"""
Find the k-cores of a graph.
The k-core is found by recursively pruning nodes with degrees less than k.
See the following references for details:
An O(m) Algorithm for Cores Decomposition of Networks
Vladimir Batagelj and Matjaz Zaversnik, 2003.
https://arxiv.org/abs/cs.DS/0310049
Generalized Cores
Vladimir Batagelj and Matjaz Zaversnik, 2002.
https://arxiv.org/pdf/cs/0202039
For directed graphs a more general notion is that of D-cores which
looks at (k, l) restrictions on (in, out) degree. The (k, k) D-core
is the k-core.
D-cores: Measuring Collaboration of Directed Graphs Based on Degeneracy
Christos Giatsidis, Dimitrios M. Thilikos, Michalis Vazirgiannis, ICDM 2011.
http://www.graphdegeneracy.org/dcores_ICDM_2011.pdf
Multi-scale structure and topological anomaly detection via a new network \
statistic: The onion decomposition
L. Hébert-Dufresne, J. A. Grochow, and A. Allard
Scientific Reports 6, 31708 (2016)
http://doi.org/10.1038/srep31708
"""
import networkx as nx
__all__ = [
"core_number",
"k_core",
"k_shell",
"k_crust",
"k_corona",
"k_truss",
"onion_layers",
]
[docs]
@nx.utils.not_implemented_for("multigraph")
@nx._dispatchable
def core_number(G):
"""Returns the core number for each node.
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
An undirected or directed graph
Returns
-------
core_number : dictionary
A dictionary keyed by node to the core number.
Raises
------
NetworkXNotImplemented
If `G` is a multigraph or contains self loops.
Notes
-----
For directed graphs the node degree is defined to be the
in-degree + out-degree.
Examples
--------
>>> degrees = [0, 1, 2, 2, 2, 2, 3]
>>> H = nx.havel_hakimi_graph(degrees)
>>> nx.core_number(H)
{0: 1, 1: 2, 2: 2, 3: 2, 4: 1, 5: 2, 6: 0}
>>> G = nx.DiGraph()
>>> G.add_edges_from([(1, 2), (2, 1), (2, 3), (2, 4), (3, 4), (4, 3)])
>>> nx.core_number(G)
{1: 2, 2: 2, 3: 2, 4: 2}
References
----------
.. [1] An O(m) Algorithm for Cores Decomposition of Networks
Vladimir Batagelj and Matjaz Zaversnik, 2003.
https://arxiv.org/abs/cs.DS/0310049
"""
if nx.number_of_selfloops(G) > 0:
msg = (
"Input graph has self loops which is not permitted; "
"Consider using G.remove_edges_from(nx.selfloop_edges(G))."
)
raise nx.NetworkXNotImplemented(msg)
degrees = dict(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 = {v: pos for pos, v in enumerate(nodes)}
# The initial guess for the core number of a node is its degree.
core = degrees
nbrs = {v: list(nx.all_neighbors(G, 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
def _core_subgraph(G, k_filter, k=None, core=None):
"""Returns the subgraph induced by nodes passing filter `k_filter`.
Parameters
----------
G : NetworkX graph
The graph or directed graph to process
k_filter : filter function
This function filters the nodes chosen. It takes three inputs:
A node of G, the filter's cutoff, and the core dict of the graph.
The function should return a Boolean value.
k : int, optional
The order of the core. If not specified use the max core number.
This value is used as the cutoff for the filter.
core : dict, optional
Precomputed core numbers keyed by node for the graph `G`.
If not specified, the core numbers will be computed from `G`.
"""
if core is None:
core = core_number(G)
if k is None:
k = max(core.values())
nodes = (v for v in core if k_filter(v, k, core))
return G.subgraph(nodes).copy()
[docs]
@nx._dispatchable(preserve_all_attrs=True, returns_graph=True)
def k_core(G, k=None, core_number=None):
"""Returns the k-core of G.
A k-core is a maximal subgraph that contains nodes of degree `k` or more.
