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

```
estimate_betweenness(
graph,
vids = V(graph),
directed = TRUE,
cutoff,
weights = NULL,
nobigint = TRUE
)
```betweenness(
graph,
v = V(graph),
directed = TRUE,
weights = NULL,
nobigint = TRUE,
normalized = FALSE
)

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

graph

The graph to analyze.

vids

The vertices for which the vertex betweenness estimation will be calculated.

directed

Logical, whether directed paths should be considered while determining the shortest paths.

cutoff

The maximum path length to consider when calculating the betweenness. If zero or negative then there is no such limit.

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.

nobigint

Logical scalar, whether to use big integers during the
calculation. This is only required for lattice-like graphs that have very
many shortest paths between a pair of vertices. If `TRUE`

(the
default), then big integers are not used.

v

The vertices for which the vertex betweenness will be calculated.

normalized

Logical scalar, whether to normalize the betweenness
scores. If `TRUE`

, then the results are normalized according to
$$B^n=\frac{2B}{n^2-3n+2}$$, where
\(B^n\) is the normalized, \(B\) the raw betweenness, and \(n\)
is the number of vertices in the graph.

e

The edges for which the edge betweenness will be calculated.

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`

.

`estimate_betweenness`

returns the estimated betweenness scores for
vertices in `vids`

, `estimate_edge_betweenness`

the estimated edge
betweenness score for *all* edges; both in a numeric vector.

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.

`estimate_betweenness`

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.

`estimate_edge_betweenness`

is similar, but for edges.

For calculating the betweenness a similar algorithm to the one proposed by Brandes (see References) is used.

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.

```
# NOT RUN {
g <- sample_gnp(10, 3/10)
betweenness(g)
edge_betweenness(g)
# }
```

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