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GraphDensity
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GraphDensity

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gives the graph density of the graph g.

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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

open allclose all

Basic Examples  (2)Summary of the most common use cases

Compute the density of a graph:

In[1]:=1
✖
https://wolfram.com/xid/0rs6psion-ww8nwx
Out[1]=1

Graph density distribution of the Bernoulli graph model:

In[1]:=1
✖
https://wolfram.com/xid/0rs6psion-it4tje
In[2]:=2
✖
https://wolfram.com/xid/0rs6psion-rtcihx
Out[2]=2

Scope  (6)Survey of the scope of standard use cases

GraphDensity works with undirected graphs:

In[1]:=1
✖
https://wolfram.com/xid/0rs6psion-edqvym
Out[1]=1

Directed graphs:

In[1]:=1
✖
https://wolfram.com/xid/0rs6psion-hike2
Out[1]=1

Multigraphs:

In[1]:=1
✖
https://wolfram.com/xid/0rs6psion-15kl6n
Out[1]=1

Mixed graphs:

In[1]:=1
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https://wolfram.com/xid/0rs6psion-czvddh
Out[1]=1

Use rules to specify the graph:

In[1]:=1
✖
https://wolfram.com/xid/0rs6psion-bndh30
Out[1]=1

GraphDensity works with large graphs:

In[1]:=1
✖
https://wolfram.com/xid/0rs6psion-cddhqp
In[2]:=2
✖
https://wolfram.com/xid/0rs6psion-nur2pn
Out[2]=2

Applications  (4)Sample problems that can be solved with this function

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

In[1]:=1
✖
https://wolfram.com/xid/0rs6psion-v3do44
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0rs6psion-ea0utm
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0rs6psion-h6tn6q
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0rs6psion-yhv13k
Out[2]=2

Model a social network:

In[1]:=1
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https://wolfram.com/xid/0rs6psion-1p08xf
In[2]:=2
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https://wolfram.com/xid/0rs6psion-chi8pv

Simulate the model:

In[3]:=3
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https://wolfram.com/xid/0rs6psion-ca7p7
Out[3]=3

Analyze the model:

In[4]:=4
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https://wolfram.com/xid/0rs6psion-q3sm2y
Out[4]=4

The expected number of edges matches the original graph:

In[5]:=5
✖
https://wolfram.com/xid/0rs6psion-c1w5e
Out[5]=5

Distribution of density in BernoulliGraphDistribution[n,p]:

In[1]:=1
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https://wolfram.com/xid/0rs6psion-3snv7e
In[2]:=2
✖
https://wolfram.com/xid/0rs6psion-2wa8us
Out[2]=2

The expected value is p:

In[3]:=3
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https://wolfram.com/xid/0rs6psion-xpq61t
Out[3]=3

Properties & Relations  (6)Properties of the function, and connections to other functions

GraphDensity measures the density of the AdjacencyMatrix:

In[1]:=1
✖
https://wolfram.com/xid/0rs6psion-h8zjco
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0rs6psion-i2mc3d
Out[2]=2

Get the density from AdjacencyMatrix:

In[3]:=3
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https://wolfram.com/xid/0rs6psion-j70azq
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0rs6psion-lpbc7l
Out[4]=4

The graph density is between 0 and 1:

In[1]:=1
✖
https://wolfram.com/xid/0rs6psion-trv57o
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0rs6psion-226qoy
Out[2]=2

The density of an empty graph is 0:

In[1]:=1
✖
https://wolfram.com/xid/0rs6psion-71w3aj
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0rs6psion-nu8h3x
Out[2]=2

Use EmptyGraphQ to test for emptiness:

In[3]:=3
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https://wolfram.com/xid/0rs6psion-13odbi
Out[3]=3

The density of a complete graph is 1:

In[1]:=1
✖
https://wolfram.com/xid/0rs6psion-dy2r3c
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0rs6psion-lvs8lo
Out[2]=2

Use CompleteGraphQ to test for complete graphs:

In[3]:=3
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https://wolfram.com/xid/0rs6psion-7idkfw
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0rs6psion-j74kcy
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0rs6psion-g3kn0c
Out[2]=2

Unless self-loops are taken into consideration:

In[3]:=3
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https://wolfram.com/xid/0rs6psion-7yf9id
Out[3]=3

LocalClusteringCoefficient gives a local measure of density:

In[1]:=1
✖
https://wolfram.com/xid/0rs6psion-lbgmc3
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0rs6psion-2fo6ui
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0rs6psion-r0zmf5
Out[3]=3
Wolfram Research (2012), GraphDensity, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphDensity.html (updated 2015).
✖
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).

✖
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.

✖
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

✖
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: 29-March-2025 ]}

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@misc{reference.wolfram_2025_graphdensity, author="Wolfram Research", title="{GraphDensity}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/GraphDensity.html}", note=[Accessed: 29-March-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: 29-March-2025 ]}

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@online{reference.wolfram_2025_graphdensity, organization={Wolfram Research}, title={GraphDensity}, year={2015}, url={https://reference.wolfram.com/language/ref/GraphDensity.html}, note=[Accessed: 29-March-2025 ]}