The vertex and edge betweenness are (roughly) defined by the number of geodesics (shortest paths) going through a vertex or an edge.
Arguments
- graph
The graph to analyze.
- v
The vertices for which the vertex betweenness will be calculated.
- directed
Logical, whether directed paths should be considered while determining the shortest paths.
- weights
Optional positive weight vector for calculating weighted betweenness. If the graph has a
weight
edge attribute, then this is used by default. Weights are used to calculate weighted shortest paths, so they are interpreted as distances.- normalized
Logical scalar, whether to normalize the betweenness scores. If
TRUE
, then the results are normalized by the number of ordered or unordered vertex pairs in directed and undirected graphs, respectively. In an undirected graph, Bn=2B(n−1)(n−2), where Bn is the normalized, B the raw betweenness, and n is the number of vertices in the graph.- cutoff
The maximum path length to consider when calculating the betweenness. If zero or negative then there is no such limit.
- e
The edges for which the edge betweenness will be calculated.
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()
.
Details
The vertex betweenness of vertex v
is defined by
∑i≠j,i≠v,j≠vgivj/gij
The edge betweenness of edge e
is defined by
∑i≠jgiej/gij.
betweenness()
calculates vertex betweenness, edge_betweenness()
calculates edge betweenness.
Here gij is the total number of shortest paths between vertices i and j while givj is the number of those shortest paths which pass though vertex v.
Both functions allow you to consider only 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 negative (the default),
then the function calculates the exact betweenness scores. Since igraph 1.6.0,
a cutoff
value of zero is treated literally, i.e. paths of length larger
than zero are ignored.
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()
, harmonic_centrality()
Centrality measures
alpha_centrality()
,
closeness()
,
diversity()
,
eigen_centrality()
,
harmonic_centrality()
,
hub_score()
,
page_rank()
,
power_centrality()
,
spectrum()
,
strength()
,
subgraph_centrality()
Author
Gabor Csardi csardi.gabor@gmail.com
Examples
g <- sample_gnp(10, 3 / 10)
betweenness(g)
#> [1] 0 0 0 14 0 6 6 0 0 0
edge_betweenness(g)
#> [1] 4 2 7 8 12 1 4 2 7 7