# GraphReciprocity

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:

Distribution of graph reciprocity:

## Scope(6)

GraphReciprocity works with undirected graphs:

Directed graphs:

Weighted graphs:

Multigraphs:

Use rules to specify the graph:

GraphReciprocity works with large graphs:

## Applications(3)

GraphReciprocity measures the number of directed edges that are bidirectional:

Test whether a square matrix is structurally symmetric:

Distribution of reciprocity in :

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

## Properties & Relations(3)

The graph reciprocity is between 0 and 1:

A bidirectional directed graph has reciprocity 1:

An undirected graph also has reciprocity 1: