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MeanGraphDistance[g]

gives the mean distance between all pairs of vertices in the graph g.

MeanGraphDistance[{vw,…}]

uses rules vw to specify the graph g.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
Method  
Applications  
Properties & Relations  
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MeanGraphDistance

MeanGraphDistance[g]

gives the mean distance between all pairs of vertices in the graph g.

MeanGraphDistance[{vw,…}]

uses rules vw to specify the graph g.

Details and Options

Examples

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Basic Examples  (2)

Find the mean distance between vertices in a graph:

Wolfram Language code: MeanGraphDistance[[image]]

Mean graph distance of the Watts–Strogatz model as a function of rewiring probability:

Wolfram Language code: Plot[MeanGraphDistance[RandomGraph[WattsStrogatzGraphDistribution[1000, p, 5]]], {p, 0, 1}, MaxRecursion -> 0]

Scope  (7)

MeanGraphDistance works with undirected graphs:

Wolfram Language code: MeanGraphDistance[[image]]

Directed graphs:

Wolfram Language code: MeanGraphDistance[[image]]

Weighted graphs:

Wolfram Language code: MeanGraphDistance[[image]]

Multigraphs:

Wolfram Language code: MeanGraphDistance[[image]]

Mixed graphs:

Wolfram Language code: MeanGraphDistance[[image]]

Use rules to specify the graph:

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

MeanGraphDistance works with large graphs:

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

Options  (2)

Method  (2)

By default, edge weights are used to determine graph distance:

Wolfram Language code: MeanGraphDistance[[image]]

Use Method->"UnitWeight" to ignore edge weights:

Wolfram Language code: MeanGraphDistance[[image], Method -> "UnitWeight"]

Applications  (4)

A network of American football games between Division IA colleges during regular season fall 2000. The average number of games linking one team to another is about 2.5:

Wolfram Language code: ExampleData[{"NetworkGraph", "AmericanCollegeFootball"}]
Wolfram Language code: N[MeanGraphDistance[%]]

It takes a few more than five clicks on average to navigate from one symbol to another using the See Also links in the Wolfram Language documentation:

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

Distribution of the number of co-appearances linking two randomly selected actors in a small-size Kevin Bacon game:

Wolfram Language code: g = \!\(\*GraphicsBox[«5»]\);
Wolfram Language code: data = Table[GraphDistance[g, RandomChoice[VertexList[g]], RandomChoice[VertexList[g]]], {10000}];
Wolfram Language code: Histogram[data]

Average number of co-appearances linking two actors:

Wolfram Language code: N[MeanGraphDistance[g]]

Distribution of the average number of relations connecting two people in a WattsStrogatzGraphDistribution social network model:

Wolfram Language code: f[g_] := MeanGraphDistance[g] /. {∞ -> VertexCount[g] + 1}
Wolfram Language code: 𝒟 = GraphPropertyDistribution[f[g], gWattsStrogatzGraphDistribution[50, 0.1, 3]];
Wolfram Language code: Histogram[RandomVariate[𝒟, 1000]]

Compute the probability that the average exceeds 3:

Wolfram Language code: NProbability[x > 3, x𝒟, PrecisionGoal -> 1]

Properties & Relations  (5)

MeanGraphDistance gives the off-diagonal mean of GraphDistanceMatrix:

Wolfram Language code: g = KnightTourGraph[4, 3]
Wolfram Language code: MeanGraphDistance[g]

Compute the off-diagonal mean when the diagonal of GraphDistanceMatrix is 0:

Wolfram Language code: a = GraphDistanceMatrix[g];
Wolfram Language code: Diagonal[a]
Wolfram Language code: Total[Flatten[a]] / (Length[Flatten[a]] - Length[Diagonal[a]])

The mean graph distance is greater than or equal to 1:

Wolfram Language code: CirculantGraph[8, {1, 2, 3}]
Wolfram Language code: N[MeanGraphDistance[%]]

It can be less than 1 for weighted graphs:

Wolfram Language code: WeightedAdjacencyGraph[(⁠| | | | | | --- | --- | --- | --- | | ∞ | 0.1 | 0.2 | 0.1 | | 0.1 | ∞ | 0.2 | ∞ | | 0.2 | 0.2 | ∞ | 0.1 | | 0.1 | ∞ | 0.1 | ∞ |⁠), EdgeLabels -> "EdgeWeight"]
Wolfram Language code: MeanGraphDistance[%]

The mean graph distance of g is 1 iff g is a complete graph:

Wolfram Language code: {g = Graph[{12, 23, 31}], h = Graph[{12, 13, 21, 31, 32}]}
Wolfram Language code: {MeanGraphDistance[g], MeanGraphDistance[h]}

Use CompleteGraphQ to test for complete graphs:

Wolfram Language code: {CompleteGraphQ[g], CompleteGraphQ[h]}

The mean graph distance of a disconnected graph is Infinity:

Wolfram Language code: g = Graph[{12, 34}]
Wolfram Language code: MeanGraphDistance[g]

Use ConnectedGraphQ to test for connected graphs:

Wolfram Language code: ConnectedGraphQ[g]

MeanGraphDistance can be used to find the GraphLinkEfficiency:

Wolfram Language code: g = CompleteKaryTree[3, 3]
Wolfram Language code: 1 - MeanGraphDistance[g] / EdgeCount[g]
Wolfram Language code: % == GraphLinkEfficiency[g]
Wolfram Research (2012), MeanGraphDistance, Wolfram Language function, https://reference.wolfram.com/language/ref/MeanGraphDistance.html (updated 2015).

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_meangraphdistance, author="Wolfram Research", title="{MeanGraphDistance}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/MeanGraphDistance.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_meangraphdistance, organization={Wolfram Research}, title={MeanGraphDistance}, year={2015}, url={https://reference.wolfram.com/language/ref/MeanGraphDistance.html}, note=[Accessed: 12-June-2026]}