Warning

This documents an unmaintained version of NetworkX. Please upgrade to a maintained version and see the current NetworkX documentation.

# Source code for networkx.algorithms.core

"""
Find the k-cores of a graph.

The k-core is found by recursively pruning nodes with degrees less than k.

See the following reference for details:

An O(m) Algorithm for Cores Decomposition of Networks
Vladimir Batagelj and Matjaz Zaversnik, 2003.
http://arxiv.org/abs/cs.DS/0310049

"""
__author__ = "\n".join(['Dan Schult (dschult@colgate.edu)',
'Jason Grout (jason-sage@creativetrax.com)',
'Aric Hagberg (hagberg@lanl.gov)'])

#    Aric Hagberg <hagberg@lanl.gov>
#    Dan Schult <dschult@colgate.edu>
#    Pieter Swart <swart@lanl.gov>
__all__ = ['core_number','k_core','k_shell','k_crust','k_corona','find_cores']

import networkx as nx

[docs]def core_number(G):
"""Return the core number for each vertex.

A k-core is a maximal subgraph that contains nodes of degree k or more.

The core number of a node is the largest value k of a k-core containing
that node.

Parameters
----------
G : NetworkX graph
A graph or directed graph

Returns
-------
core_number : dictionary
A dictionary keyed by node to the core number.

Raises
------
NetworkXError
The k-core is not defined for graphs with self loops or parallel edges.

Notes
-----
Not implemented for graphs with parallel edges or self loops.

For directed graphs the node degree is defined to be the
in-degree + out-degree.

References
----------
.. [1] An O(m) Algorithm for Cores Decomposition of Networks
Vladimir Batagelj and Matjaz Zaversnik, 2003.
http://arxiv.org/abs/cs.DS/0310049
"""
if G.is_multigraph():
raise nx.NetworkXError(
'MultiGraph and MultiDiGraph types not supported.')

if G.number_of_selfloops()>0:
raise nx.NetworkXError(
'Input graph has self loops; the core number is not defined.',
'Consider using G.remove_edges_from(G.selfloop_edges()).')

if G.is_directed():
import itertools
def neighbors(v):
return itertools.chain.from_iterable([G.predecessors_iter(v),
G.successors_iter(v)])
else:
neighbors=G.neighbors_iter
degrees=G.degree()
# sort nodes by degree
nodes=sorted(degrees,key=degrees.get)
bin_boundaries=[0]
curr_degree=0
for i,v in enumerate(nodes):
if degrees[v]>curr_degree:
bin_boundaries.extend([i]*(degrees[v]-curr_degree))
curr_degree=degrees[v]
node_pos = dict((v,pos) for pos,v in enumerate(nodes))
# initial guesses for core is degree
core=degrees
nbrs=dict((v,set(neighbors(v))) for v in G)
for v in nodes:
for u in nbrs[v]:
if core[u] > core[v]:
nbrs[u].remove(v)
pos=node_pos[u]
bin_start=bin_boundaries[core[u]]
node_pos[u]=bin_start
node_pos[nodes[bin_start]]=pos
nodes[bin_start],nodes[pos]=nodes[pos],nodes[bin_start]
bin_boundaries[core[u]]+=1
core[u]-=1
return core

find_cores=core_number

[docs]def k_core(G,k=None,core_number=None):
"""Return the k-core of G.

A k-core is a maximal subgraph that contains nodes of degree k or more.

Parameters
----------
G : NetworkX graph
A graph or directed graph
k : int, optional
The order of the core.  If not specified return the main core.
core_number : dictionary, optional
Precomputed core numbers for the graph G.

Returns
-------
G : NetworkX graph
The k-core subgraph

Raises
------
NetworkXError
The k-core is not defined for graphs with self loops or parallel edges.

Notes
-----
The main core is the core with the largest degree.

Not implemented for graphs with parallel edges or self loops.

For directed graphs the node degree is defined to be the
in-degree + out-degree.

Graph, node, and edge attributes are copied to the subgraph.

