WattsStrogatzGraphDistribution
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WattsStrogatzGraphDistribution
represents the Watts–Strogatz graph distribution for n-vertex graphs with rewiring probability p.
represents the Watts–Strogatz graph distribution for n-vertex graphs with rewiring probability p starting from a 2k-regular graph.
Details
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- WattsStrogatzGraphDistribution is also known as small-world graph distribution.
- WattsStrogatzGraphDistribution[n,p] is equivalent to WattsStrogatzGraphDistribution[n,p,2].
- The WattsStrogatzGraphDistribution is constructed starting from CirculantGraph[n,Range[k]] and rewiring each edge with probability p. Each edge is rewired by changing one of the vertices, making sure that no loop or multiple edge is created.
- WattsStrogatzGraphDistribution can be used with such functions as RandomGraph and GraphPropertyDistribution.
Examples
open allclose allBasic Examples (2)Summary of the most common use cases
Generate a pseudorandom graph:
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-yg5284
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GlobalClusteringCoefficient as a function of rewiring probability:
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-6ay7jz
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Scope (3)Survey of the scope of standard use cases
Generate simple undirected graphs:
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-2cq0pg
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Generate a set of pseudorandom graphs:
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-y6wkjj
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Compute probabilities and statistical properties:
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-hso99s
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-3ac0m9
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Applications (3)Sample problems that can be solved with this function
The Western States Power Grid can be modeled with WattsStrogatzGraphDistribution:
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-d603lt
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-n3s43x
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The model captures the small-world characteristics of the empirical network, with short mean graph distance and high clustering:
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-td9s4a
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-gigk93
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A social network in a village of 100 people where the average number of relations per person is 20 can be modeled using a WattsStrogatzGraphDistribution. Find the expected number of relations for the least-connected person:
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-xti07a
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-jhjbsf
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The expected number of relations for the least-connected person:
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-nj4bxl
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-dv8zrg
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This represents a simplified model for the spread of an infectious disease in a social network. The disease spreads in each step with probability 0.4 from infected individuals to some of their susceptible neighbors, while infected individuals recover and become immune:
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-bv7qhr
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Simulate an infection and find infected persons:
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-bkmook
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-hai8wc
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-r36kt
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The fraction of infected persons as a function of the transmission probability:
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-dxhmy
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-nhy61d
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Properties & Relations (5)Properties of the function, and connections to other functions
Distribution of the number of vertices:
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-nrpf1s
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Distribution of the number of edges:
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-c8umns
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Distribution of the vertex degree:
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-il6f2k
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-gwu8se
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Approximate with a sum of BinomialDistribution and PoissonDistribution:
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-oo44wu
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-padwem
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The mean distance decreases quickly as the rewiring probability increases:
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-bhfup2
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The clustering coefficient decreases slowly:
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-0hpmj8
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WattsStrogatzGraphDistribution[n,0,k] is a 2k-regular graph:
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-pp6fgq
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-fifabg
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-t9onkr
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https://wolfram.com/xid/0hu9l02zbw9f9f7nm2a6oq-18s54o
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Wolfram Research (2010), WattsStrogatzGraphDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/WattsStrogatzGraphDistribution.html.
Text
Wolfram Research (2010), WattsStrogatzGraphDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/WattsStrogatzGraphDistribution.html.
Wolfram Research (2010), WattsStrogatzGraphDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/WattsStrogatzGraphDistribution.html.
CMS
Wolfram Language. 2010. "WattsStrogatzGraphDistribution." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/WattsStrogatzGraphDistribution.html.
Wolfram Language. 2010. "WattsStrogatzGraphDistribution." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/WattsStrogatzGraphDistribution.html.
APA
Wolfram Language. (2010). WattsStrogatzGraphDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WattsStrogatzGraphDistribution.html
Wolfram Language. (2010). WattsStrogatzGraphDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WattsStrogatzGraphDistribution.html
BibTeX
@misc{reference.wolfram_2025_wattsstrogatzgraphdistribution, author="Wolfram Research", title="{WattsStrogatzGraphDistribution}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/WattsStrogatzGraphDistribution.html}", note=[Accessed: 28-February-2025
]}
BibLaTeX
@online{reference.wolfram_2025_wattsstrogatzgraphdistribution, organization={Wolfram Research}, title={WattsStrogatzGraphDistribution}, year={2010}, url={https://reference.wolfram.com/language/ref/WattsStrogatzGraphDistribution.html}, note=[Accessed: 28-February-2025
]}