.. deprecated:: 3.3
`k_core` will not accept `MultiGraph` objects in version 3.5.
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
------
NetworkXNotImplemented
The k-core is not defined for multigraphs or graphs with self loops.
Notes
-----
The main core is the core with `k` as the largest core_number.
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.
Examples
--------
>>> degrees = [0, 1, 2, 2, 2, 2, 3]
>>> H = nx.havel_hakimi_graph(degrees)
>>> H.degree
DegreeView({0: 1, 1: 2, 2: 2, 3: 2, 4: 2, 5: 3, 6: 0})
>>> nx.k_core(H).nodes
NodeView((1, 2, 3, 5))
See Also
--------
core_number
References
----------
.. [1] An O(m) Algorithm for Cores Decomposition of Networks
Vladimir Batagelj and Matjaz Zaversnik, 2003.
https://arxiv.org/abs/cs.DS/0310049
"""
import warnings
if G.is_multigraph():
warnings.warn(
(
"\n\n`k_core` will not accept `MultiGraph` objects in version 3.5.\n"
"Convert it to an undirected graph instead, using::\n\n"
"\tG = nx.Graph(G)\n"
),
category=DeprecationWarning,
stacklevel=5,
)
def k_filter(v, k, c):
return c[v] >= k
return _core_subgraph(G, k_filter, k, core_number)
[docs]
@nx._dispatchable(preserve_all_attrs=True, returns_graph=True)
def k_shell(G, k=None, core_number=None):
"""Returns the k-shell of G.
The k-shell is the subgraph induced by nodes with core number k.
That is, nodes in the k-core that are not in the (k+1)-core.
.. deprecated:: 3.3
`k_shell` will not accept `MultiGraph` objects in version 3.5.
Parameters
----------
G : NetworkX graph
A graph or directed graph.
k : int, optional
The order of the shell. If not specified return the outer shell.
core_number : dictionary, optional
Precomputed core numbers for the graph G.
Returns
-------
G : NetworkX graph
The k-shell subgraph
Raises
------
NetworkXNotImplemented
The k-shell is not implemented for multigraphs or graphs with self loops.
Notes
-----
This is similar to k_corona but in that case only neighbors in the
k-core are considered.
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.
Examples
--------
>>> degrees = [0, 1, 2, 2, 2, 2, 3]
>>> H = nx.havel_hakimi_graph(degrees)
>>> H.degree
DegreeView({0: 1, 1: 2, 2: 2, 3: 2, 4: 2, 5: 3, 6: 0})
>>> nx.k_shell(H, k=1).nodes
NodeView((0, 4))
See Also
--------
core_number
k_corona
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
"""
import warnings
if G.is_multigraph():
warnings.warn(
(
"\n\n`k_shell` will not accept `MultiGraph` objects in version 3.5.\n"
"Convert it to an undirected graph instead, using::\n\n"
"\tG = nx.Graph(G)\n"
),
category=DeprecationWarning,
stacklevel=5,
)
def k_filter(v, k, c):
return c[v] == k
return _core_subgraph(G, k_filter, k, core_number)
[docs]
@nx._dispatchable(preserve_all_attrs=True, returns_graph=True)
def k_crust(G, k=None, core_number=None):
"""Returns the k-crust of G.
The k-crust is the graph G with the edges of the k-core removed
and isolated nodes found after the removal of edges are also removed.
.. deprecated:: 3.3
`k_crust` will not accept `MultiGraph` objects in version 3.5.
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
------
NetworkXNotImplemented
The k-crust is not implemented for multigraphs or graphs with self loops.
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.
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.
