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GraphHub

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gives the set of vertices with maximum vertex degree in the underlying simple graph of g.

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GraphHub[g,"In"]

gives the set of vertices with maximum vertex in-degree.

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GraphHub[g,"Out"]

gives the set of vertices with maximum vertex out-degree.

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GraphHub[{vw,…},…]

uses rules vw to specify the graph g.

Details

  • The vertex degree for a vertex v is the number of edges incident to v.
  • For a directed graph, the in-degree is the number of incoming edges and the out-degree is the number of outgoing edges.
  • For an undirected graph, an edge is taken to be both an in-edge and out-edge.
  • GraphHub works with undirected graphs, directed graphs, multigraphs and mixed graphs.

Examples

open allclose all

Basic Examples  (1)Summary of the most common use cases

Find the hub for a graph:

In[1]:=1
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https://wolfram.com/xid/0tz1rth7-ze5j75
In[2]:=2
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https://wolfram.com/xid/0tz1rth7-1i13ql
Out[2]=2

Highlight:

In[3]:=3
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https://wolfram.com/xid/0tz1rth7-nemwv1
Out[3]=3

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

GraphHub works with undirected graphs:

In[1]:=1
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https://wolfram.com/xid/0tz1rth7-rvhr2v
Out[1]=1

Directed graphs:

In[1]:=1
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https://wolfram.com/xid/0tz1rth7-og1ays
Out[1]=1

Use rules to specify the graph:

In[1]:=1
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https://wolfram.com/xid/0tz1rth7-bndh30
Out[1]=1

Compute the hubs with respect to vertex in-degree and vertex out-degree:

In[1]:=1
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https://wolfram.com/xid/0tz1rth7-o535o0
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0tz1rth7-8l42tk
Out[2]=2

GraphHub works with large graphs:

In[1]:=1
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https://wolfram.com/xid/0tz1rth7-pq9ae
In[2]:=2
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https://wolfram.com/xid/0tz1rth7-7r0z3f
Out[2]=2

Applications  (7)Sample problems that can be solved with this function

The administrator is the hub of the friendship network between members of a karate club:

In[1]:=1
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https://wolfram.com/xid/0tz1rth7-2rxdfs
In[2]:=2
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https://wolfram.com/xid/0tz1rth7-9lt98a
Out[2]=2

Find the people with the most family members present at the family gathering:

In[1]:=1
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https://wolfram.com/xid/0tz1rth7-xuzztf
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0tz1rth7-htatos
Out[2]=2

Find the publications with the most citations in a citation network:

In[1]:=1
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https://wolfram.com/xid/0tz1rth7-oqzcw5
In[2]:=2
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https://wolfram.com/xid/0tz1rth7-mujly6
Out[2]=2

With the most references to other publications:

In[3]:=3
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https://wolfram.com/xid/0tz1rth7-yjnfb1
Out[3]=3

The terrorist network linked to the tragic events of September 11, 2001. The ringleader of the conspiracy is the hub of the network:

In[1]:=1
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https://wolfram.com/xid/0tz1rth7-lbwuve
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0tz1rth7-hhyzfu
Out[2]=2

The Medici family is the hub of the marriage network of the ruling families of Florence:

In[1]:=1
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https://wolfram.com/xid/0tz1rth7-wecsv2
In[2]:=2
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https://wolfram.com/xid/0tz1rth7-eilbv4
Out[2]=2

It is the most powerful family and has the highest betweenness centrality:

In[3]:=3
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https://wolfram.com/xid/0tz1rth7-282ha1
Out[3]=3

Find the stations with the largest number of neighboring stations in the London Underground network:

In[1]:=1
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https://wolfram.com/xid/0tz1rth7-ezj0zr
Out[1]=1

Find the messages receiving the largest number of replies in the network of email sent to the MathGroup list in November 2011:

In[1]:=1
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https://wolfram.com/xid/0tz1rth7-hrqjma
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0tz1rth7-omw8er
Out[2]=2

The most interesting subject of the month:

In[3]:=3
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https://wolfram.com/xid/0tz1rth7-gcs5ks
Out[3]=3

Compute the message generating the largest total number of messages:

In[4]:=4
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https://wolfram.com/xid/0tz1rth7-go8k52
In[5]:=5
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https://wolfram.com/xid/0tz1rth7-7r9swl
Out[5]=5

Properties & Relations  (11)Properties of the function, and connections to other functions

GraphHub gives the center of a graph with respect to DegreeCentrality:

In[3]:=3
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https://wolfram.com/xid/0tz1rth7-w8z5rk
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0tz1rth7-kwwfkt
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0tz1rth7-1xc8bj
In[6]:=6
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https://wolfram.com/xid/0tz1rth7-05d73j
Out[6]=6

With respect to in-degree centrality:

In[7]:=7
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https://wolfram.com/xid/0tz1rth7-ob1rb9
Out[7]=7
In[8]:=8
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https://wolfram.com/xid/0tz1rth7-1dflif
In[9]:=9
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https://wolfram.com/xid/0tz1rth7-voaw5j
Out[9]=9

With respect to out-degree centrality:

