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GraphPropertyDistribution
✖
GraphPropertyDistribution

✖
GraphPropertyDistribution[expr,xgdist]

represents the distribution of the property expr where the random variable x follows the graph distribution gdist.

✖
GraphPropertyDistribution[expr,{x1gdist1,x2gdist2,…}]

represents the distribution where x1, x2, … are independent and follow the graph distributions gdist1, gdist2, ….

Details and Options

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

Obtain property distributions of graph models:

In[1]:=1
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-g100dd
Out[1]=1

Simulate the distribution of a graph property:

In[1]:=1
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-1yhivg
In[2]:=2
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-mypox7
Out[2]=2

Compute the mean:

In[3]:=3
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-g5i30z
Out[3]=3

Plot probability density functions:

In[1]:=1
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-2l7uvb
Out[1]=1

Scope  (14)Survey of the scope of standard use cases

Basic Uses  (5)

Obtain symbolic property distributions of graph models:

In[1]:=1
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-9qrowp
Out[1]=1

Compute the probability of an event for a graph property distribution:

In[1]:=1
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-bjd8us
In[2]:=2
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-hdqpjy
Out[2]=2

Compute the expectation of an expression for a graph property distribution:

In[1]:=1
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-fnqwmy
In[2]:=2
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-bc6qmi
Out[2]=2

Simulate a property distribution:

In[1]:=1
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-pnqfmj
In[2]:=2
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-u998n
Out[2]=2

Generate a probability histogram:

In[3]:=3
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-jm4m9
In[4]:=4
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-bjfhkr
Out[4]=4

Create an empirical distribution of graph property data:

In[1]:=1
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-g7khph
In[2]:=2
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-8681v
Out[2]=2

Visualize distribution functions:

In[3]:=3
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-dcg2mw
Out[3]=3

Compute stochastic properties:

In[4]:=4
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-fhf4rs
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-lgm1pw
Out[5]=5

Compute moments and quantiles:

In[6]:=6
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-tv9xr
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-bbzf3
Out[7]=7

Distribution Properties  (4)

GraphPropertyDistribution works with basic graph properties, such as EdgeCount:

In[1]:=1
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-5hhqug
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-33lold
Out[2]=2

VertexCount:

In[3]:=3
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-mq9xw1
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-btgrtw
Out[4]=4

VertexDegree:

In[5]:=5
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-gy5bh1
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-uv347j
Out[6]=6

Predicates, such as ConnectedGraphQ:

In[1]:=1
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-kmt2w9
In[2]:=2
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-j5q2cw
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-pia21w
Out[3]=3

EulerianGraphQ:

In[4]:=4
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-sojwf2
In[5]:=5
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-y0oiv3
Out[5]=5

EdgeQ:

In[6]:=6
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-jhincz
In[7]:=7
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-fhi2bv
Out[7]=7

Graph measures and metrics, such as GraphDiameter:

In[1]:=1
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-peyfe7
In[2]:=2
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-yc3d61
Out[2]=2

GraphAssortativity:

In[3]:=3
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-4yl4xf
In[4]:=4
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-lggt2b
Out[4]=4

GlobalClusteringCoefficient:

In[5]:=5
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-3iki28
In[6]:=6
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-y9a6wv
Out[6]=6

GraphPropertyDistribution works with any expression, such as maximum eigenvector centrality:

In[1]:=1
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-0k9ova
In[2]:=2
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-noifni
Out[2]=2

Size of giant component:

In[3]:=3
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-gsmrck
In[4]:=4
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-tyf03c
Out[4]=4

Graph Distributions  (3)

GraphPropertyDistribution works with all graph distributions, such as BarabasiAlbertGraphDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-cqblso
Out[1]=1

BernoulliGraphDistribution:

In[2]:=2
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-f74dj8
Out[2]=2

DegreeGraphDistribution:

In[3]:=3
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-myuha
Out[3]=3

PriceGraphDistribution:

In[4]:=4
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-lg1vdh
Out[4]=4

SpatialGraphDistribution:

In[5]:=5
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-idxb3t
Out[5]=5

UniformGraphDistribution:

In[6]:=6
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-jdlv0
Out[6]=6

WattsStrogatzGraphDistribution:

In[7]:=7
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-oi40a
Out[7]=7

Symbolic graph distributions:

In[1]:=1
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-gwu7jh
Out[1]=1

Combine independent distributions:

In[1]:=1
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-v0ur0
In[2]:=2
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-e6fki1
Out[2]=2

Automatic Simplifications  (2)

GraphPropertyDistribution will simplify to known distributions whenever possible:

In[1]:=1
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-c1lpps
Out[1]=1

Special transformations of graph property distributions:

In[1]:=1
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-cqpje4
Out[1]=1

Options  (1)Common values & functionality for each option

Assumptions  (1)

Compute the edge count of the Price distribution:

