Closeness centrality measures how many steps is required to access every other vertex from a given vertex.

## Arguments

- graph
The graph to analyze.

- vids
The vertices for which closeness will be calculated.

- mode
Character string, defined the types of the paths used for measuring the distance in directed graphs. “in” measures the paths

*to*a vertex, “out” measures paths*from*a vertex,*all*uses undirected paths. This argument is ignored for undirected graphs.- weights
Optional positive weight vector for calculating weighted closeness. If the graph has a

`weight`

edge attribute, then this is used by default. Weights are used for calculating weighted shortest paths, so they are interpreted as distances.- normalized
Logical scalar, whether to calculate the normalized closeness, i.e. the inverse average distance to all reachable vertices. The non-normalized closeness is the inverse of the sum of distances to all reachable vertices.

- cutoff
The maximum path length to consider when calculating the closeness. If zero or negative then there is no such limit.

## Details

The closeness centrality of a vertex is defined as the inverse of the sum of distances to all the other vertices in the graph:

$$\frac{1}{\sum_{i\ne v} d_{vi}}$$

If there is no (directed) path between vertex `v`

and `i`

, then
`i`

is omitted from the calculation. If no other vertices are reachable
from `v`

, then its closeness is returned as NaN.

`cutoff`

or smaller. This can be run for larger graphs, as the running
time is not quadratic (if `cutoff`

is small). If `cutoff`

is
negative (which is the default), then the function calculates the exact
closeness scores. Since igraph 1.6.0, a `cutoff`

value of zero is treated
literally, i.e. path with a length greater than zero are ignored.

Closeness centrality is meaningful only for connected graphs. In disconnected
graphs, consider using the harmonic centrality with
`harmonic_centrality()`

## References

Freeman, L.C. (1979). Centrality in Social Networks I:
Conceptual Clarification. *Social Networks*, 1, 215-239.

## See also

Centrality measures
`alpha_centrality()`

,
`authority_score()`

,
`betweenness()`

,
`diversity()`

,
`eigen_centrality()`

,
`harmonic_centrality()`

,
`hits_scores()`

,
`page_rank()`

,
`power_centrality()`

,
`spectrum()`

,
`strength()`

,
`subgraph_centrality()`

## Author

Gabor Csardi csardi.gabor@gmail.com

## Examples

```
g <- make_ring(10)
g2 <- make_star(10)
closeness(g)
#> [1] 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04
closeness(g2, mode = "in")
#> [1] 0.1111111 NaN NaN NaN NaN NaN NaN
#> [8] NaN NaN NaN
closeness(g2, mode = "out")
#> [1] NaN 1 1 1 1 1 1 1 1 1
closeness(g2, mode = "all")
#> [1] 0.11111111 0.05882353 0.05882353 0.05882353 0.05882353 0.05882353
#> [7] 0.05882353 0.05882353 0.05882353 0.05882353
```