In[10]:=10
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https://wolfram.com/xid/0tz1rth7-lqabmr
Out[10]=10
In[11]:=11
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https://wolfram.com/xid/0tz1rth7-w5ytdl
In[12]:=12
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https://wolfram.com/xid/0tz1rth7-taapji
Out[12]=12

For simple graphs, GraphHub gives the center with respect to VertexDegree:

In[1]:=1
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https://wolfram.com/xid/0tz1rth7-uuzwfj
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0tz1rth7-ox3wlp
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0tz1rth7-lrav6k
In[4]:=4
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https://wolfram.com/xid/0tz1rth7-4yk36e
Out[4]=4

Or with respect to VertexInDegree:

In[5]:=5
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https://wolfram.com/xid/0tz1rth7-nw2ahc
Out[5]=5
In[6]:=6
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https://wolfram.com/xid/0tz1rth7-yssxqc
In[7]:=7
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https://wolfram.com/xid/0tz1rth7-c7x57a
Out[7]=7

Or with respect to VertexOutDegree:

In[8]:=8
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https://wolfram.com/xid/0tz1rth7-b8clsw
Out[8]=8
In[9]:=9
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https://wolfram.com/xid/0tz1rth7-tcviwp
In[10]:=10
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https://wolfram.com/xid/0tz1rth7-nhfpmw
Out[10]=10

Self-loops are ignored:

In[1]:=1
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https://wolfram.com/xid/0tz1rth7-4na5su
In[2]:=2
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https://wolfram.com/xid/0tz1rth7-estqme
Out[2]=2

For a CompleteGraph, every vertex is a hub:

In[1]:=1
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https://wolfram.com/xid/0tz1rth7-kq6mmo
Out[1]=1

For a PathGraph, all vertices except for the endpoints are hubs:

In[1]:=1
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https://wolfram.com/xid/0tz1rth7-w6ws08
Out[1]=1

For a CycleGraph, every vertex is a hub:

In[1]:=1
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https://wolfram.com/xid/0tz1rth7-wyf682
Out[1]=1

For a WheelGraph of size 5 or more, the hub of the wheel is the graph hub:

In[1]:=1
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https://wolfram.com/xid/0tz1rth7-6f0pih
Out[1]=1

For a GridGraph, all vertices that are not at an edge of the grid are hubs:

In[1]:=1
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https://wolfram.com/xid/0tz1rth7-0ieuul
Out[1]=1

For a CompleteKaryTree, all vertices except for the leaves and the root are hubs:

In[1]:=1
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https://wolfram.com/xid/0tz1rth7-0qqbr3
Out[1]=1

The center of a graph with respect to EccentricityCentrality is obtained with GraphCenter:

In[1]:=1
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https://wolfram.com/xid/0tz1rth7-5z6idu
In[2]:=2
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https://wolfram.com/xid/0tz1rth7-7bk3yc
Out[2]=2

The set of vertices with maximum VertexEccentricity is obtained with GraphPeriphery:

In[3]:=3
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https://wolfram.com/xid/0tz1rth7-qu35os
Out[3]=3

The BetweennessCentrality center of a graph:

In[1]:=1
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https://wolfram.com/xid/0tz1rth7-fp458p
In[2]:=2
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https://wolfram.com/xid/0tz1rth7-tysf0c
In[3]:=3
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https://wolfram.com/xid/0tz1rth7-krdaqx
Out[3]=3

The ClosenessCentrality center:

In[4]:=4
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https://wolfram.com/xid/0tz1rth7-v8saw5
Out[4]=4

The EigenvectorCentrality center:

In[5]:=5
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https://wolfram.com/xid/0tz1rth7-p2cyi9
Out[5]=5

Possible Issues  (1)Common pitfalls and unexpected behavior

Self-loops are not accounted for:

In[1]:=1
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https://wolfram.com/xid/0tz1rth7-fadc17
In[2]:=2
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https://wolfram.com/xid/0tz1rth7-j315jg
Out[2]=2

Use VertexDegree to find the center with self-loops included:

In[3]:=3
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https://wolfram.com/xid/0tz1rth7-dxmdqk
Out[4]=4
Wolfram Research (2012), GraphHub, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphHub.html (updated 2015).
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Wolfram Research (2012), GraphHub, Wolfram Language function, https://reference.wolfram.com/language/ref/GraphHub.html (updated 2015).

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_graphhub, author="Wolfram Research", title="{GraphHub}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/GraphHub.html}", note=[Accessed: 07-July-2025 ]}

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@misc{reference.wolfram_2025_graphhub, author="Wolfram Research", title="{GraphHub}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/GraphHub.html}", note=[Accessed: 07-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_graphhub, organization={Wolfram Research}, title={GraphHub}, year={2015}, url={https://reference.wolfram.com/language/ref/GraphHub.html}, note=[Accessed: 07-July-2025 ]}

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@online{reference.wolfram_2025_graphhub, organization={Wolfram Research}, title={GraphHub}, year={2015}, url={https://reference.wolfram.com/language/ref/GraphHub.html}, note=[Accessed: 07-July-2025 ]}