# Difference between revisions of "Closeness"

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

− | On directed, unweighted graphs <math>G=(V,E)</math> that are [[Connectivity|strongly connected]], the closeness centrality <math>c_C(v)</math> of a node <math>v\in V</math> is defined as | + | On directed, unweighted graphs <math>\displaystyle{G=(V,E)}</math> that are [[Connectivity|strongly connected]], the closeness centrality <math>\displaystyle{c_C(v)}</math> of a node <math>v\in V</math> is defined as |

<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>. 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]]. | + | where <math>\displaystyle{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>\displaystyle{d_G(v,t)}</math> being defined as the length of a [[Shortest path|shortest path]]. |

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

## Revision as of 07:55, 11 April 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.