GraphDensity

GraphDensity[g]

gives the graph density of the graph g.

GraphDensity[{vw,…}]

uses rules vw to specify the graph g.

Details and Options

  • GraphDensity is the ratio of the number of edges divided by the number of edges of a complete graph with the same number of vertices.
  • A simple undirected graph with vertices and edges has graph density .
  • A simple directed graph with vertices and edges has graph density .

Examples

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Basic Examples  (2)

Compute the density of a graph:

Graph density distribution of the Bernoulli graph model:

Scope  (6)

GraphDensity works with undirected graphs:

Directed graphs:

Multigraphs:

Mixed graphs:

Use rules to specify the graph:

GraphDensity works with large graphs:

Applications  (4)

Find the proportion of games between teams during an American college football season:

Compute the probability that two randomly chosen friends in a network are connected:

Model a social network:

Simulate the model:

Analyze the model:

The expected number of edges matches the original graph:

Distribution of density in BernoulliGraphDistribution[n,p]:

The expected value is p:

Properties & Relations  (6)

GraphDensity measures the density of the AdjacencyMatrix:

Get the density from AdjacencyMatrix:

The graph density is between 0 and 1:

The density of an empty graph is 0:

Use EmptyGraphQ to test for emptiness:

The density of a complete graph is 1:

Use CompleteGraphQ to test for complete graphs:

Converting an undirected graph to a directed graph does not change the density:

Unless self-loops are taken into consideration:

LocalClusteringCoefficient gives a local measure of density:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2025_graphdensity, author="Wolfram Research", title="{GraphDensity}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/GraphDensity.html}", note=[Accessed: 18-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_graphdensity, organization={Wolfram Research}, title={GraphDensity}, year={2015}, url={https://reference.wolfram.com/language/ref/GraphDensity.html}, note=[Accessed: 18-April-2025 ]}