# Betweenness centrality

> Betweenness centrality measures how often a person sits on the shortest path between two others. The formula, why the Medici top it, and how to find brokers.

Source: https://netgraf.isik.co/learn/betweenness-centrality
Updated: 2026-09-24

Betweenness centrality measures how often a person lies on the shortest path between two other people. A person with high betweenness is a broker: information, money or influence moving between parts of the network tends to pass through them, and removing them makes everyone else further apart.

## How it's calculated

```
betweenness(v) = sum over pairs s, t (s != v != t) of  paths(s, t through v) / paths(s, t)
normalised     = betweenness(v) / ((n - 1)(n - 2) / 2)
```

paths(s, t) is the number of shortest paths from s to t. The normalised form runs from 0 (on nobody's path) to 1 (on every path, the centre of a star).

Linton Freeman defined the measure in 1977. Computing it naively takes a very long time on large networks; Ulrik Brandes's 2001 algorithm, which Netgraf uses, does it with one breadth-first search from every person.

## The Medici

The best-known result in the field comes from the [Florentine families](https://netgraf.isik.co/m/florentine-families). Padgett and Ansell mapped the marriage and business ties of fifteenth-century Florence's elite and found that the Medici were neither the wealthiest family nor the most established politically. They were the family that sat between the others: many of the families they married into had no ties to each other. That brokerage position, not raw size, is what the Medici turned into power. Their betweenness is by far the highest on the map.

## In Netgraf

In the Insights panel, betweenness is **Holding it together**: people who "sit between people who would otherwise be far apart". Choose it and the dots on the map are resized by betweenness, so the brokers stand out. Related: [brokers and cut points](https://netgraf.isik.co/learn/brokers-and-bridges), the people whose removal would actually split the map.

## Questions

### What does high betweenness centrality mean?

It means the person lies on many of the shortest routes between other people. They are well placed to broker, filter or block what flows through the network, and the network depends on them more than their number of ties suggests.

### How is betweenness different from degree?

Degree counts a person's own ties. Betweenness looks at everyone else's routes. A person with only two ties can have the highest betweenness if those ties are the only link between two groups, as Heather does in Krackhardt's kite.

## Example maps

- https://netgraf.isik.co/m/florentine-families
- https://netgraf.isik.co/m/krackhardt-kite

## References

- Freeman, L. C. (1977). A set of measures of centrality based on betweenness. Sociometry, 40(1), 35-41. https://doi.org/10.2307/3033543
- Brandes, U. (2001). A faster algorithm for betweenness centrality. Journal of Mathematical Sociology, 25(2), 163-177. https://doi.org/10.1080/0022250X.2001.9990249
- Padgett, J. F., & Ansell, C. K. (1993). Robust action and the rise of the Medici, 1400-1434. American Journal of Sociology, 98(6), 1259-1319. https://doi.org/10.1086/230190