# 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 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
from networkx.exception import NetworkXError
from networkx.utils import not_implemented_for
__all__ = [
"core_number",
"find_cores",
"k_core",
"k_shell",
"k_crust",
"k_corona",
"k_truss",
"onion_layers",
]
[docs]@not_implemented_for("multigraph")
def core_number(G):
"""Returns 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 implemented 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.
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 NetworkXError(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
find_cores = core_number
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]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.
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.
https://arxiv.org/abs/cs.DS/0310049
"""
def k_filter(v, k, c):
return c[v] >= k
return _core_subgraph(G, k_filter, k, core_number)
[docs]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.
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
------
NetworkXError
The k-shell is not implemented 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
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
"""
def k_filter(v, k, c):
return c[v] == k
return _core_subgraph(G, k_filter, k, core_number)
[docs]def k_crust(G, k=None, core_number=None):
"""Returns 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 implemented 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
"""
# 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 = find_cores(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]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 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
"""
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]@not_implemented_for("directed")
@not_implemented_for("multigraph")
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
------
NetworkXError
The k-truss is not defined for graphs with self loops or parallel edges
or directed graphs.
Notes
-----
A k-clique is a (k-2)-truss and a k-truss is a (k+1)-core.
Not implemented for digraphs or graphs with parallel edges or self loops.
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.
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.
"""
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]@not_implemented_for("multigraph")
@not_implemented_for("directed")
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
A simple graph without self loops or parallel edges
Returns
-------
od_layers : dictionary
A dictionary keyed by vertex to the onion layer. The layers are
contiguous integers starting at 1.
Raises
------
NetworkXError
The onion decomposition is not implemented for graphs with self loops
or parallel edges or for directed graphs.
Notes
-----
Not implemented for graphs with parallel edges or self loops.
Not implemented for directed graphs.
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 NetworkXError(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 = [v for v in 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
```