betweenness {igraph} R Documentation

## Vertex and edge betweenness centrality

### Description

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

### Usage

```betweenness(graph, v=V(graph), directed = TRUE)
edge.betweenness(graph, e=E(graph), directed = TRUE)
betweenness.estimate(graph, vids = V(graph), directed = TRUE, cutoff)
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. `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( g_ivj / g_ij, i!=j,i!=v,j!=v)

The edge betweenness of edge `e` is defined by

sum( g_iej / g_ij, i!=j)

.

`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.

### Author(s)

Gabor Csardi csardi@rmki.kfki.hu

### 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.

`closeness`, `degree`
```g <- random.graph.game(10, 3/10)