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GraphLinkEfficiency[g]

gives the link efficiency of the graph g.

GraphLinkEfficiency[{vw,…}]

uses rules vw to specify the graph g.

Details
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Examples  
Basic Examples  
Scope  
Applications  
Properties & Relations  
Possible Issues  
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GraphLinkEfficiency

GraphLinkEfficiency[g]

gives the link efficiency of the graph g.

GraphLinkEfficiency[{vw,…}]

uses rules vw to specify the graph g.

Details

Examples

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Basic Examples  (2)

Find the link efficiency in the graph:

Wolfram Language code: GraphLinkEfficiency[[image]]

Graph link efficiency distribution of the Watts–Strogatz graph model:

Wolfram Language code: 𝒟 = GraphPropertyDistribution[GraphLinkEfficiency[g], gWattsStrogatzGraphDistribution[50, 0.1, 3]];
Wolfram Language code: Histogram[RandomVariate[𝒟, 1000]]

Scope  (4)

GraphLinkEfficiency works with undirected graphs:

Wolfram Language code: GraphLinkEfficiency[[image]]

Directed graphs:

Wolfram Language code: GraphLinkEfficiency[[image]]

Use rules to specify the graph:

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

GraphLinkEfficiency works with large graphs:

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

Applications  (2)

Find how tightly connected the overground line of the London Underground is with respect to the number of lines between stations:

Wolfram Language code: N[GraphLinkEfficiency[[image]]]

Analyze the distribution of link efficiency in the Watts–Strogatz graph model:

Wolfram Language code: 𝒢 = WattsStrogatzGraphDistribution[50, 0.1, 3];
Wolfram Language code: RandomGraph[𝒢]

Distribution of link efficiency:

Wolfram Language code: 𝒟 = GraphPropertyDistribution[GraphLinkEfficiency[g], g𝒢];
Wolfram Language code: Histogram[RandomVariate[𝒟, 1000]]

Expected value:

Wolfram Language code: NExpectation[x, x𝒟]

Properties & Relations  (5)

GraphLinkEfficiency is related to MeanGraphDistance:

Wolfram Language code: g = KnightTourGraph[7, 4]
Wolfram Language code: GraphLinkEfficiency[g]
Wolfram Language code: 1 - MeanGraphDistance[g] / EdgeCount[g]

The GraphLinkEfficiency is always less than 1:

Wolfram Language code: N[GraphLinkEfficiency[RandomGraph[{10, 20}]]]

The GraphLinkEfficiency of a complete graph is close to 1:

Wolfram Language code: g = CompleteGraph[9]
Wolfram Language code: N[GraphLinkEfficiency[g]]

Test if a graph is complete using CompleteGraphQ:

Wolfram Language code: CompleteGraphQ[g]

The GraphLinkEfficiency of a path graph of length 1 is 0:

Wolfram Language code: PathGraph[Range[2]]
Wolfram Language code: GraphLinkEfficiency[%]

The GraphLinkEfficiency of a disconnected graph is -∞:

Wolfram Language code: graphs = {Graph[{12, 34}], Graph[{12, 34}], Graph[{12, 23, 13}]}
Wolfram Language code: Table[GraphLinkEfficiency[g], {g, graphs}]

Use ConnectedGraphQ to test for connected graphs:

Wolfram Language code: Table[ConnectedGraphQ[g], {g, graphs}]

Possible Issues  (1)

Self-loops are ignored:

Wolfram Language code: g = Graph[{12, 31, 33}]; graphs = {g, SimpleGraph[g]}
Wolfram Language code: Table[GraphLinkEfficiency[g], {g, graphs}]
Wolfram Research (2012), GraphLinkEfficiency, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphLinkEfficiency.html (updated 2015).

Text

Wolfram Research (2012), GraphLinkEfficiency, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphLinkEfficiency.html (updated 2015).

CMS

Wolfram Language. 2012. "GraphLinkEfficiency." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/GraphLinkEfficiency.html.

APA

Wolfram Language. (2012). GraphLinkEfficiency. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GraphLinkEfficiency.html

BibTeX

@misc{reference.wolfram_2026_graphlinkefficiency, author="Wolfram Research", title="{GraphLinkEfficiency}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/GraphLinkEfficiency.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_graphlinkefficiency, organization={Wolfram Research}, title={GraphLinkEfficiency}, year={2015}, url={https://reference.wolfram.com/language/ref/GraphLinkEfficiency.html}, note=[Accessed: 12-June-2026]}