nodes of a network. The family exists because central has no single meaning: a node can be central because it touches many others, because it sits near everyone on average, or because traffic between others must cross it. Each meaning has a measure, and each measure has graphs where it disagrees with the other two.
Degree centrality is the count of a node's links. It is local, cheap to compute, and honest about one thing: how much of the network is within one step. High degree marks the busy and the visible. It cannot see the wider map, so it regularly over-ranks nodes whose many neighbors lead nowhere.
Closeness centrality measures how far a node is, on average, from every other node, counting steps along shortest paths. High closeness marks a good broadcaster: anything starting there arrives everywhere soon. It is a global measure, so it needs the whole map — and it rewards being in the middle of the web rather than on top of a pile of links.
Betweenness centrality counts how often a node lies on the shortest path between pairs of other nodes. High betweenness marks a broker or a bottleneck: the bridge between two communities, the only road between two districts. A node can score high here with very few links, which is precisely the case degree misses.
The practical lesson is to choose the question before the measure. Spreading something fast argues for closeness; guarding a flow argues for betweenness; finding the merely popular argues for degree. On one and the same graph the three can name three different nodes as the most central — and each answer is correct for its question.
Further reading