"""Algorithms to characterize the number of triangles in a graph."""
from itertools import chain
from itertools import combinations
from collections import Counter
from networkx.utils import not_implemented_for
__all__ = [
"triangles",
"average_clustering",
"clustering",
"transitivity",
"square_clustering",
"generalized_degree",
]
[docs]@not_implemented_for("directed")
def triangles(G, nodes=None):
"""Compute the number of triangles.
Finds the number of triangles that include a node as one vertex.
Parameters
----------
G : graph
A networkx graph
nodes : container of nodes, optional (default= all nodes in G)
Compute triangles for nodes in this container.
Returns
-------
out : dictionary
Number of triangles keyed by node label.
Examples
--------
>>> G = nx.complete_graph(5)
>>> print(nx.triangles(G, 0))
6
>>> print(nx.triangles(G))
{0: 6, 1: 6, 2: 6, 3: 6, 4: 6}
>>> print(list(nx.triangles(G, (0, 1)).values()))
[6, 6]
Notes
-----
When computing triangles for the entire graph each triangle is counted
three times, once at each node. Self loops are ignored.
"""
# If `nodes` represents a single node in the graph, return only its number
# of triangles.
if nodes in G:
return next(_triangles_and_degree_iter(G, nodes))[2] // 2
# Otherwise, `nodes` represents an iterable of nodes, so return a
# dictionary mapping node to number of triangles.
return {v: t // 2 for v, d, t, _ in _triangles_and_degree_iter(G, nodes)}
@not_implemented_for("multigraph")
def _triangles_and_degree_iter(G, nodes=None):
"""Return an iterator of (node, degree, triangles, generalized degree).
This double counts triangles so you may want to divide by 2.
See degree(), triangles() and generalized_degree() for definitions
and details.
"""
if nodes is None:
nodes_nbrs = G.adj.items()
else:
nodes_nbrs = ((n, G[n]) for n in G.nbunch_iter(nodes))
for v, v_nbrs in nodes_nbrs:
vs = set(v_nbrs) - {v}
gen_degree = Counter(len(vs & (set(G[w]) - {w})) for w in vs)
ntriangles = sum(k * val for k, val in gen_degree.items())
yield (v, len(vs), ntriangles, gen_degree)
@not_implemented_for("multigraph")
def _weighted_triangles_and_degree_iter(G, nodes=None, weight="weight"):
"""Return an iterator of (node, degree, weighted_triangles).
Used for weighted clustering.
Note: this returns the geometric average weight of edges in the triangle.
Also, each triangle is counted twice (each direction).
So you may want to divide by 2.
"""
import numpy as np
if weight is None or G.number_of_edges() == 0:
max_weight = 1
else:
max_weight = max(d.get(weight, 1) for u, v, d in G.edges(data=True))
if nodes is None:
nodes_nbrs = G.adj.items()
else:
nodes_nbrs = ((n, G[n]) for n in G.nbunch_iter(nodes))
def wt(u, v):
return G[u][v].get(weight, 1) / max_weight
for i, nbrs in nodes_nbrs:
inbrs = set(nbrs) - {i}
weighted_triangles = 0
seen = set()
for j in inbrs:
seen.add(j)
# This avoids counting twice -- we double at the end.
jnbrs = set(G[j]) - seen
# Only compute the edge weight once, before the inner inner
# loop.
wij = wt(i, j)
weighted_triangles += sum(
np.cbrt([(wij * wt(j, k) * wt(k, i)) for k in inbrs & jnbrs])
)
yield (i, len(inbrs), 2 * weighted_triangles)
@not_implemented_for("multigraph")
def _directed_triangles_and_degree_iter(G, nodes=None):
"""Return an iterator of
(node, total_degree, reciprocal_degree, directed_triangles).
Used for directed clustering.
Note that unlike `_triangles_and_degree_iter()`, this function counts
directed triangles so does not count triangles twice.
"""
nodes_nbrs = ((n, G._pred[n], G._succ[n]) for n in G.nbunch_iter(nodes))
for i, preds, succs in nodes_nbrs:
ipreds = set(preds) - {i}
isuccs = set(succs) - {i}
directed_triangles = 0
for j in chain(ipreds, isuccs):
jpreds = set(G._pred[j]) - {j}
jsuccs = set(G._succ[j]) - {j}
directed_triangles += sum(
1
for k in chain(
(ipreds & jpreds),
(ipreds & jsuccs),
(isuccs & jpreds),
(isuccs & jsuccs),
)
)
dtotal = len(ipreds) + len(isuccs)
dbidirectional = len(ipreds & isuccs)
yield (i, dtotal, dbidirectional, directed_triangles)
@not_implemented_for("multigraph")
def _directed_weighted_triangles_and_degree_iter(G, nodes=None, weight="weight"):
"""Return an iterator of
(node, total_degree, reciprocal_degree, directed_weighted_triangles).
