WOLFRAM

Wolfram Language & System Documentation Center
Wolfram Language Home Page »

BarabasiAlbertGraphDistribution
✖
BarabasiAlbertGraphDistribution

✖

represents a Barabasi–Albert graph distribution for n-vertex graphs where a new vertex with k edges is added at each step.

Details

Examples

open allclose all

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

Generate a pseudorandom graph:

In[1]:=1
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-yg5284
Out[1]=1

Degree distribution:

In[1]:=1
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-sy45hm
In[2]:=2
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-6kpmxy
Out[2]=2

Probability density function:

In[3]:=3
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-o0xrtw
Out[3]=3

Scope  (3)Survey of the scope of standard use cases

Generate simple undirected graphs:

In[1]:=1
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-2cq0pg
Out[1]=1

Generate a set of pseudorandom graphs:

In[1]:=1
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-y6wkjj
Out[1]=1

Compute probabilities and statistical properties:

In[1]:=1
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-hso99s
In[2]:=2
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-h18ttv
Out[2]=2

Applications  (3)Sample problems that can be solved with this function

The internet at the level of autonomous systems can be modeled with BarabasiAlbertGraphDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-ni36gl
In[2]:=2
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-qbehy8
Out[2]=2

The model captures the power-law nature of the empirical degree distribution:

In[3]:=3
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-czisbn
Out[3]=3

The model has a lower clustering coefficient:

In[4]:=4
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-nihcv
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-fvh2ge
Out[5]=5

Use the Barabasi–Albert graph distribution as a model of the Western States Power Grid network:

In[1]:=1
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-j93wjw
In[2]:=2
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-8x68dd
Out[2]=2

The model captures the power-law nature of the empirical degree distribution:

In[3]:=3
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-bxel6h
In[4]:=4
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-yd3kkm
Out[4]=4

A social network with 400 people and prominent hubs is modeled with BarabasiAlbertGraphDistribution. Find the expected number of ties separating a person at the hub from the most remote person in the network:

In[1]:=1
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-t0wttd
In[2]:=2
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-d9domo
Out[2]=2

Properties & Relations  (5)Properties of the function, and connections to other functions

Distribution of the number of vertices:

In[1]:=1
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-nrpf1s
Out[1]=1

Distribution of the number of edges:

In[1]:=1
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-c8umns
Out[1]=1

Degree distribution:

In[1]:=1
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-ug7002

The distribution can be approximated by ZipfDistribution:

In[2]:=2
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-ces1w7
In[3]:=3
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-1iz79a
Out[3]=3

The degree distribution follows a power law:

In[4]:=4
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-bhkvm8
Out[4]=4

Use RandomSample to simulate a BarabasiAlbertGraphDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-o7pszz
In[2]:=2
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-ys2col

Pseudorandom graphs:

In[3]:=3
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-ofze7f
Out[3]=3

In BarabasiAlbertGraphDistribution[n,k], there is a maximum clique of size k+1:

In[1]:=1
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-82cqji
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-e3kd3w
Out[2]=2

Neat Examples  (1)Surprising or curious use cases

Randomly colored vertices:

In[1]:=1
✖
https://wolfram.com/xid/0x8wrak8962usdivvlicpaca-y4ddd2
Out[1]=1
Wolfram Research (2010), BarabasiAlbertGraphDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BarabasiAlbertGraphDistribution.html.
✖
Wolfram Research (2010), BarabasiAlbertGraphDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BarabasiAlbertGraphDistribution.html.

Text

Wolfram Research (2010), BarabasiAlbertGraphDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BarabasiAlbertGraphDistribution.html.

✖
Wolfram Research (2010), BarabasiAlbertGraphDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/BarabasiAlbertGraphDistribution.html.

CMS

Wolfram Language. 2010. "BarabasiAlbertGraphDistribution." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BarabasiAlbertGraphDistribution.html.

✖
Wolfram Language. 2010. "BarabasiAlbertGraphDistribution." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BarabasiAlbertGraphDistribution.html.

APA

Wolfram Language. (2010). BarabasiAlbertGraphDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BarabasiAlbertGraphDistribution.html

✖
Wolfram Language. (2010). BarabasiAlbertGraphDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BarabasiAlbertGraphDistribution.html

BibTeX

@misc{reference.wolfram_2025_barabasialbertgraphdistribution, author="Wolfram Research", title="{BarabasiAlbertGraphDistribution}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/BarabasiAlbertGraphDistribution.html}", note=[Accessed: 29-March-2025 ]}

✖
@misc{reference.wolfram_2025_barabasialbertgraphdistribution, author="Wolfram Research", title="{BarabasiAlbertGraphDistribution}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/BarabasiAlbertGraphDistribution.html}", note=[Accessed: 29-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_barabasialbertgraphdistribution, organization={Wolfram Research}, title={BarabasiAlbertGraphDistribution}, year={2010}, url={https://reference.wolfram.com/language/ref/BarabasiAlbertGraphDistribution.html}, note=[Accessed: 29-March-2025 ]}

✖
@online{reference.wolfram_2025_barabasialbertgraphdistribution, organization={Wolfram Research}, title={BarabasiAlbertGraphDistribution}, year={2010}, url={https://reference.wolfram.com/language/ref/BarabasiAlbertGraphDistribution.html}, note=[Accessed: 29-March-2025 ]}