# KatzCentrality

KatzCentrality[g,α]

gives a list of Katz centralities for the vertices in the graph g and weight α.

KatzCentrality[g,α,β]

gives a list of Katz centralities using weight α and initial centralities β.

KatzCentrality[{vw,},]

uses rules vw to specify the graph g.

# Details and Options

• KatzCentrality gives a list of centralities that satisfy , where is the adjacency matrix of g.
• If β is a scalar, it is taken to mean {β,β,}.
• KatzCentrality[g,α] is equivalent to KatzCentrality[g,α,1].
• The option can be used to control the precision used in internal computations.
• KatzCentrality works with undirected graphs, directed graphs, multigraphs, and mixed graphs.

# Examples

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## Basic Examples(2)

Compute Katz centralities:

Highlight:

Rank vertices by centrality; higher value means more influence:

## Scope(8)

KatzCentrality works with undirected graphs:

Directed graphs:

Multigraphs:

Mixed graphs:

Use rules to specify the graph:

Use weights:

Nondefault initial centralities:

KatzCentrality works with large graphs:

## Options(3)

### WorkingPrecision(3)

By default, KatzCentrality finds centralities using machine-precision computations:

Specify a higher working precision:

Infinite working precision corresponds to exact computation:

## Applications(6)

Rank vertices of a graph by their importance in their reachable neighborhood:

Highlight the Katz centrality for CycleGraph:

Simulate a citation network:

Find the top five most important papers and highlight them:

Predict a partition of the Zachary karate club in case of a conflict between influential members:

Show the partition:

Find the most common ancestor in a family tree:

Find descendants at the bottom of the tree:

In a trust network among employees, select employees who could efficiently spread the corporate culture:

Employees who are less likely to influence others:

## Properties & Relations(4)

The centrality vector satisfies the equation :

EigenvectorCentrality is a special case of KatzCentrality:

Take and with the largest eigenvalue of the adjacency matrix:

As , all vertices get the same centrality:

Use VertexIndex to obtain the centrality of a specific vertex:

Wolfram Research (2010), KatzCentrality, Wolfram Language function, https://reference.wolfram.com/language/ref/KatzCentrality.html (updated 2015).

#### Text

Wolfram Research (2010), KatzCentrality, Wolfram Language function, https://reference.wolfram.com/language/ref/KatzCentrality.html (updated 2015).

#### CMS

Wolfram Language. 2010. "KatzCentrality." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/KatzCentrality.html.

#### APA

Wolfram Language. (2010). KatzCentrality. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/KatzCentrality.html

#### BibTeX

@misc{reference.wolfram_2024_katzcentrality, author="Wolfram Research", title="{KatzCentrality}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/KatzCentrality.html}", note=[Accessed: 21-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_katzcentrality, organization={Wolfram Research}, title={KatzCentrality}, year={2015}, url={https://reference.wolfram.com/language/ref/KatzCentrality.html}, note=[Accessed: 21-June-2024 ]}