betweenness
Compute the Betweenness Centrality Scores of Network Positions
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, ignore.eval=TRUE)
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.
gmode
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
isFALSE
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, or a variant form  see below).
 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.
 ignore.eval
 logical; ignore edge values when computing shortest paths?
Details
The shortestpath betweenness of a vertex, $v$, is given by
$$ C_B(v) = \sum_{i,j : i \neq j, i \neq v, j \neq v} \frac{g_{ivj}}{g_{ij}}$$
where $g_ijk$ is the number of geodesics from $i$ to $k$ through $j$. Conceptually, highbetweenness vertices lie on a large number of nonredundant shortest paths between other vertices; they can thus be thought of as ``bridges'' or ``boundary spanners.''
Several variant forms of shortestpath betweenness exist, and can be selected using the cmode
argument. Supported options are as follows:
directed
 Standard betweenness (see above), calculated on directed pairs. (This is the default option.)
undirected
endpoints
proximalsrc
proximaltar
proximalsum
lengthscaled
linearscaled
See Brandes (2008) for details and additional references. Geodesics for all of the above can be calculated using valued edges by setting ignore.eval=TRUE
. Edge values are interpreted as distances for this purpose; proximity data should be transformed accordingly before invoking this routine.
Value

A vector, matrix, or list containing the betweenness scores (depending on the number and size of the input graphs).
Note
Judicious use of geodist.precomp
can save a great deal of time when computing multiple pathbased indices on the same network.
Warning
Rescale may cause unexpected results if all actors have zero betweenness.
References
Borgatti, S.P. and Everett, M.G. (2006). “A GraphTheoretic Perspective on Centrality.” Social Networks, 28, 466484.
Brandes, U. (2008). “On Variants of ShortestPath Betweenness Centrality and their Generic Computation.” Social Networks, 30, 136145.
Freeman, L.C. (1979). “Centrality in Social Networks I: Conceptual Clarification.” Social Networks, 1, 215239.
Geisberger, R., Sanders, P., and Schultes, D. (2008). “Better Approximation of Betweenness Centrality.” In Proceedings of the 10th Workshop on Algorithm Engineering and Experimentation (ALENEX'08), 90100. SIAM.
See Also
Examples
g<rgraph(10) #Draw a random graph with 10 members
betweenness(g) #Compute betweenness scores