Used for directed weighted clustering.
Note that unlike `_weighted_triangles_and_degree_iter()`, this function counts
directed triangles so does not count triangles twice.
"""
import numpy as np
if weight is None or G.number_of_edges() == 0:
max_weight = 1
else:
max_weight = max(d.get(weight, 1) for u, v, d in G.edges(data=True))
nodes_nbrs = ((n, G._pred[n], G._succ[n]) for n in G.nbunch_iter(nodes))
def wt(u, v):
return G[u][v].get(weight, 1) / max_weight
for i, preds, succs in nodes_nbrs:
ipreds = set(preds) - {i}
isuccs = set(succs) - {i}
directed_triangles = 0
for j in ipreds:
jpreds = set(G._pred[j]) - {j}
jsuccs = set(G._succ[j]) - {j}
directed_triangles += sum(
np.cbrt([(wt(j, i) * wt(k, i) * wt(k, j)) for k in ipreds & jpreds])
)
directed_triangles += sum(
np.cbrt([(wt(j, i) * wt(k, i) * wt(j, k)) for k in ipreds & jsuccs])
)
directed_triangles += sum(
np.cbrt([(wt(j, i) * wt(i, k) * wt(k, j)) for k in isuccs & jpreds])
)
directed_triangles += sum(
np.cbrt([(wt(j, i) * wt(i, k) * wt(j, k)) for k in isuccs & jsuccs])
)
for j in isuccs:
jpreds = set(G._pred[j]) - {j}
jsuccs = set(G._succ[j]) - {j}
directed_triangles += sum(
np.cbrt([(wt(i, j) * wt(k, i) * wt(k, j)) for k in ipreds & jpreds])
)
directed_triangles += sum(
np.cbrt([(wt(i, j) * wt(k, i) * wt(j, k)) for k in ipreds & jsuccs])
)
directed_triangles += sum(
np.cbrt([(wt(i, j) * wt(i, k) * wt(k, j)) for k in isuccs & jpreds])
)
directed_triangles += sum(
np.cbrt([(wt(i, j) * wt(i, k) * wt(j, k)) for k in isuccs & jsuccs])
)
dtotal = len(ipreds) + len(isuccs)
dbidirectional = len(ipreds & isuccs)
yield (i, dtotal, dbidirectional, directed_triangles)
[docs]def average_clustering(G, nodes=None, weight=None, count_zeros=True):
r"""Compute the average clustering coefficient for the graph G.
The clustering coefficient for the graph is the average,
.. math::
C = \frac{1}{n}\sum_{v \in G} c_v,
where :math:`n` is the number of nodes in `G`.
Parameters
----------
G : graph
nodes : container of nodes, optional (default=all nodes in G)
Compute average clustering for nodes in this container.
weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight.
If None, then each edge has weight 1.
count_zeros : bool
If False include only the nodes with nonzero clustering in the average.
Returns
-------
avg : float
Average clustering
Examples
--------
>>> G = nx.complete_graph(5)
>>> print(nx.average_clustering(G))
1.0
Notes
-----
This is a space saving routine; it might be faster
to use the clustering function to get a list and then take the average.
Self loops are ignored.
References
----------
.. [1] Generalizations of the clustering coefficient to weighted
complex networks by J. Saramäki, M. Kivelä, J.-P. Onnela,
K. Kaski, and J. Kertész, Physical Review E, 75 027105 (2007).
http://jponnela.com/web_documents/a9.pdf
.. [2] Marcus Kaiser, Mean clustering coefficients: the role of isolated
nodes and leafs on clustering measures for small-world networks.
https://arxiv.org/abs/0802.2512
"""
c = clustering(G, nodes, weight=weight).values()
if not count_zeros:
c = [v for v in c if abs(v) > 0]
return sum(c) / len(c)
[docs]def clustering(G, nodes=None, weight=None):
r"""Compute the clustering coefficient for nodes.