In[3]:=3
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-flmxd5
Out[3]=3

Use Assumptions to specify the condition :

In[2]:=2
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-yvxi9
Out[2]=2

Applications  (5)Sample problems that can be solved with this function

After 20 children have spent their first week in kindergarten, the probability that two children have made friends is 0.2. Find the probability that the social network is connected:

In[1]:=1
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-x25twn
In[2]:=2
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-5xmvj4
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-kqht0c

Directly compute the probability:

In[4]:=4
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-2cevl5
Out[4]=4

This represents a social network of 100 persons in a small village where the average number of relations per person is 20. Find the expected number of relations of the least-connected person:

In[1]:=1
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-xti07a
In[2]:=2
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-jhjbsf
Out[2]=2

The number of relations is given by the VertexDegree:

In[3]:=3
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-gzoolm
Out[3]=3

A frog in a lily pond is able to jump 1.5 feet to get from one of the 25 lily pads to another. Model the frog's jumping network from the lily leaf density and SpatialGraphDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-ekfp0
In[2]:=2
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-e2crlz

Sample a random pond:

In[3]:=3
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-kh8x34
Out[3]=3

Find the largest collection of lily pads the frog can jump between:

In[4]:=4
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-dhcq16
Out[4]=4

Use simulation to find the sizes of the largest collections of lily pads for similar ponds:

In[5]:=5
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-fhimm1
In[6]:=6
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-h6cf47
Out[6]=6

Find the number of times the frog would have to swim to visit all lily pads:

In[7]:=7
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-qtlf91
Out[7]=7

Simulate to get results for similar lily ponds:

In[8]:=8
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-b7d2ny
In[9]:=9
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-nl42v
Out[9]=9

In a medical study of an outbreak of influenza in a group of seven subjects, each subject has reported his or her number of potentially contagious interactions within the group. Model the interactions as a DegreeGraphDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-j01ul3
In[2]:=2
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-la21hk
Out[2]=2

Simulate to see whether the first two subjects have interacted:

In[3]:=3
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-fpr17c
In[4]:=4
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-cfqd7t
Out[4]=4

Find the probability that the first two subjects have interacted:

In[5]:=5
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-q3oky0
Out[5]=5

In a piece of brain cortex with 100 neurons, the neurons are connected by synapses if they are at a distance less than 0.2:

In[1]:=1
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-gkki97
In[2]:=2
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-uu3fd3
Out[2]=2

Probability that the network is connected:

In[3]:=3
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-iyjb5a
Out[3]=3

Properties & Relations  (4)Properties of the function, and connections to other functions

GraphPropertyDistribution uses local names for the variables in the input:

In[1]:=1
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-fd9zu8
Out[1]=1

Use NProbability to compute the probability of an event:

In[1]:=1
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-bom7zk
In[2]:=2
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-01868
Out[2]=2

Use NExpectation to compute the expectation of an expression:

In[1]:=1
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-b9buo0
In[2]:=2
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-f3obtw
Out[2]=2

Use RandomVariate to simulate a property distribution:

In[1]:=1
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-dannj0
In[2]:=2
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-jpku6b
Out[2]=2

Possible Issues  (1)Common pitfalls and unexpected behavior

Monte Carlo methods are used to numerically approximate statistical properties:

In[1]:=1
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-pqfk4y
In[2]:=2
✖
https://wolfram.com/xid/0pnjdl1clt6iuo3u-cdackj
Out[2]=2
Wolfram Research (2012), GraphPropertyDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphPropertyDistribution.html.
✖
Wolfram Research (2012), GraphPropertyDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphPropertyDistribution.html.

Text

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

✖
Wolfram Research (2012), GraphPropertyDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphPropertyDistribution.html.

CMS

Wolfram Language. 2012. "GraphPropertyDistribution." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/GraphPropertyDistribution.html.

✖
Wolfram Language. 2012. "GraphPropertyDistribution." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/GraphPropertyDistribution.html.

APA

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

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

BibTeX

@misc{reference.wolfram_2025_graphpropertydistribution, author="Wolfram Research", title="{GraphPropertyDistribution}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/GraphPropertyDistribution.html}", note=[Accessed: 19-June-2025 ]}

✖
@misc{reference.wolfram_2025_graphpropertydistribution, author="Wolfram Research", title="{GraphPropertyDistribution}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/GraphPropertyDistribution.html}", note=[Accessed: 19-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_graphpropertydistribution, organization={Wolfram Research}, title={GraphPropertyDistribution}, year={2012}, url={https://reference.wolfram.com/language/ref/GraphPropertyDistribution.html}, note=[Accessed: 19-June-2025 ]}

✖
@online{reference.wolfram_2025_graphpropertydistribution, organization={Wolfram Research}, title={GraphPropertyDistribution}, year={2012}, url={https://reference.wolfram.com/language/ref/GraphPropertyDistribution.html}, note=[Accessed: 19-June-2025 ]}