KatzCentrality
✖
KatzCentrality
gives a list of Katz centralities for the vertices in the graph g and weight α.
gives a list of Katz centralities using weight α and initial centralities β.
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 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
open allclose allBasic Examples (2)Summary of the most common use cases

https://wolfram.com/xid/0bnjdqup2gq0e-dprrh4

https://wolfram.com/xid/0bnjdqup2gq0e-3cvhh


https://wolfram.com/xid/0bnjdqup2gq0e-cut8tc

Rank vertices by centrality; higher value means more influence:

https://wolfram.com/xid/0bnjdqup2gq0e-gtfc0a

https://wolfram.com/xid/0bnjdqup2gq0e-dlqxo3

Scope (8)Survey of the scope of standard use cases
KatzCentrality works with undirected graphs:

https://wolfram.com/xid/0bnjdqup2gq0e-u3xie


https://wolfram.com/xid/0bnjdqup2gq0e-bgzrn4


https://wolfram.com/xid/0bnjdqup2gq0e-15kl6n


https://wolfram.com/xid/0bnjdqup2gq0e-czvddh

Use rules to specify the graph:

https://wolfram.com/xid/0bnjdqup2gq0e-bndh30


https://wolfram.com/xid/0bnjdqup2gq0e-blth2j

Nondefault initial centralities:

https://wolfram.com/xid/0bnjdqup2gq0e-coe4mi

KatzCentrality works with large graphs:

https://wolfram.com/xid/0bnjdqup2gq0e-pq9ae

https://wolfram.com/xid/0bnjdqup2gq0e-ed6xxu

Options (3)Common values & functionality for each option
WorkingPrecision (3)
By default, KatzCentrality finds centralities using machine-precision computations:

https://wolfram.com/xid/0bnjdqup2gq0e-bg8b8

Specify a higher working precision:

https://wolfram.com/xid/0bnjdqup2gq0e-e6isci

Infinite working precision corresponds to exact computation:

https://wolfram.com/xid/0bnjdqup2gq0e-jsrmu

Applications (6)Sample problems that can be solved with this function
Rank vertices of a graph by their importance in their reachable neighborhood:

https://wolfram.com/xid/0bnjdqup2gq0e-61b1r

https://wolfram.com/xid/0bnjdqup2gq0e-gi24yp

Highlight the Katz centrality for CycleGraph:

https://wolfram.com/xid/0bnjdqup2gq0e-gzipus

https://wolfram.com/xid/0bnjdqup2gq0e-baipzx

https://wolfram.com/xid/0bnjdqup2gq0e-g2btke

https://wolfram.com/xid/0bnjdqup2gq0e-h351s2


https://wolfram.com/xid/0bnjdqup2gq0e-jbz7i6

https://wolfram.com/xid/0bnjdqup2gq0e-brg6kn

https://wolfram.com/xid/0bnjdqup2gq0e-hyiutu


https://wolfram.com/xid/0bnjdqup2gq0e-gca8u7

https://wolfram.com/xid/0bnjdqup2gq0e-gcylrp

https://wolfram.com/xid/0bnjdqup2gq0e-bp9wqg


https://wolfram.com/xid/0bnjdqup2gq0e-6ri66

https://wolfram.com/xid/0bnjdqup2gq0e-fh33ql

https://wolfram.com/xid/0bnjdqup2gq0e-iop80


https://wolfram.com/xid/0bnjdqup2gq0e-g7rx5j
Find the top five most important papers and highlight them:

https://wolfram.com/xid/0bnjdqup2gq0e-bd3s0w

https://wolfram.com/xid/0bnjdqup2gq0e-bu54c9


https://wolfram.com/xid/0bnjdqup2gq0e-cla5c9

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

https://wolfram.com/xid/0bnjdqup2gq0e-qdja1e


https://wolfram.com/xid/0bnjdqup2gq0e-g1rv5s


https://wolfram.com/xid/0bnjdqup2gq0e-xq0r9

Find the most common ancestor in a family tree:

https://wolfram.com/xid/0bnjdqup2gq0e-d0a99b

https://wolfram.com/xid/0bnjdqup2gq0e-ezawh7

https://wolfram.com/xid/0bnjdqup2gq0e-fol7w6

Find descendants at the bottom of the tree:

https://wolfram.com/xid/0bnjdqup2gq0e-evkuo3

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

https://wolfram.com/xid/0bnjdqup2gq0e-ccfrz

https://wolfram.com/xid/0bnjdqup2gq0e-buk49l

Employees who are less likely to influence others:

https://wolfram.com/xid/0bnjdqup2gq0e-dqlp66

Properties & Relations (4)Properties of the function, and connections to other functions
The centrality vector satisfies the equation
:

https://wolfram.com/xid/0bnjdqup2gq0e-lc2wfi

https://wolfram.com/xid/0bnjdqup2gq0e-c1iwez

https://wolfram.com/xid/0bnjdqup2gq0e-mxrd2o

EigenvectorCentrality is a special case of KatzCentrality:

https://wolfram.com/xid/0bnjdqup2gq0e-jb9j3c

https://wolfram.com/xid/0bnjdqup2gq0e-hlzxw7

Take and
with
the largest eigenvalue of the adjacency matrix:

https://wolfram.com/xid/0bnjdqup2gq0e-g5w1yu


https://wolfram.com/xid/0bnjdqup2gq0e-fxy593

As , all vertices get the same centrality:

https://wolfram.com/xid/0bnjdqup2gq0e-i0eknx


https://wolfram.com/xid/0bnjdqup2gq0e-kvbm7

Use VertexIndex to obtain the centrality of a specific vertex:

https://wolfram.com/xid/0bnjdqup2gq0e-szwpd

https://wolfram.com/xid/0bnjdqup2gq0e-bbshqt

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
]}
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
]}