Centrality
A family of measures that score which nodes matter most in a network — by connections, by bridging, or by closeness.
Centrality
In any Graph, some nodes matter more than others — but matter how? Centrality is not one number but a family of them, each formalizing a different intuition about importance. The most popular person, the airport whose closure strands the most travelers, and the gossip best placed to spread news fastest are all "central," yet they can be entirely different nodes. Choosing a centrality measure means choosing what kind of importance you mean.
Three classic answers
- Degree centrality — how many connections? Simply count a node's edges. The person with the most friends, the website with the most links. It is instant to compute and often a fine first cut, but it is purely local: it cannot tell a hub wired into the global network from one buried in an isolated clique.
- Betweenness centrality — how often are you on the path between others? For every pair of nodes, find the shortest path between them; a node's betweenness is the share of those paths that run through it. High-betweenness nodes are bridges — remove one and distant regions of the network fall apart. They control flow even with few connections of their own.
- Closeness centrality — how near are you to everyone? The inverse of a node's average shortest-path distance to all others. High closeness means short reach to the whole network — ideal for spreading something fast, computed with the help of Breadth-First Search from each node.
A bridge node lights up
The graph below is two tight clusters joined by a single bridge node. By degree the bridge is unremarkable. By betweenness it is the most important node in the network: pulse-by-pulse, almost every path that crosses from one side to the other must pass through it. Watch the traffic — sized to flow through each node — and the bridge glows brightest.
A spectrum of importance
PageRank is itself a centrality measure — a recursive, eigenvector-flavored cousin of degree centrality in which a connection's worth depends on the importance of whoever it comes from. That is the deep idea uniting the family: importance can be counted (degree), positional (betweenness, closeness), or self-referential (PageRank). Which one to trust depends entirely on what flows through your network and what you are trying to protect, spread, or rank.