# closeness

##### Closeness centrality of vertices

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

- Keywords
- graphs

##### Usage

```
closeness(graph, vids = V(graph), mode = c("out", "in", "all", "total"),
weights = NULL, normalized = FALSE)
```estimate_closeness(graph, vids = V(graph), mode = c("out", "in", "all",
"total"), cutoff, weights = NULL, normalized = FALSE)

##### 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. Normalization is performed by multiplying the raw closeness by \(n-1\), where \(n\) is the number of vertices in the graph.

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

##### Details

The closeness centrality of a vertex is defined by the inverse of the average length of the shortest paths to/from 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 the total number of vertices is used in the formula instead of the path length.

`estimate_closeness`

only considers paths of length `cutoff`

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

is small). If `cutoff`

is zero or negative
then the function calculates the exact closeness scores.

##### Value

Numeric vector with the closeness values of all the vertices in
`v`

.

##### References

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

##### See Also

##### Examples

`library(igraph)`

```
# NOT RUN {
g <- make_ring(10)
g2 <- make_star(10)
closeness(g)
closeness(g2, mode="in")
closeness(g2, mode="out")
closeness(g2, mode="all")
# }
```

*Documentation reproduced from package igraph, version 1.1.2, License: GPL (>= 2)*