# Difference between revisions of "Closeness"

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where <math>d_G(v,t)</math> denotes the length of a shortest directed path from <math>v</math> to <math>t</math>. | where <math>d_G(v,t)</math> denotes the length of a shortest directed path from <math>v</math> to <math>t</math>. | ||

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. | 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. | ||

+ | |||

+ | |||

+ | == 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 | ||

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

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+ | 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>. | ||

+ | |||

+ | == Example == | ||

+ | |||

+ | == Special cases == | ||

+ | |||

+ | === Unconnected graphs === | ||

+ | |||

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

+ | |||

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

+ | |||

+ | === Normalization and treatment of special cases === | ||

+ | |||

+ | === Algorithmic runtime === | ||

+ | |||

+ | == Related measures == | ||

+ | |||

+ | == References == |

## Revision as of 12:25, 30 March 2011

**Closeness** is a radial measure of centrality that favors actors who are connected with many others via short paths.

The raw closeness score of a node is defined as the reciprocal of its total distance to those it can reach via directed paths,

where denotes the length of a shortest directed path from to . 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.

## 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 .