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GraphReciprocity[g]

gives the reciprocity of a graph g.

GraphReciprocity[{vw,…}]

uses rules vw to specify the graph g.

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

GraphReciprocity[g]

gives the reciprocity of a graph g.

GraphReciprocity[{vw,…}]

uses rules vw to specify the graph g.

Details and Options

  • The reciprocity of a graph g is the fraction of reciprocal edges over all edges of g.
  • For a directed graph, the edges and are reciprocal and form a cycle of length 2.
  • For an undirected graph, all edges are reciprocal.
  • GraphReciprocity works with undirected graphs, directed graphs, and weighted graphs.

Examples

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Basic Examples  (2)

Compute the reciprocity of a directed graph:

Wolfram Language code: GraphReciprocity[[image]]

Distribution of graph reciprocity:

Wolfram Language code: Histogram[GraphReciprocity /@ RandomGraph[{10, 20}, 99, DirectedEdges -> True]]

Scope  (6)

GraphReciprocity works with undirected graphs:

Wolfram Language code: GraphReciprocity[[image]]

Directed graphs:

Wolfram Language code: GraphReciprocity[[image]]

Weighted graphs:

Wolfram Language code: GraphReciprocity[[image]]

Multigraphs:

Wolfram Language code: GraphReciprocity[[image]]

Use rules to specify the graph:

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

GraphReciprocity works with large graphs:

Wolfram Language code: g = RandomGraph[UniformGraphDistribution[10 ^ 4, 10 ^ 5, DirectedEdges -> True]];
Wolfram Language code: Timing[GraphReciprocity[g]]

Applications  (3)

GraphReciprocity measures the number of directed edges that are bidirectional:

Wolfram Language code: g = CycleGraph[4, DirectedEdges -> True];
Wolfram Language code: Table[Labeled[g = EdgeAdd[g, iMod[i - 1, 4, 1]], N@GraphReciprocity[g]], {i, 4}]

Test whether a square matrix is structurally symmetric:

Wolfram Language code: m = SparseArray[{Band[{1, 2}] -> 1, Band[{2, 1}] -> -1}, {25, 25}];
Wolfram Language code: GraphReciprocity[AdjacencyGraph[Unitize[m]]] == 1

Distribution of reciprocity in UniformGraphDistribution[n,m,DirectedEdges->True]:

Wolfram Language code: 𝒢 = UniformGraphDistribution[9, 40, DirectedEdges -> True];
Wolfram Language code: data = GraphReciprocity /@ RandomGraph[𝒢, 1000];
Wolfram Language code: Histogram[data]

The expected value is (m-1)/(n(n-1)-1):

Wolfram Language code: N[{Mean[data], 39 / 71}]

Properties & Relations  (3)

The graph reciprocity is between 0 and 1:

Wolfram Language code: RandomGraph[UniformGraphDistribution[5, 10, DirectedEdges -> True]]
Wolfram Language code: N[GraphReciprocity[%]]

A bidirectional directed graph has reciprocity 1:

Wolfram Language code: DirectedGraph[StarGraph[4]]
Wolfram Language code: GraphReciprocity[%]

An undirected graph also has reciprocity 1:

Wolfram Language code: StarGraph[8]
Wolfram Language code: GraphReciprocity[%]
Wolfram Research (2012), GraphReciprocity, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphReciprocity.html (updated 2015).

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_graphreciprocity, author="Wolfram Research", title="{GraphReciprocity}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/GraphReciprocity.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_graphreciprocity, organization={Wolfram Research}, title={GraphReciprocity}, year={2015}, url={https://reference.wolfram.com/language/ref/GraphReciprocity.html}, note=[Accessed: 13-June-2026]}