Vertex and edge betweenness centrality

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

betweenness(
  graph,
  v = V(graph),
  directed = TRUE,
  weights = NULL,
  normalized = FALSE,
  cutoff = -1
)

edge_betweenness(
  graph,
  e = E(graph),
  directed = TRUE,
  weights = NULL,
  cutoff = -1
)

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, $$B^n=\frac{2B}{(n-1)(n-2)},$$ where \(B^n\) is the normalized, \(B\) the raw betweenness, and \(n\) is the number of vertices in the graph. Note that the same normalization factor is used even when setting a cutoff on the considered shortest path lengths, even though the number of vertex pairs reachable from each other may be less than \((n-1)(n-2)/2\).

cutoff

The maximum shortest path length to consider when calculating betweenness. If 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

$$\sum_{i\ne j, i\ne v, j\ne v} g_{ivj}/g_{ij}$$

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.

Here \(g_{ij}\) is the total number of shortest paths between vertices \(i\) and \(j\) while \(g_{ivj}\) 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.

Note

edge_betweenness() might give false values for graphs with multiple edges.

References

Freeman, L.C. (1979). Centrality in Social Networks I: Conceptual Clarification. Social Networks, 1, 215-239. doi:10.1016/0378-8733(78)90021-7

Ulrik Brandes, A Faster Algorithm for Betweenness Centrality. Journal of Mathematical Sociology 25(2):163-177, 2001. doi:10.1080/0022250X.2001.9990249

See also

closeness(), degree(), harmonic_centrality()

Centrality measures alpha_centrality(), authority_score(), closeness(), diversity(), eigen_centrality(), harmonic_centrality(), hits_scores(), 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