# Difference between revisions of "Closeness"

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− | '''Closeness''' is a radial measure of centrality that favors actors who are connected with many others via short paths. | + | '''Closeness''' is a radial measure of centrality that favors actors who are connected with many others via short paths. Intuitively, if the graph represents a transportation network, then a node with high closeness would make a good location for a warehouse since the average distance to all other locations (i.e., all other nodes in the graph) is relatively short. In information-spreading networks, a node with high closeness centrality would be a good choice to start a rumor since many others can be reached with relatively few intermediates. |

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== Definition (simple case) == | == Definition (simple case) == | ||

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<math>c_C(v)=\frac{|V|-1}{\sum\limits_{t\in V\setminus v} d_G(v,t)}</math>, | <math>c_C(v)=\frac{|V|-1}{\sum\limits_{t\in V\setminus v} d_G(v,t)}</math>, | ||

− | where <math>d_G(v,t)</math> denotes the length of a [[Shortest path|shortest directed path]] from <math>v</math> to <math>t</math>. | + | where <math>d_G(v,t)</math> denotes the length of a [[Shortest path|shortest directed path]] from <math>v</math> to <math>t</math>. The definition for [[Connectivity|connected]] undirected graphs is identical with <math>d_G(v,t)</math> being defined as the length of a [[Shortest path|shortest path]]. |

== Example == | == Example == | ||

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=== Edge weights and distances === | === Edge weights and distances === | ||

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+ | If a [[link strength]] has been selected, the length of an <math>(v,t)</math>-path is the sum of the corresponding attribute values of all links in the path. | ||

== Implementation in visone == | == Implementation in visone == |

## Revision as of 13:29, 30 March 2011

**Closeness** is a radial measure of centrality that favors actors who are connected with many others via short paths. Intuitively, if the graph represents a transportation network, then a node with high closeness would make a good location for a warehouse since the average distance to all other locations (i.e., all other nodes in the graph) is relatively short. In information-spreading networks, a node with high closeness centrality would be a good choice to start a rumor since many others can be reached with relatively few intermediates.

## Definition (simple case)

On directed, unweighted graphs that are strongly connected, the closeness centrality of a node is defined as

,

where denotes the length of a shortest directed path from to . The definition for connected undirected graphs is identical with being defined as the length of a shortest path.

## Example

## Special cases

### Unconnected graphs

### Edge weights and distances

If a link strength has been selected, the length of an -path is the sum of the corresponding attribute values of all links in the path.