# 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)
closeness.estimate(graph, vids=V(graph), mode = c("out", "in", "all",
"total"), cutoff, weights = NULL)
```

##### 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 und - 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.
- weights
- Optional positive weight vector for calculating
weighted closeness. If the graph has a
`weight`

edge attribute, then this is used by default.

##### 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.

`closeness.estimate`

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`

.

##### concept

Closeness centrality

##### References

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

##### See Also

##### Examples

```
g <- graph.ring(10)
g2 <- graph.star(10)
closeness(g)
closeness(g2, mode="in")
closeness(g2, mode="out")
closeness(g2, mode="all")
```

*Documentation reproduced from package igraph, version 0.6.5-2, License: GPL (>= 2)*