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HITSCentrality[g]

gives a list of authority and hub centralities for the vertices in the graph g.

HITSCentrality[{vw,…}]

uses rules vw to specify the graph g.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
WorkingPrecision  
Applications  
Properties & Relations  
See Also
Related Guides
History
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HITSCentrality

HITSCentrality[g]

gives a list of authority and hub centralities for the vertices in the graph g.

HITSCentrality[{vw,…}]

uses rules vw to specify the graph g.

Details and Options

  • HITSCentrality is also known as Kleinberg centrality.
  • HITSCentrality gives two lists of authority centralities and hub centralities for each vertex.
  • If the graph g has adjacency matrix , then authority centralities are given by TemplateBox[{a}, Transpose].a.x=lambda_1 x, where is the largest eigenvalue of TemplateBox[{a}, Transpose].a, and hub centralities are given by . »
  • The authority and hub centralities satisfy and x=1/lambda_1TemplateBox[{a}, Transpose].y. »
  • The option WorkingPrecision->p can be used to control the precision used in internal computations.
  • HITSCentrality works with undirected graphs, directed graphs, multigraphs, and mixed graphs.

Examples

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Basic Examples  (2)

Compute HITS hub and authority centralities:

Wolfram Language code: g = RandomGraph[{10, 20}];
Wolfram Language code: HITSCentrality[g]

Chart of the data:

Wolfram Language code: BarChart[Transpose[%], ChartLegends -> {"Authority", "Hub"}]

Rate web pages using hyperlink-induced topic search:

Wolfram Language code: g = [image];
Wolfram Language code: {aut, hub} = HITSCentrality[g]

Web pages with high hub centralities are connected to many other web pages:

Wolfram Language code: Part[VertexList[g], Ordering[hub, All, Greater]] //Short

Web pages with high authority centralities are connected from many other web pages:

Wolfram Language code: Part[VertexList[g], Ordering[aut, All, Greater]] //Short

Scope  (6)

HITSCentrality works with undirected graphs:

Wolfram Language code: HITSCentrality[[image]]

Directed graphs:

Wolfram Language code: HITSCentrality[[image]]

Multigraphs:

Wolfram Language code: HITSCentrality[[image]]

Mixed graphs:

Wolfram Language code: HITSCentrality[[image]]

Use rules to specify the graph:

Wolfram Language code: HITSCentrality[{1 -> 3, 2 -> 1, 3 -> 6, 4 -> 6, 1 -> 5, 5 -> 4, 6 -> 1}]

HITSCentrality works with large graphs:

Wolfram Language code: g = GridGraph[{10, 10, 10, 10}];
Wolfram Language code: HITSCentrality[g]//Short//Timing

Options  (3)

WorkingPrecision  (3)

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

Wolfram Language code: HITSCentrality[[image]]

Specify a higher working precision:

Wolfram Language code: HITSCentrality[[image], WorkingPrecision -> 50]

Infinite working precision corresponds to exact computation:

Wolfram Language code: HITSCentrality[[image], WorkingPrecision -> ∞]

Applications  (3)

Highlight the HITS authority and hub centralities for CycleGraph:

Wolfram Language code: HighlightCentrality[g_, cc_] := HighlightGraph[g, Table[Style[VertexList[g][[i]], ColorData["TemperatureMap"][cc[[i]] / Max[cc]]], {i, VertexCount[g]}]];
Wolfram Language code: g = CycleGraph[8, VertexSize -> Large];
Wolfram Language code: {ahc, hhc} = HITSCentrality[g];
Wolfram Language code: HighlightCentrality[g, #]& /@ %

GridGraph:

Wolfram Language code: g = GridGraph[{5, 5}, VertexSize -> 0.5];
Wolfram Language code: {ahc, hhc} = HITSCentrality[g];
Wolfram Language code: HighlightCentrality[g, #]& /@ %

CompleteKaryTree:

Wolfram Language code: g = CompleteKaryTree[3, 3, VertexSize -> Large];
Wolfram Language code: {ahc, hhc} = HITSCentrality[g];
Wolfram Language code: HighlightCentrality[g, #]& /@ %

PathGraph:

Wolfram Language code: g = PathGraph[Range[20], VertexSize -> Large];
Wolfram Language code: {ahc, hhc} = HITSCentrality[g];
Wolfram Language code: HighlightCentrality[g, #]& /@ %

A network of web pages linked via hyperlinks. Find the top five informative web pages:

Wolfram Language code: g = DynamicModule[«3»];
Wolfram Language code: {aut, hub} = HITSCentrality[g];

Find the top five pages containing authoritative information:

Wolfram Language code: Table[VertexList[g][[i]], {i, Ordering[aut, -5]}]

HITS authority ranking is highly correlated with in-degree ranking:

Wolfram Language code: Correlation[aut, DegreeCentrality[g, "In"]]

Find the top five pages containing links to authoritative pages:

Wolfram Language code: Table[VertexList[g][[i]], {i, Ordering[hub, -5]}]

HITS hub ranking is highly correlated with out-degree ranking:

Wolfram Language code: Correlation[hub, DegreeCentrality[g, "Out"]]

Properties & Relations  (4)

The authority and hub centrality can be found using the first eigenvector of :

Wolfram Language code: g = [image];
Wolfram Language code: {x, y} = HITSCentrality[g];
Wolfram Language code: a = N@AdjacencyMatrix[g];

Authority centrality:

Wolfram Language code: With[{u = Abs[First[Eigenvectors[a.a, 1]]]}, u / Norm[u, 1]]
Wolfram Language code: x

Hub centrality:

Wolfram Language code: a.x
Wolfram Language code: y

The authority and hub centralities satisfy and x=TemplateBox[{{{1, /, {lambda, _, 1}}, a}}, Transpose].y:

Wolfram Language code: g = [image];
Wolfram Language code: {x, y} = HITSCentrality[g]; a = N@AdjacencyMatrix[g]; λ1 = First@Eigenvalues[a.a, 1];
Wolfram Language code: {y == a.x, x == 1 / λ1 a.y}

The authority centrality is normalized:

Wolfram Language code: g = RandomGraph[{10, 20}];
Wolfram Language code: Total[First[HITSCentrality[g]]]

Use VertexIndex to obtain the authority centrality and hub centrality of a specific vertex:

Wolfram Language code: g = ExampleData[{"NetworkGraph", "Friendship"}];
Wolfram Language code: HITSCentrality[g][[All, VertexIndex[g, "Anna"]]]
Wolfram Research (2010), HITSCentrality, Wolfram Language function, https://reference.wolfram.com/language/ref/HITSCentrality.html (updated 2015).

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_hitscentrality, author="Wolfram Research", title="{HITSCentrality}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/HITSCentrality.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_hitscentrality, organization={Wolfram Research}, title={HITSCentrality}, year={2015}, url={https://reference.wolfram.com/language/ref/HITSCentrality.html}, note=[Accessed: 12-June-2026]}