# Closeness centrality

> Closeness centrality measures how few steps a person needs to reach everyone else. Classic and harmonic closeness, why they differ, and a worked example.

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

Closeness centrality measures how near a person is to everyone else in the network, counted in steps along ties. A person with high closeness can reach the whole group quickly, which makes them well placed to spread news or notice what is happening anywhere in the network.

## How it's calculated

```
classic closeness(v)  = (n - 1) / sum of distance(v, u) over all u
harmonic closeness(v) = (1 / (n - 1)) * sum of 1 / distance(v, u) over all u
```

distance(v, u) is the number of steps on the shortest path. Both forms run from 0 to 1.

The classic form breaks the moment a network has two separate islands, because the distance to an unreachable person is infinite. Harmonic closeness, proposed by Marchiori and Latora in 2000, sums 1 / distance instead, so an unreachable person simply adds 0. Netgraf uses the harmonic form because real relationship maps very often have islands in them.

## The kite, and why the measures disagree

Krackhardt drew his kite so that three people would each be "most central" by a different measure: Diane by degree, Heather by betweenness, and Fernando and Garth by classic closeness, since they have the shortest total distance to everyone. Harmonic closeness rewards many one-step ties more heavily, so on the kite it puts Diane narrowly ahead of Fernando and Garth. The live figures below show how close it is. Neither is wrong; they weigh near and far contacts differently.

## In Netgraf

Closeness is exported for every person in the People CSV and in GraphML, next to degree, betweenness, PageRank (as influence) and clustering. The Insights panel's **Steps apart** figure is the related average: the mean shortest path between any two people in the largest connected group.

## Questions

### What is the difference between closeness and betweenness centrality?

Closeness asks how quickly a person can reach everyone. Betweenness asks how often a person is on other people's routes. A person at the centre of a dense group has high closeness; a person joining two groups has high betweenness.

### Why use harmonic closeness?

Because classic closeness is undefined on a network with disconnected parts, and many real networks have them. Harmonic closeness treats unreachable people as contributing nothing instead of breaking the whole calculation.

## Example maps

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

## References

- Freeman, L. C. (1978). Centrality in social networks: Conceptual clarification. Social Networks, 1(3), 215-239. https://doi.org/10.1016/0378-8733(78)90021-7
- Marchiori, M., & Latora, V. (2000). Harmony in the small-world. Physica A, 285(3-4), 539-546. https://doi.org/10.1016/S0378-4371(00)00311-3
- Krackhardt, D. (1990). Assessing the political landscape: Structure, cognition, and power in organizations. Administrative Science Quarterly, 35(2), 342-369. https://doi.org/10.2307/2393394