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GraphLinkEfficiency
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GraphLinkEfficiency

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gives the link efficiency of the graph g.

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GraphLinkEfficiency[{vw,…}]

uses rules vw to specify the graph g.

Details

Examples

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

Find the link efficiency in the graph:

In[3]:=3
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https://wolfram.com/xid/0c0q3mbaute94-q559fq
Out[3]=3

Graph link efficiency distribution of the Watts–Strogatz graph model:

In[1]:=1
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https://wolfram.com/xid/0c0q3mbaute94-6fbar
In[2]:=2
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https://wolfram.com/xid/0c0q3mbaute94-cfc1bw
Out[2]=2

Scope  (4)Survey of the scope of standard use cases

GraphLinkEfficiency works with undirected graphs:

In[3]:=3
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https://wolfram.com/xid/0c0q3mbaute94-dy6dt4
Out[3]=3

Directed graphs:

In[1]:=1
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https://wolfram.com/xid/0c0q3mbaute94-yo56q0
Out[1]=1

Use rules to specify the graph:

In[1]:=1
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https://wolfram.com/xid/0c0q3mbaute94-bndh30
Out[1]=1

GraphLinkEfficiency works with large graphs:

In[1]:=1
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https://wolfram.com/xid/0c0q3mbaute94-r64pi
In[2]:=2
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https://wolfram.com/xid/0c0q3mbaute94-qbj9kq
Out[2]=2

Applications  (2)Sample problems that can be solved with this function

Find how tightly connected the overground line of the London Underground is with respect to the number of lines between stations:

In[6]:=6
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https://wolfram.com/xid/0c0q3mbaute94-33whzp
Out[6]=6

Analyze the distribution of link efficiency in the Watts–Strogatz graph model:

In[1]:=1
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https://wolfram.com/xid/0c0q3mbaute94-xtw9fw
In[2]:=2
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https://wolfram.com/xid/0c0q3mbaute94-qfnq1q
Out[2]=2

Distribution of link efficiency:

In[3]:=3
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https://wolfram.com/xid/0c0q3mbaute94-3snv7e
In[4]:=4
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https://wolfram.com/xid/0c0q3mbaute94-2wa8us
Out[4]=4

Expected value:

In[5]:=5
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https://wolfram.com/xid/0c0q3mbaute94-xpq61t
Out[5]=5

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

GraphLinkEfficiency is related to MeanGraphDistance:

In[1]:=1
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https://wolfram.com/xid/0c0q3mbaute94-bvrefv
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0c0q3mbaute94-1oehyg
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0c0q3mbaute94-2kzhu3
Out[3]=3

The GraphLinkEfficiency is always less than 1:

In[1]:=1
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https://wolfram.com/xid/0c0q3mbaute94-yjgmrp
Out[1]=1

The GraphLinkEfficiency of a complete graph is close to 1:

In[1]:=1
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https://wolfram.com/xid/0c0q3mbaute94-y373rb
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0c0q3mbaute94-drmhzc
Out[2]=2

Test if a graph is complete using CompleteGraphQ:

In[3]:=3
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https://wolfram.com/xid/0c0q3mbaute94-ro8zil
Out[3]=3

The GraphLinkEfficiency of a path graph of length 1 is 0:

In[1]:=1
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https://wolfram.com/xid/0c0q3mbaute94-6ueoci
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0c0q3mbaute94-j32y0o
Out[2]=2

The GraphLinkEfficiency of a disconnected graph is -∞:

In[1]:=1
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https://wolfram.com/xid/0c0q3mbaute94-ry6a3b
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0c0q3mbaute94-shyicc
Out[2]=2

Use ConnectedGraphQ to test for connected graphs:

In[3]:=3
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https://wolfram.com/xid/0c0q3mbaute94-i47s6p
Out[3]=3

Possible Issues  (1)Common pitfalls and unexpected behavior

Self-loops are ignored:

In[1]:=1
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https://wolfram.com/xid/0c0q3mbaute94-5cfkbm
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0c0q3mbaute94-wnmkti
Out[3]=3
Wolfram Research (2012), GraphLinkEfficiency, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphLinkEfficiency.html (updated 2015).
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Wolfram Research (2012), GraphLinkEfficiency, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphLinkEfficiency.html (updated 2015).

Text

Wolfram Research (2012), GraphLinkEfficiency, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphLinkEfficiency.html (updated 2015).

✖
Wolfram Research (2012), GraphLinkEfficiency, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphLinkEfficiency.html (updated 2015).

CMS

Wolfram Language. 2012. "GraphLinkEfficiency." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/GraphLinkEfficiency.html.

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Wolfram Language. 2012. "GraphLinkEfficiency." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/GraphLinkEfficiency.html.

APA

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

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

BibTeX

@misc{reference.wolfram_2025_graphlinkefficiency, author="Wolfram Research", title="{GraphLinkEfficiency}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/GraphLinkEfficiency.html}", note=[Accessed: 22-April-2025 ]}

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@misc{reference.wolfram_2025_graphlinkefficiency, author="Wolfram Research", title="{GraphLinkEfficiency}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/GraphLinkEfficiency.html}", note=[Accessed: 22-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_graphlinkefficiency, organization={Wolfram Research}, title={GraphLinkEfficiency}, year={2015}, url={https://reference.wolfram.com/language/ref/GraphLinkEfficiency.html}, note=[Accessed: 22-April-2025 ]}

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@online{reference.wolfram_2025_graphlinkefficiency, organization={Wolfram Research}, title={GraphLinkEfficiency}, year={2015}, url={https://reference.wolfram.com/language/ref/GraphLinkEfficiency.html}, note=[Accessed: 22-April-2025 ]}