Examples
--------
>>> degrees = [0, 1, 2, 2, 2, 2, 3]
>>> H = nx.havel_hakimi_graph(degrees)
>>> H.degree
DegreeView({0: 1, 1: 2, 2: 2, 3: 2, 4: 2, 5: 3, 6: 0})
>>> nx.k_crust(H, k=1).nodes
NodeView((0, 4, 6))
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
"""
import warnings
if G.is_multigraph():
warnings.warn(
(
"\n\n`k_crust` will not accept `MultiGraph` objects in version 3.5.\n"
"Convert it to an undirected graph instead, using::\n\n"
"\tG = nx.Graph(G)\n"
),
category=DeprecationWarning,
stacklevel=5,
)
# Default for k is one less than in _core_subgraph, so just inline.
# Filter is c[v] <= k
if core_number is None:
core_number = nx.core_number(G)
if k is None:
k = max(core_number.values()) - 1
nodes = (v for v in core_number if core_number[v] <= k)
return G.subgraph(nodes).copy()
[docs]
@nx._dispatchable(preserve_all_attrs=True, returns_graph=True)
def k_corona(G, k, core_number=None):
"""Returns the k-corona of G.
The k-corona is the subgraph of nodes in the k-core which have
exactly k neighbors in the k-core.
.. deprecated:: 3.3
`k_corona` will not accept `MultiGraph` objects in version 3.5.
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
------
NetworkXNotImplemented
The k-corona is not defined for multigraphs or graphs with self loops.
Notes
-----
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.
Examples
--------
>>> degrees = [0, 1, 2, 2, 2, 2, 3]
>>> H = nx.havel_hakimi_graph(degrees)
>>> H.degree
DegreeView({0: 1, 1: 2, 2: 2, 3: 2, 4: 2, 5: 3, 6: 0})
>>> nx.k_corona(H, k=2).nodes
NodeView((1, 2, 3, 5))
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
"""
import warnings
if G.is_multigraph():
warnings.warn(
(
"\n\n`k_corona` will not accept `MultiGraph` objects in version 3.5.\n"
"Convert it to an undirected graph instead, using::\n\n"
"\tG = nx.Graph(G)\n"
),
category=DeprecationWarning,
stacklevel=5,
)
def func(v, k, c):
return c[v] == k and k == sum(1 for w in G[v] if c[w] >= k)
return _core_subgraph(G, func, k, core_number)
[docs]
@nx.utils.not_implemented_for("directed")
@nx.utils.not_implemented_for("multigraph")
@nx._dispatchable(preserve_all_attrs=True, returns_graph=True)
def k_truss(G, k):
"""Returns the k-truss of `G`.
The k-truss is the maximal induced subgraph of `G` which contains at least
three vertices where every edge is incident to at least `k-2` triangles.
Parameters
----------
G : NetworkX graph
An undirected graph
k : int
The order of the truss
Returns
-------
H : NetworkX graph
The k-truss subgraph
Raises
------
NetworkXNotImplemented
If `G` is a multigraph or directed graph or if it contains self loops.
Notes
-----
A k-clique is a (k-2)-truss and a k-truss is a (k+1)-core.
Graph, node, and edge attributes are copied to the subgraph.
K-trusses were originally defined in [2] which states that the k-truss
is the maximal induced subgraph where each edge belongs to at least
`k-2` triangles. A more recent paper, [1], uses a slightly different
definition requiring that each edge belong to at least `k` triangles.
This implementation uses the original definition of `k-2` triangles.
Examples
--------
>>> degrees = [0, 1, 2, 2, 2, 2, 3]
>>> H = nx.havel_hakimi_graph(degrees)
>>> H.degree
DegreeView({0: 1, 1: 2, 2: 2, 3: 2, 4: 2, 5: 3, 6: 0})
>>> nx.k_truss(H, k=2).nodes
NodeView((0, 1, 2, 3, 4, 5))
References
----------
.. [1] Bounds and Algorithms for k-truss. Paul Burkhardt, Vance Faber,
David G. Harris, 2018. https://arxiv.org/abs/1806.05523v2
.. [2] Trusses: Cohesive Subgraphs for Social Network Analysis. Jonathan
Cohen, 2005.
