# 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)
edge.betweenness(graph, e=E(graph), directed = TRUE)
```

##### 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.

##### 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

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

##### 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`

.For calculating the betweenness a similar algorithm to the one proposed by Brandes is used.

##### Note

`edge.betweenness`

might give false values for graphs with
multiple edges.

##### 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 0.4.4, License: GPL version 2 or later (June, 1991)*