For unweighted graphs, the clustering of a node :math:`u`
is the fraction of possible triangles through that node that exist,
.. math::
c_u = \frac{2 T(u)}{deg(u)(deg(u)-1)},
where :math:`T(u)` is the number of triangles through node :math:`u` and
:math:`deg(u)` is the degree of :math:`u`.
For weighted graphs, there are several ways to define clustering [1]_.
the one used here is defined
as the geometric average of the subgraph edge weights [2]_,
.. math::
c_u = \frac{1}{deg(u)(deg(u)-1))}
\sum_{vw} (\hat{w}_{uv} \hat{w}_{uw} \hat{w}_{vw})^{1/3}.
The edge weights :math:`\hat{w}_{uv}` are normalized by the maximum weight
in the network :math:`\hat{w}_{uv} = w_{uv}/\max(w)`.
The value of :math:`c_u` is assigned to 0 if :math:`deg(u) < 2`.
Additionally, this weighted definition has been generalized to support negative edge weights [3]_.
For directed graphs, the clustering is similarly defined as the fraction
of all possible directed triangles or geometric average of the subgraph
edge weights for unweighted and weighted directed graph respectively [4]_.
.. math::
c_u = \frac{2}{deg^{tot}(u)(deg^{tot}(u)-1) - 2deg^{\leftrightarrow}(u)}
T(u),
where :math:`T(u)` is the number of directed triangles through node
:math:`u`, :math:`deg^{tot}(u)` is the sum of in degree and out degree of
:math:`u` and :math:`deg^{\leftrightarrow}(u)` is the reciprocal degree of
:math:`u`.
Parameters
----------
G : graph
nodes : container of nodes, optional (default=all nodes in G)
Compute clustering for nodes in this container.
weight : string or None, optional (default=None)
The edge attribute that holds the numerical value used as a weight.
If None, then each edge has weight 1.
Returns
-------
out : float, or dictionary
Clustering coefficient at specified nodes
Examples
--------
>>> G = nx.complete_graph(5)
>>> print(nx.clustering(G, 0))
1.0
>>> print(nx.clustering(G))
{0: 1.0, 1: 1.0, 2: 1.0, 3: 1.0, 4: 1.0}
Notes
-----
Self loops are ignored.
References
----------
.. [1] Generalizations of the clustering coefficient to weighted
complex networks by J. Saramäki, M. Kivelä, J.-P. Onnela,
K. Kaski, and J. Kertész, Physical Review E, 75 027105 (2007).
http://jponnela.com/web_documents/a9.pdf
.. [2] Intensity and coherence of motifs in weighted complex
networks by J. P. Onnela, J. Saramäki, J. Kertész, and K. Kaski,
Physical Review E, 71(6), 065103 (2005).
.. [3] Generalization of Clustering Coefficients to Signed Correlation Networks
by G. Costantini and M. Perugini, PloS one, 9(2), e88669 (2014).
.. [4] Clustering in complex directed networks by G. Fagiolo,
Physical Review E, 76(2), 026107 (2007).
"""
if G.is_directed():
if weight is not None:
td_iter = _directed_weighted_triangles_and_degree_iter(G, nodes, weight)
clusterc = {
v: 0 if t == 0 else t / ((dt * (dt - 1) - 2 * db) * 2)
for v, dt, db, t in td_iter
}
else:
td_iter = _directed_triangles_and_degree_iter(G, nodes)
clusterc = {
v: 0 if t == 0 else t / ((dt * (dt - 1) - 2 * db) * 2)
for v, dt, db, t in td_iter
}
else:
# The formula 2*T/(d*(d-1)) from docs is t/(d*(d-1)) here b/c t==2*T
if weight is not None:
td_iter = _weighted_triangles_and_degree_iter(G, nodes, weight)
clusterc = {v: 0 if t == 0 else t / (d * (d - 1)) for v, d, t in td_iter}
else:
td_iter = _triangles_and_degree_iter(G, nodes)
clusterc = {v: 0 if t == 0 else t / (d * (d - 1)) for v, d, t, _ in td_iter}
if nodes in G:
# Return the value of the sole entry in the dictionary.
return clusterc[nodes]
return clusterc
[docs]def transitivity(G):
r"""Compute graph transitivity, the fraction of all possible triangles
present in G.
Possible triangles are identified by the number of "triads"
(two edges with a shared vertex).
The transitivity is
.. math::
T = 3\frac{\#triangles}{\#triads}.
