# closeness: Closeness centrality of vertices In igraph: Network Analysis and Visualization

## Description

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

## Usage

 ```1 2 3 4 5``` ```closeness(graph, vids = V(graph), mode = c("out", "in", "all", "total"), weights = NULL, normalized = FALSE) estimate_closeness(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 undirected paths. This argument is ignored for undirected graphs. `weights` Optional positive weight vector for calculating weighted closeness. If the graph has a `weight` edge attribute, then this is used by default. Weights are used for calculating weighted shortest paths, so they are interpreted as distances. `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.

## 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:

1/sum( d(v,i), i != v)

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

`estimate_closeness` 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.

## Value

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

## Author(s)

Gabor Csardi [email protected]

## References

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

`betweenness`, `degree`
 ```1 2 3 4 5 6``` ```g <- make_ring(10) g2 <- make_star(10) closeness(g) closeness(g2, mode="in") closeness(g2, mode="out") closeness(g2, mode="all") ```