--------
core_number

References
----------
.. [1] An O(m) Algorithm for Cores Decomposition of Networks
Vladimir Batagelj and Matjaz Zaversnik,  2003.
http://arxiv.org/abs/cs.DS/0310049
"""
if core_number is None:
core_number=nx.core_number(G)
if k is None:
k=max(core_number.values()) # max core
nodes=(n for n in core_number if core_number[n]>=k)
return G.subgraph(nodes).copy()

[docs]def k_shell(G,k=None,core_number=None):
"""Return the k-shell of G.

The k-shell is the subgraph of nodes in the k-core but not in the (k+1)-core.

Parameters
----------
G : NetworkX graph
A graph or directed graph.
k : int, optional
The order of the shell.  If not specified return the main shell.
core_number : dictionary, optional
Precomputed core numbers for the graph G.

Returns
-------
G : NetworkX graph
The k-shell subgraph

Raises
------
NetworkXError
The k-shell is not defined for graphs with self loops or parallel edges.

Notes
-----
This is similar to k_corona but in that case only neighbors in the
k-core are considered.

Not implemented for graphs with parallel edges or self loops.

For directed graphs the node degree is defined to be the
in-degree + out-degree.

Graph, node, and edge attributes are copied to the subgraph.

--------
core_number
k_corona

----------
.. [1] A model of Internet topology using k-shell decomposition
Shai Carmi, Shlomo Havlin, Scott Kirkpatrick, Yuval Shavitt,
and Eran Shir, PNAS  July 3, 2007   vol. 104  no. 27  11150-11154
http://www.pnas.org/content/104/27/11150.full
"""
if core_number is None:
core_number=nx.core_number(G)
if k is None:
k=max(core_number.values()) # max core
nodes=(n for n in core_number if core_number[n]==k)
return G.subgraph(nodes).copy()

[docs]def k_crust(G,k=None,core_number=None):
"""Return the k-crust of G.

The k-crust is the graph G with the k-core removed.

Parameters
----------
G : NetworkX graph
A graph or directed graph.
k : int, optional
The order of the shell.  If not specified return the main crust.
core_number : dictionary, optional
Precomputed core numbers for the graph G.

Returns
-------
G : NetworkX graph
The k-crust subgraph

Raises
------
NetworkXError
The k-crust is not defined for graphs with self loops or parallel edges.

Notes
-----
This definition of k-crust is different than the definition in [1]_.
The k-crust in [1]_ is equivalent to the k+1 crust of this algorithm.

Not implemented for graphs with parallel edges or self loops.

For directed graphs the node degree is defined to be the
in-degree + out-degree.

Graph, node, and edge attributes are copied to the subgraph.

--------
core_number

References
----------
.. [1] A model of Internet topology using k-shell decomposition
Shai Carmi, Shlomo Havlin, Scott Kirkpatrick, Yuval Shavitt,
and Eran Shir, PNAS  July 3, 2007   vol. 104  no. 27  11150-11154
http://www.pnas.org/content/104/27/11150.full
"""
if core_number is None:
core_number=nx.core_number(G)
if k is None:
k=max(core_number.values())-1
nodes=(n for n in core_number if core_number[n]<=k)
return G.subgraph(nodes).copy()

[docs]def k_corona(G, k, core_number=None):
"""Return the k-corona of G.

The k-corona is the subgraph of nodes in the k-core which have
exactly k neighbours in the k-core.

Parameters
----------
G : NetworkX graph
A graph or directed graph
k : int
The order of the corona.
core_number : dictionary, optional
Precomputed core numbers for the graph G.

Returns
-------
G : NetworkX graph
The k-corona subgraph

Raises
------
NetworkXError
The k-cornoa is not defined for graphs with self loops or
parallel edges.

Notes
-----
Not implemented for graphs with parallel edges or self loops.

For directed graphs the node degree is defined to be the
in-degree + out-degree.

Graph, node, and edge attributes are copied to the subgraph.

--------
core_number

References
----------
.. [1]  k -core (bootstrap) percolation on complex networks:
Critical phenomena and nonlocal effects,
A. V. Goltsev, S. N. Dorogovtsev, and J. F. F. Mendes,
Phys. Rev. E 73, 056101 (2006)