WOLFRAM

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 c=alpha TemplateBox[{a}, Transpose].c+beta, 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 WorkingPrecision->p 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)Summary of the most common use cases

Compute Katz centralities:

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Highlight:

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Rank vertices by centrality; higher value means more influence:

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Scope  (8)Survey of the scope of standard use cases

KatzCentrality works with undirected graphs:

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Directed graphs:

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Multigraphs:

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Mixed graphs:

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Use rules to specify the graph:

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Use weights:

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Nondefault initial centralities:

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KatzCentrality works with large graphs:

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Options  (3)Common values & functionality for each option

WorkingPrecision  (3)

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

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Specify a higher working precision:

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Infinite working precision corresponds to exact computation:

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Applications  (6)Sample problems that can be solved with this function

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

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Highlight the Katz centrality for CycleGraph:

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GridGraph:

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CompleteKaryTree:

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PathGraph:

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Simulate a citation network:

Find the top five most important papers and highlight them:

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

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Show the partition:

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Find the most common ancestor in a family tree:

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Find descendants at the bottom of the tree:

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

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Employees who are less likely to influence others:

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Properties & Relations  (4)Properties of the function, and connections to other functions

The centrality vector satisfies the equation c=alpha TemplateBox[{a}, Transpose].c+beta:

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EigenvectorCentrality is a special case of KatzCentrality:

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Take and with the largest eigenvalue of the adjacency matrix:

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As , all vertices get the same centrality:

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Use VertexIndex to obtain the centrality of a specific vertex:

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

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.

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

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

BibTeX

@misc{reference.wolfram_2025_katzcentrality, author="Wolfram Research", title="{KatzCentrality}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/KatzCentrality.html}", note=[Accessed: 11-July-2025 ]}

@misc{reference.wolfram_2025_katzcentrality, author="Wolfram Research", title="{KatzCentrality}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/KatzCentrality.html}", note=[Accessed: 11-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_katzcentrality, organization={Wolfram Research}, title={KatzCentrality}, year={2015}, url={https://reference.wolfram.com/language/ref/KatzCentrality.html}, note=[Accessed: 11-July-2025 ]}

@online{reference.wolfram_2025_katzcentrality, organization={Wolfram Research}, title={KatzCentrality}, year={2015}, url={https://reference.wolfram.com/language/ref/KatzCentrality.html}, note=[Accessed: 11-July-2025 ]}