igraph (version 0.5.2-2)

betweenness: 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, 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.

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.

concept

  • Betweenness centrality
  • Edge betweenness

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.

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

closeness, degree

Examples

Run this code
g <- random.graph.game(10, 3/10)
betweenness(g)
edge.betweenness(g)

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