"""
if nx.number_of_selfloops(G) > 0:
msg = (
"Input graph has self loops which is not permitted; "
"Consider using G.remove_edges_from(nx.selfloop_edges(G))."
)
raise nx.NetworkXNotImplemented(msg)
H = G.copy()
n_dropped = 1
while n_dropped > 0:
n_dropped = 0
to_drop = []
seen = set()
for u in H:
nbrs_u = set(H[u])
seen.add(u)
new_nbrs = [v for v in nbrs_u if v not in seen]
for v in new_nbrs:
if len(nbrs_u & set(H[v])) < (k - 2):
to_drop.append((u, v))
H.remove_edges_from(to_drop)
n_dropped = len(to_drop)
H.remove_nodes_from(list(nx.isolates(H)))
return H
[docs]
@nx.utils.not_implemented_for("multigraph")
@nx.utils.not_implemented_for("directed")
@nx._dispatchable
def onion_layers(G):
"""Returns the layer of each vertex in an onion decomposition of the graph.
The onion decomposition refines the k-core decomposition by providing
information on the internal organization of each k-shell. It is usually
used alongside the `core numbers`.
Parameters
----------
G : NetworkX graph
An undirected graph without self loops.
Returns
-------
od_layers : dictionary
A dictionary keyed by node to the onion layer. The layers are
contiguous integers starting at 1.
Raises
------
NetworkXNotImplemented
If `G` is a multigraph or directed graph or if it contains self loops.
Examples
--------
>>> degrees = [0, 1, 2, 2, 2, 2, 3]
>>> H = nx.havel_hakimi_graph(degrees)
>>> H.degree
DegreeView({0: 1, 1: 2, 2: 2, 3: 2, 4: 2, 5: 3, 6: 0})
>>> nx.onion_layers(H)
{6: 1, 0: 2, 4: 3, 1: 4, 2: 4, 3: 4, 5: 4}
See Also
--------
core_number
References
----------
.. [1] Multi-scale structure and topological anomaly detection via a new
network statistic: The onion decomposition
L. Hébert-Dufresne, J. A. Grochow, and A. Allard
Scientific Reports 6, 31708 (2016)
http://doi.org/10.1038/srep31708
.. [2] Percolation and the effective structure of complex networks
A. Allard and L. Hébert-Dufresne
Physical Review X 9, 011023 (2019)
http://doi.org/10.1103/PhysRevX.9.011023
"""
if nx.number_of_selfloops(G) > 0:
msg = (
"Input graph contains self loops which is not permitted; "
"Consider using G.remove_edges_from(nx.selfloop_edges(G))."
)
raise nx.NetworkXNotImplemented(msg)
# Dictionaries to register the k-core/onion decompositions.
od_layers = {}
# Adjacency list
neighbors = {v: list(nx.all_neighbors(G, v)) for v in G}
# Effective degree of nodes.
degrees = dict(G.degree())
# Performs the onion decomposition.
current_core = 1
current_layer = 1
# Sets vertices of degree 0 to layer 1, if any.
isolated_nodes = list(nx.isolates(G))
if len(isolated_nodes) > 0:
for v in isolated_nodes:
od_layers[v] = current_layer
degrees.pop(v)
current_layer = 2
# Finds the layer for the remaining nodes.
while len(degrees) > 0:
# Sets the order for looking at nodes.
nodes = sorted(degrees, key=degrees.get)
# Sets properly the current core.
min_degree = degrees[nodes[0]]
if min_degree > current_core:
current_core = min_degree
# Identifies vertices in the current layer.
this_layer = []
for n in nodes:
if degrees[n] > current_core:
break
this_layer.append(n)
# Identifies the core/layer of the vertices in the current layer.
for v in this_layer:
od_layers[v] = current_layer
for n in neighbors[v]:
neighbors[n].remove(v)
degrees[n] = degrees[n] - 1
degrees.pop(v)
# Updates the layer count.
current_layer = current_layer + 1
# Returns the dictionaries containing the onion layer of each vertices.
return od_layers
