WOLFRAM

Wolfram Language & System Documentation Center
Wolfram Language Home Page »

KCoreComponents
✖
KCoreComponents

✖

gives the k-core components of the underlying simple graph of g.

✖
KCoreComponents[g,k,"In"]

gives the k-core components with vertex in-degrees at least k.

✖
KCoreComponents[g,k,"Out"]

gives the k-core components with vertex out-degrees at least k.

✖
KCoreComponents[{vw,…},…]

uses rules vw to specify the graph g.

Details

  • A k-core component is a maximal weakly connected subgraph in which all vertices have degree at least k.
  • KCoreComponents returns a list of components {c1,c2,…}, where each component ci is given as a list of vertices.
  • For a directed graph g, KCoreComponents[g,k] gives the k-core components of the underlying undirected simple graph of g.
  • KCoreComponents works with undirected graphs, directed graphs, multigraphs, and mixed graphs.

Examples

open allclose all

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

Find the 3-core components of a graph:

In[1]:=1
✖
https://wolfram.com/xid/0ywuc1asntar-b26eb7
In[2]:=2
✖
https://wolfram.com/xid/0ywuc1asntar-jnuttl
Out[2]=2

Show the 3-core components:

In[3]:=3
✖
https://wolfram.com/xid/0ywuc1asntar-x33u6
Out[3]=3

Find the 4-core components in a social network:

In[1]:=1
✖
https://wolfram.com/xid/0ywuc1asntar-i8ys3s
In[2]:=2
✖
https://wolfram.com/xid/0ywuc1asntar-bofdft
Out[2]=2

Scope  (10)Survey of the scope of standard use cases

KCoreComponents works with undirected graphs:

In[1]:=1
✖
https://wolfram.com/xid/0ywuc1asntar-bv2jib
Out[1]=1

Directed graphs:

In[1]:=1
✖
https://wolfram.com/xid/0ywuc1asntar-b4ybz0
Out[1]=1

Multigraphs:

In[1]:=1
✖
https://wolfram.com/xid/0ywuc1asntar-djehuq
Out[1]=1

Mixed graphs:

In[1]:=1
✖
https://wolfram.com/xid/0ywuc1asntar-czvddh
Out[1]=1

KCoreComponents finds core components of any size:

In[1]:=1
✖
https://wolfram.com/xid/0ywuc1asntar-2jfqc
Out[1]=1

Find k-core components with vertex in-degrees at least k:

In[1]:=1
✖
https://wolfram.com/xid/0ywuc1asntar-dutfr4
Out[1]=1

Find k-core components with vertex out-degrees at least k:

In[1]:=1
✖
https://wolfram.com/xid/0ywuc1asntar-gr8kj0
Out[1]=1

Use rules to specify the graph:

In[1]:=1
✖
https://wolfram.com/xid/0ywuc1asntar-t96r7
Out[1]=1

KCoreComponents gives an empty list if there is no k-core:

In[1]:=1
✖
https://wolfram.com/xid/0ywuc1asntar-gqw14k
Out[1]=1

KCoreComponents works with large graphs:

In[1]:=1
✖
https://wolfram.com/xid/0ywuc1asntar-y605q8
In[2]:=2
✖
https://wolfram.com/xid/0ywuc1asntar-rdadr
Out[2]=2

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

Highlight the k-cores of a graph:

In[1]:=1
✖
https://wolfram.com/xid/0ywuc1asntar-etupyc
In[2]:=2
✖
https://wolfram.com/xid/0ywuc1asntar-dcimw6
Out[2]=2

Find the degeneracy of a graph g, being the largest k such that g has a k-core:

In[1]:=1
✖
https://wolfram.com/xid/0ywuc1asntar-45cp1o
In[2]:=2
✖
https://wolfram.com/xid/0ywuc1asntar-5daysf
Out[2]=2

Trees and forests are 1-degenerate graphs:

In[3]:=3
✖
https://wolfram.com/xid/0ywuc1asntar-h1llma
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0ywuc1asntar-g0js8k
Out[4]=4

The Barabasi–Albert model with k edges added at each step is k-degenerate:

In[5]:=5
✖
https://wolfram.com/xid/0ywuc1asntar-eavg8q
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0ywuc1asntar-ki1xzp
Out[6]=6

A social network:

In[1]:=1
✖
https://wolfram.com/xid/0ywuc1asntar-cmlbzy

Group actors:

In[2]:=2
✖
https://wolfram.com/xid/0ywuc1asntar-mh0acz
Out[2]=2

Highlight groups:

In[3]:=3
✖
https://wolfram.com/xid/0ywuc1asntar-eugq8
Out[3]=3

Properties & Relations  (8)Properties of the function, and connections to other functions

Find k-core components by repeatedly removing vertices of out-degree less than k:

In[1]:=1
✖
https://wolfram.com/xid/0ywuc1asntar-8evpm7
In[2]:=2
✖
https://wolfram.com/xid/0ywuc1asntar-s7fsx2
In[3]:=3
✖
https://wolfram.com/xid/0ywuc1asntar-vjzb8e
In[4]:=4
✖
https://wolfram.com/xid/0ywuc1asntar-db9mxn

First iteration:

