# betweenness

##### Vertex and edge betweenness centrality

The vertex and edge betweenness are (roughly) defined by the number of geodesics (shortest paths) going through a vertex or an edge.

- Keywords
- graphs

##### Usage

```
betweenness(graph, v=V(graph), directed = TRUE, verbose = igraph.par("verbose"))
edge.betweenness(graph, e=E(graph), directed = TRUE)
betweenness.estimate(graph, vids = V(graph), directed = TRUE, cutoff,
verbose = igraph.par("verbose"))
edge.betweenness.estimate(graph, directed = TRUE, cutoff)
```

##### Arguments

- graph
- The graph to analyze.
- v
- The vertices for which the vertex betweenness will be calculated.
- e
- The edges for which the edge betweenness will be calculated.
- directed
- Logical, whether directed paths should be considered while determining the shortest paths.
- verbose
- Logical, whether to show a progress bar.
- vids
- The vertices for which the vertex betweenness estimation will be calculated.
- cutoff
- The maximum path length to consider when calculating the betweenness. If zero or negative then there is no such limit.

##### Details

The vertex betweenness of vertex $v$ is defined by

$$\sum_{i\ne j, i\ne v, j\ne v} g_{ivj}/g_{ij}$$ The edge betweenness of edge $e$ is defined by

$$\sum_{i\ne j} g{iej}/g_{ij}$$.

`betweenness`

calculates vertex betweenness,
`edge.betweenness`

calculates edge.betweenness.

`betweenness.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 betweenness scores.

`edge.betweenness.estimate`

is similar, but for edges.

For calculating the betweenness a similar algorithm to the one proposed by Brandes (see References) is used.

##### Value

- A numeric vector with the betweenness score for each vertex in
`v`

for`betweenness`

.A numeric vector with the edge betweenness score for each edge in

`e`

for`edge.betweenness`

.`betweenness.estimate`

returns the estimated betweenness scores for vertices in`vids`

,`edge.betweenness.estimate`

the estimated edge betweenness score for*all*edges; both in a numeric vector.

##### Note

`edge.betweenness`

might give false values for graphs with
multiple edges.

##### concept

- Betweenness centrality
- Edge betweenness

##### References

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

Ulrik Brandes, A Faster Algorithm for Betweenness Centrality. *Journal
of Mathematical Sociology* 25(2):163-177, 2001.

##### See Also

##### Examples

```
g <- random.graph.game(10, 3/10)
betweenness(g)
edge.betweenness(g)
```

*Documentation reproduced from package igraph, version 0.5.5-3, License: GPL (>= 2)*