GlobalClusteringCoefficient

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`GlobalClusteringCoefficient`

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GlobalClusteringCoefficient[g]

gives the global clustering coefficient of the graph g.

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GlobalClusteringCoefficient[{vw,…}]

uses rules vw to specify the graph g.

Details

GlobalClusteringCoefficient is also known as clustering coefficient.
GlobalClusteringCoefficient is typically used to quantify the level of transitivity in a graph.
The global clustering coefficient of g is the fraction of paths of length two in g that are closed over all paths of length two in g.
GlobalClusteringCoefficient works with undirected graphs, directed graphs, and multigraphs.

Examples

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

Find the global clustering coefficient of a graph:

In[3]:=3

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https://wolfram.com/xid/0pnjtekg1or8fgya-5b9umz

Out[3]=3

Global clustering coefficient of the Watts–Strogatz model as a function of rewiring probability:

In[1]:=1

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https://wolfram.com/xid/0pnjtekg1or8fgya-rtcihx

Out[1]=1

Scope (5)Survey of the scope of standard use cases

GlobalClusteringCoefficient works with undirected graphs:

In[3]:=3

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https://wolfram.com/xid/0pnjtekg1or8fgya-b1cxtq

Out[3]=3

Directed graphs:

In[1]:=1

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https://wolfram.com/xid/0pnjtekg1or8fgya-cy689y

Out[1]=1

Multigraphs:

In[1]:=1

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https://wolfram.com/xid/0pnjtekg1or8fgya-15kl6n

Out[1]=1

Use rules to specify the graph:

In[1]:=1

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https://wolfram.com/xid/0pnjtekg1or8fgya-bndh30

Out[1]=1

GlobalClusteringCoefficient works with large graphs:

In[1]:=1

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https://wolfram.com/xid/0pnjtekg1or8fgya-cddhqp

In[2]:=2

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https://wolfram.com/xid/0pnjtekg1or8fgya-bioxh0

Out[2]=2

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

The global clustering coefficient is between 0 and 1:

In[1]:=1

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https://wolfram.com/xid/0pnjtekg1or8fgya-w1hdto

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0pnjtekg1or8fgya-zrslmi

Out[2]=2

The global clustering coefficient for a graph with no paths of length two is 0:

In[1]:=1

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https://wolfram.com/xid/0pnjtekg1or8fgya-0zinzo

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0pnjtekg1or8fgya-g2qcfc

Out[2]=2

The global clustering coefficient of a complete graph with at least three vertices is 1:

In[1]:=1

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https://wolfram.com/xid/0pnjtekg1or8fgya-6rjps4

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0pnjtekg1or8fgya-nis1k1

Out[2]=2

Distribution of global clustering coefficient in BernoulliGraphDistribution:

In[1]:=1

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https://wolfram.com/xid/0pnjtekg1or8fgya-b6zk05

In[2]:=2

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https://wolfram.com/xid/0pnjtekg1or8fgya-2wa8us

Out[2]=2

Expected value:

In[3]:=3

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https://wolfram.com/xid/0pnjtekg1or8fgya-3w1lp0

Out[3]=3

Distribution of a global clustering coefficient in WattsStrogatzGraphDistribution:

In[1]:=1

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https://wolfram.com/xid/0pnjtekg1or8fgya-hsxdys

In[2]:=2

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https://wolfram.com/xid/0pnjtekg1or8fgya-xwz21h

Out[2]=2

With low rewiring probability and high mean vertex degree, the expected value is near :

In[3]:=3

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https://wolfram.com/xid/0pnjtekg1or8fgya-0qub8l

Out[3]=3

With high rewiring probability, the expected value is near 0:

In[4]:=4

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https://wolfram.com/xid/0pnjtekg1or8fgya-mw85vu

Out[4]=4

Distribution of a global clustering coefficient in BarabasiAlbertGraphDistribution:

In[1]:=1

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https://wolfram.com/xid/0pnjtekg1or8fgya-xb3x76

In[2]:=2

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https://wolfram.com/xid/0pnjtekg1or8fgya-kafjru

In[3]:=3

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https://wolfram.com/xid/0pnjtekg1or8fgya-id0xd0

Out[3]=3

Compare with MeanClusteringCoefficient:

In[4]:=4

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https://wolfram.com/xid/0pnjtekg1or8fgya-nyso6x

In[5]:=5

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https://wolfram.com/xid/0pnjtekg1or8fgya-w928v8

Out[5]=5

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