In[5]:=5
✖
https://wolfram.com/xid/0ywuc1asntar-5x7xt5
Out[5]=5

Second iteration:

In[6]:=6
✖
https://wolfram.com/xid/0ywuc1asntar-i6v07e
Out[6]=6

No more vertices are removed by further iteration:

In[7]:=7
✖
https://wolfram.com/xid/0ywuc1asntar-yfdtq2
Out[7]=7

Use ConnectedComponents to obtain the components of the k-core:

In[8]:=8
✖
https://wolfram.com/xid/0ywuc1asntar-rer7xc
Out[8]=8

The obtained k-cores of undirected graphs are connected:

In[1]:=1
✖
https://wolfram.com/xid/0ywuc1asntar-nys1go
In[2]:=2
✖
https://wolfram.com/xid/0ywuc1asntar-saorci
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ywuc1asntar-kksg0h
Out[3]=3

A vertex in a k-core component ci has at least k neighbors in ci:

In[1]:=1
✖
https://wolfram.com/xid/0ywuc1asntar-jhlksp
In[2]:=2
✖
https://wolfram.com/xid/0ywuc1asntar-la7oi9
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ywuc1asntar-g72l24
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0ywuc1asntar-fhbuub
Out[4]=4

For an undirected graph, the vertex in-degree and out-degree are equal to the vertex degree:

In[1]:=1
✖
https://wolfram.com/xid/0ywuc1asntar-qaga4
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ywuc1asntar-m13mg
Out[2]=2

The k-core vertex in-degree and out-degree components are equal to the k-core components:

In[3]:=3
✖
https://wolfram.com/xid/0ywuc1asntar-b7tef
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0ywuc1asntar-cmw7wt
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0ywuc1asntar-buftm0
Out[5]=5

The maximum clique of size k+1 is contained in a k-core component:

In[1]:=1
✖
https://wolfram.com/xid/0ywuc1asntar-bkiavs
In[2]:=2
✖
https://wolfram.com/xid/0ywuc1asntar-bge618
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ywuc1asntar-xi3ik
Out[3]=3

A k-core component is contained in a (k-1)-core component:

In[1]:=1
✖
https://wolfram.com/xid/0ywuc1asntar-mpj5t7
In[2]:=2
✖
https://wolfram.com/xid/0ywuc1asntar-gk1hgd
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ywuc1asntar-b8sofq
Out[3]=3

The adjacency matrix of a k-core component has at least k nonzero entries in each row:

In[1]:=1
✖
https://wolfram.com/xid/0ywuc1asntar-ctqp1j
In[2]:=2
✖
https://wolfram.com/xid/0ywuc1asntar-eiscnm
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ywuc1asntar-eoh9xz
In[4]:=4
✖
https://wolfram.com/xid/0ywuc1asntar-cb5vqh
Out[4]=4

The adjacency matrix of a k-core in-degree component has at least k nonzero entries in each column:

In[1]:=1
✖
https://wolfram.com/xid/0ywuc1asntar-gz6v0n
In[2]:=2
✖
https://wolfram.com/xid/0ywuc1asntar-iabe27
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ywuc1asntar-x4c9b
In[4]:=4
✖
https://wolfram.com/xid/0ywuc1asntar-hyx1hr
Out[4]=4

The adjacency matrix of a k-core out-degree component has at least k nonzero entries in each row:

In[5]:=5
✖
https://wolfram.com/xid/0ywuc1asntar-dlrr3l
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0ywuc1asntar-by0dx0
In[7]:=7
✖
https://wolfram.com/xid/0ywuc1asntar-k9qsee
Out[7]=7
Wolfram Research (2010), KCoreComponents, Wolfram Language function, https://reference.wolfram.com/language/ref/KCoreComponents.html (updated 2015).
✖
Wolfram Research (2010), KCoreComponents, Wolfram Language function, https://reference.wolfram.com/language/ref/KCoreComponents.html (updated 2015).

Text

Wolfram Research (2010), KCoreComponents, Wolfram Language function, https://reference.wolfram.com/language/ref/KCoreComponents.html (updated 2015).

✖
Wolfram Research (2010), KCoreComponents, Wolfram Language function, https://reference.wolfram.com/language/ref/KCoreComponents.html (updated 2015).

CMS

Wolfram Language. 2010. "KCoreComponents." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/KCoreComponents.html.

✖
Wolfram Language. 2010. "KCoreComponents." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/KCoreComponents.html.

APA

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

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

BibTeX

@misc{reference.wolfram_2025_kcorecomponents, author="Wolfram Research", title="{KCoreComponents}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/KCoreComponents.html}", note=[Accessed: 30-May-2025 ]}

✖
@misc{reference.wolfram_2025_kcorecomponents, author="Wolfram Research", title="{KCoreComponents}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/KCoreComponents.html}", note=[Accessed: 30-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_kcorecomponents, organization={Wolfram Research}, title={KCoreComponents}, year={2015}, url={https://reference.wolfram.com/language/ref/KCoreComponents.html}, note=[Accessed: 30-May-2025 ]}

✖
@online{reference.wolfram_2025_kcorecomponents, organization={Wolfram Research}, title={KCoreComponents}, year={2015}, url={https://reference.wolfram.com/language/ref/KCoreComponents.html}, note=[Accessed: 30-May-2025 ]}