GraphUtilities`
GraphUtilities`

# ClosenessCentrality

As of Version 10, all the functionality of the GraphUtilities package is built into the Wolfram System. »

ClosenessCentrality[g]

finds the closeness centrality.

# Details and Options

• ClosenessCentrality functionality is now available in the built-in Wolfram Language function ClosenessCentrality.
• To use ClosenessCentrality, you first need to load the Graph Utilities Package using Needs["GraphUtilities`"].
• The closeness centrality of a vertex u is defined as the inverse of the sum of the distance from u to all other vertices. The closeness centrality of a vertex in a disconnected graph is based on the closeness centrality of the component where this vertex belongs.
• The following options can be given:
•  Weighted True whether edge weight is to be used in calculating distance Normalize False whether to normalize the output

# Examples

open allclose all

## Basic Examples(2)

This defines a small graph:

Compute the closeness centrality:

This function has been superseded by ClosenessCentrality in the Wolfram System:

## Scope(1)

This defines a disconnected graph and finds the closeness centrality:

## Options(1)

### Weighted(1)

This defines a graph with edge weights:

By default, edge weights are taken into account:

This gives the closeness centrality if edge weights are assumed to be 1.:

## Applications(1)

A plot of a grid graph with vertices of high centrality in red:

## Properties & Relations(1)

The centrality of a vertex that cannot reach all other vertices in its component is zero:

The centrality of a disconnected graph is calculated by treating each component separately:

Wolfram Research (2007), ClosenessCentrality, Wolfram Language function, https://reference.wolfram.com/language/GraphUtilities/ref/ClosenessCentrality.html.

#### Text

Wolfram Research (2007), ClosenessCentrality, Wolfram Language function, https://reference.wolfram.com/language/GraphUtilities/ref/ClosenessCentrality.html.

#### CMS

Wolfram Language. 2007. "ClosenessCentrality." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/GraphUtilities/ref/ClosenessCentrality.html.

#### APA

Wolfram Language. (2007). ClosenessCentrality. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/GraphUtilities/ref/ClosenessCentrality.html

#### BibTeX

@misc{reference.wolfram_2024_closenesscentrality, author="Wolfram Research", title="{ClosenessCentrality}", year="2007", howpublished="\url{https://reference.wolfram.com/language/GraphUtilities/ref/ClosenessCentrality.html}", note=[Accessed: 24-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_closenesscentrality, organization={Wolfram Research}, title={ClosenessCentrality}, year={2007}, url={https://reference.wolfram.com/language/GraphUtilities/ref/ClosenessCentrality.html}, note=[Accessed: 24-July-2024 ]}