"""
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)'])
# Copyright (C) 2004-2010 by
# Aric Hagberg <hagberg@lanl.gov>
# Dan Schult <dschult@colgate.edu>
# Pieter Swart <swart@lanl.gov>
# All rights reserved.
# BSD license.
__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.
See Also
--------
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 containing
nodes of exactly degree k.
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.
See Also
--------
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.
See Also
--------
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-crust of G.
The k-corona is the subset of vertices 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.
See Also
--------
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)
http://link.aps.org/doi/10.1103/PhysRevE.73.056101
"""
if core_number is None:
core_number = nx.core_number(G)
nodes = (n for n in core_number
if core_number[n] >= k
and len([v for v in G[n] if core_number[v] >= k]) == k)
return G.subgraph(nodes).copy()