Four kinds of "important", four different people — you don't pick the measure, what flows picks it
Linked from: Topic 20 · Structure Drives Dynamics
01The question it poses
Hand someone a relationship graph and they will always ask the same thing: who matters most here?
The trouble is that the sentence is unfinished. "Matters" can mean having the most ties, or being closest to everyone, or being someone others cannot route around, or being connected to important people. On one and the same graph those four pick different nodes, often very different ones.
The centrality family exists to pull those four apart and define each one precisely — which forces you to answer a question before computing anything: what actually flows on this graph? Borgatti's widely cited 2005 paper put it most bluntly: every centrality measure implies a model of flow, and if you have the flow wrong, a correctly computed number is still the wrong number.
02The rules
Degree centrality = how many edges the node has. The simplest, and the hardest to get wrong.
Closeness centrality = compute the shortest distance from this node to every other node, sum them, take the reciprocal. A smaller total means the node sits nearer to the whole graph.
Betweenness centrality = count how many pairs of nodes have a shortest path running through this node. If a pair has several equally short paths, split the credit proportionally.
Eigenvector centrality = defined recursively: a node's score is proportional to the sum of its neighbours' scores. Solving that gives the eigenvector of the adjacency matrix belonging to its largest eigenvalue, which is where the name comes from. PageRank is this plus a random-jump correction.
The first three all rest on one concept, the shortest path; the fourth uses it not at all. That isn't a detail: the first three assume things travel along shortest routes, the fourth assumes they seep in all directions like water. Real flows are rarely purely either, which is the soft spot every centrality measure shares.
Also worth noting: degree and closeness are always positive in a connected graph, while betweenness can be exactly zero — a node with many neighbours who are all directly linked to each other is nobody's necessary route. That zero is informative.
03What you see when it runs
The standard example is the "kite graph" introduced by Krackhardt in 1990: ten people, eighteen ties. The only reason it gets cited over and over is that three centralities pick three different people on it, and all three are right.
Four winners on one graph. D is busiest, F and G are most central, H holds the throat.
Putting the four side by side makes the divergence sharper:
The horizontal axis is the same ten people in all four. Watch H: first by betweenness, mid-pack by eigenvector, lower half by degree.
Worth pausing on: C and E both have betweenness of exactly zero. Neither is isolated — each has three neighbours — but those neighbours are already directly linked to each other, so no pair needs to route through them. "Well connected" and "impossible to route around" are entirely different properties, and that zero separates them cleanly.
One more thing, clearer on large graphs: the four measures are strongly correlated but never coincide. High correlation means picking any one of them usually isn't disastrous; imperfect coincidence means the very top of the ranking — precisely the people you were going to act on — is the part most likely to be wrong.
04What it explains
Centrality's soundest use is taking "who should we act on" out of intuition. A few settings where it has repeatedly held up:
Vaccination and contact tracing. Who gets a scarce dose first? If transmission runs on everyday close contact, the high-degree people. If the crux is the handful of people linking otherwise separate populations, the high-betweenness people. On the same graph those two lists can barely overlap.
Supply chains and infrastructure. A node with very high betweenness and unremarkable degree is the classic invisible single point of failure — inconspicuous in normal operation, and the whole chain stops when it goes. In a cost review these are exactly the nodes cut first, for being small.
Bridge-builders in organisations. A high-betweenness person is both a source of efficiency and a bottleneck, and the most expensive kind of departure risk: what breaks when they leave isn't a role, it's the route between two departments.
Web ranking. PageRank is eigenvector centrality plus a "jump to a random page" correction, added so the scores still converge on a real network full of dead-end links.
What it cannot explain
It cannot choose the measure for you. The flow chooses: things that can be copied and spread in parallel (messages, ideas) suit degree and eigenvector; things that cannot be copied and must traverse a route (money, goods, referrals) suit betweenness and closeness. That is Borgatti's 2005 argument — most misuse isn't arithmetic error, it's the wrong flow.
It cannot give a stable ranking on an inaccurate graph. Betweenness is acutely sensitive to missing edges because it lives off the property of being the only route; one added edge can halve someone's score. Real graphs are always sampled, so rankings need a threshold sweep, and the second decimal place is meaningless.
It cannot handle time order. A—B—C on a static graph may be impassable in time: if A met B after B met C, the route does not exist. Flattening timestamped interactions into a static graph systematically overstates reachability, and overstates the middle node's betweenness along with it.
It cannot get from position to influence. Centrality says things must pass through someone. It does not say they are willing to pass them on, able to, or that the content survives the trip. Structure supplies opportunity, not behaviour. Reading a centrality ranking as an influence ranking mistakes a constraint for a motive.
It cannot compare raw values across graphs. Raw betweenness grows roughly with the square of the node count, so raw values from graphs of different sizes are incomparable. Normalisation helps, but more than one normalisation exists and the ranking can change between them — so state which one you used.
It cannot stay trustworthy once published. The moment high centrality buys resources, the people being measured start adding edges, and adding edges is far cheaper than contributing. Search ranking is the complete demonstration: link farms appeared as soon as PageRank became public.