igraph (version 0.7.0)

closeness: Closeness centrality of vertices

Description

Cloness centrality measures how many steps is required to access every other vertex from a given vertex.

Usage

closeness(graph, vids=V(graph), mode = c("out", "in", "all", "total"),
        weights = NULL, normalized = FALSE)
closeness.estimate(graph, vids=V(graph), mode = c("out", "in", "all",
       "total"), cutoff, weights = NULL, normalized = FALSE)

Arguments

graph

The graph to analyze.

vids

The vertices for which closeness will be calculated.

mode

Character string, defined the types of the paths used for measuring the distance in directed graphs. in measures the paths to a vertex, out measures paths from a vertex, all uses und

normalized

Logical scalar, whether to calculate the normalized closeness. Normalization is performed by multiplying the raw closeness by $n-1$, where $n$ 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.

weights

Optional positive weight vector for calculating weighted closeness. If the graph has a weight edge attribute, then this is used by default.

Value

• Numeric vector with the closeness values of all the vertices in v.

concept

Closeness centrality

Details

The closeness centrality of a vertex is defined by the inverse of the average length of the shortest paths to/from all the other vertices in the graph:

$$\frac{1}{\sum_{i\ne v} d_vi}$$

If there is no (directed) path between vertex $v$ and $i$ then the total number of vertices is used in the formula instead of the path length.

closeness.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 closeness scores.

References

Freeman, L.C. (1979). Centrality in Social Networks I: Conceptual Clarification. Social Networks, 1, 215-239.

See Also

betweenness, degree

Examples

g <- graph.ring(10)
g2 <- graph.star(10)
closeness(g)
closeness(g2, mode="in")
closeness(g2, mode="out")
closeness(g2, mode="all")