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LinkRankMatrix[g]

returns the link rank of the graph g, in the form of a sparse matrix. The link rank of an edge u->v is defined as the PageRanks of u, divided by the out-degree of u.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Applications  
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GraphUtilities`
GraphUtilities`

LinkRankMatrix

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

LinkRankMatrix[g]

returns the link rank of the graph g, in the form of a sparse matrix. The link rank of an edge u->v is defined as the PageRanks of u, divided by the out-degree of u.

Details and Options

  • LinkRankMatrix functionality is now available in the built-in Wolfram Language function LinkRankCentrality.
  • To use LinkRankMatrix, you first need to load the Graph Utilities Package using Needs["GraphUtilities`"].
  • The following options can be given:
  • ToleranceAutomatictolerance used for convergence check
    TeleportProbability0.15probability of visiting random nodes
    RemoveSinksTruewhether to remove sinks by linking them with every node
  • The link rank of a link from vertex i to vertex j is defined as page rank of i, as given by PageRanks[g], divided by the out-degree of i.
  • The link rank reflects the probability that a random surfer follows that link.
  • LinkRankMatrix has the same options as PageRanks.

Examples

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Basic Examples  (2)

This shows a small network of web pages:

Wolfram Language code: Needs["GraphUtilities`"]
Wolfram Language code: g = {"home" -> "contact", "home" -> "links", "home" -> "work", "links" -> "friend site", "friend site" -> "home", "work" -> "home", "contact" -> "home"};

This calculates the link ranks:

Wolfram Language code: LinkRankMatrix[g] // MatrixForm

LinkRankMatrix has been superseded by LinkRankCentrality:

Wolfram Language code: g = Graph[{"home" -> "contact", "home" -> "links", "home" -> "work", "links" -> "friend site", "friend site" -> "home", "work" -> "home", "contact" -> "home"}];
Wolfram Language code: LinkRankCentrality[g]
Wolfram Language code: WeightedAdjacencyMatrix[g, EdgeWeight -> %]//MatrixForm

Applications  (1)

This shows a small network of web pages:

Wolfram Language code: Needs["GraphUtilities`"]
Wolfram Language code: g = {"home" -> "contact", "home" -> "links", "home" -> "work", "links" -> "friend site", "friend site" -> "home", "work" -> "home", "contact" -> "home"};
Wolfram Language code: GraphPlot[g, VertexLabeling -> True, DirectedEdges -> True]

This calculates the link ranks:

Wolfram Language code: lm = LinkRankMatrix[g]

This replots the network with link rank information:

Wolfram Language code: GraphPlot[g, VertexLabeling -> True, EdgeRenderingFunction -> ({RGBColor[0.5, 0., 0.], Arrowheads[{{0.03, 0.8}}], Arrow[#1], Text[lm[[First[#4], Last[#4]]], LineScaledCoordinate[#1, .6], Background -> White]}&), DirectedEdges -> True]
Wolfram Research (2007), LinkRankMatrix, Wolfram Language function, https://reference.wolfram.com/language/GraphUtilities/ref/LinkRankMatrix.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_linkrankmatrix, author="Wolfram Research", title="{LinkRankMatrix}", year="2007", howpublished="\url{https://reference.wolfram.com/language/GraphUtilities/ref/LinkRankMatrix.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_linkrankmatrix, organization={Wolfram Research}, title={LinkRankMatrix}, year={2007}, url={https://reference.wolfram.com/language/GraphUtilities/ref/LinkRankMatrix.html}, note=[Accessed: 13-June-2026]}