betweenness {sna} | R Documentation

## Compute the Betweenness Centrality Scores of Network Positions

### Description

`betweenness` takes one or more graphs (`dat`) and returns the betweenness centralities of positions (selected by `nodes`) within the graphs indicated by `g`. Depending on the specified mode, betweenness on directed or undirected geodesics will be returned; this function is compatible with `centralization`, and will return the theoretical maximum absolute deviation (from maximum) conditional on size (which is used by `centralization` to normalize the observed centralization score).

### Usage

    betweenness(dat, g=1, nodes=NULL, gmode="digraph", diag=FALSE,
        tmaxdev=FALSE, cmode="directed", geodist.precomp=NULL,
        rescale=FALSE)

### Arguments

`dat` | one or more input graphs.
`g` | integer indicating the index of the graph for which centralities are to be calculated (or a vector thereof). By default, `g`=1.
`nodes` | vector indicating which nodes are to be included in the calculation. By default, all nodes are included.
`gmode` | string indicating the type of graph being evaluated. "digraph" indicates that edges should be interpreted as directed; "graph" indicates that edges are undirected. `dmode` is set to "digraph" by default.
`diag` | boolean indicating whether or not the diagonal should be treated as valid data. Set this true if and only if the data can contain loops. `diag` is `FALSE` by default.
`tmaxdev` | boolean indicating whether or not the theoretical maximum absolute deviation from the maximum nodal centrality should be returned. By default, `tmaxdev`==`FALSE`.
`cmode` | string indicating the type of betweenness centrality being computed (directed or undirected geodesics).
`geodist.precomp` | A `geodist` object precomputed for the graph to be analyzed (optional)
`rescale` | if true, centrality scores are rescaled such that they sum to 1.

### Details

The betweenness of a vertex, v, is given by

C_B(v) = sum( g_ivj / g_ij, i,j: i!=j,i!=v,j!=v )

where g_ijk is the number of geodesics from i to k through j. Conceptually, high-betweenness vertices lie on a large number of non-redundant shortest paths between other vertices; they can thus be thought of as ``bridges'' or ``boundary spanners.''

### Value

A vector, matrix, or list containing the betweenness scores (depending on the number and size of the input graphs).

### Warning

Rescale may cause unexpected results if all actors have zero betweenness.

### Note

Judicious use of `geodist.precomp` can save a great deal of time when computing multiple path-based indices on the same network.

### Author(s)

Carter T. Butts buttsc@uci.edu

### References

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

### See Also

`centralization`

### Examples

    g<-rgraph(10)     #Draw a random graph with 10 members
    betweenness(g)    #Compute betweenness scores

[Package sna version 1.1 Index]