Parameters
----------
G : graph
Returns
-------
out : float
Transitivity
Examples
--------
>>> G = nx.complete_graph(5)
>>> print(nx.transitivity(G))
1.0
"""
triangles_contri = [
(t, d * (d - 1)) for v, d, t, _ in _triangles_and_degree_iter(G)
]
# If the graph is empty
if len(triangles_contri) == 0:
return 0
triangles, contri = map(sum, zip(*triangles_contri))
return 0 if triangles == 0 else triangles / contri
[docs]def square_clustering(G, nodes=None):
r"""Compute the squares clustering coefficient for nodes.
For each node return the fraction of possible squares that exist at
the node [1]_
.. math::
C_4(v) = \frac{ \sum_{u=1}^{k_v}
\sum_{w=u+1}^{k_v} q_v(u,w) }{ \sum_{u=1}^{k_v}
\sum_{w=u+1}^{k_v} [a_v(u,w) + q_v(u,w)]},
where :math:`q_v(u,w)` are the number of common neighbors of :math:`u` and
:math:`w` other than :math:`v` (ie squares), and :math:`a_v(u,w) = (k_u -
(1+q_v(u,w)+\theta_{uv})) + (k_w - (1+q_v(u,w)+\theta_{uw}))`, where
:math:`\theta_{uw} = 1` if :math:`u` and :math:`w` are connected and 0
otherwise. [2]_
Parameters
----------
G : graph
nodes : container of nodes, optional (default=all nodes in G)
Compute clustering for nodes in this container.
Returns
-------
c4 : dictionary
A dictionary keyed by node with the square clustering coefficient value.
Examples
--------
>>> G = nx.complete_graph(5)
>>> print(nx.square_clustering(G, 0))
1.0
>>> print(nx.square_clustering(G))
{0: 1.0, 1: 1.0, 2: 1.0, 3: 1.0, 4: 1.0}
Notes
-----
While :math:`C_3(v)` (triangle clustering) gives the probability that
two neighbors of node v are connected with each other, :math:`C_4(v)` is
the probability that two neighbors of node v share a common
neighbor different from v. This algorithm can be applied to both
bipartite and unipartite networks.
References
----------
.. [1] Pedro G. Lind, Marta C. González, and Hans J. Herrmann. 2005
Cycles and clustering in bipartite networks.
Physical Review E (72) 056127.
.. [2] Zhang, Peng et al. Clustering Coefficient and Community Structure of
Bipartite Networks. Physica A: Statistical Mechanics and its Applications 387.27 (2008): 6869–6875.
https://arxiv.org/abs/0710.0117v1
"""
if nodes is None:
node_iter = G
else:
node_iter = G.nbunch_iter(nodes)
clustering = {}
for v in node_iter:
clustering[v] = 0
potential = 0
for u, w in combinations(G[v], 2):
squares = len((set(G[u]) & set(G[w])) - {v})
clustering[v] += squares
degm = squares + 1
if w in G[u]:
degm += 1
potential += (len(G[u]) - degm) + (len(G[w]) - degm) + squares
if potential > 0:
clustering[v] /= potential
if nodes in G:
# Return the value of the sole entry in the dictionary.
return clustering[nodes]
return clustering
[docs]@not_implemented_for("directed")
def generalized_degree(G, nodes=None):
r"""Compute the generalized degree for nodes.
For each node, the generalized degree shows how many edges of given
triangle multiplicity the node is connected to. The triangle multiplicity
of an edge is the number of triangles an edge participates in. The
generalized degree of node :math:`i` can be written as a vector
:math:`\mathbf{k}_i=(k_i^{(0)}, \dotsc, k_i^{(N-2)})` where
:math:`k_i^{(j)}` is the number of edges attached to node :math:`i` that
participate in :math:`j` triangles.
Parameters
----------
G : graph
nodes : container of nodes, optional (default=all nodes in G)
Compute the generalized degree for nodes in this container.
Returns
-------
out : Counter, or dictionary of Counters
Generalized degree of specified nodes. The Counter is keyed by edge
triangle multiplicity.
Examples
--------
>>> G = nx.complete_graph(5)
>>> print(nx.generalized_degree(G, 0))
Counter({3: 4})
>>> print(nx.generalized_degree(G))
{0: Counter({3: 4}), 1: Counter({3: 4}), 2: Counter({3: 4}), 3: Counter({3: 4}), 4: Counter({3: 4})}
To recover the number of triangles attached to a node:
>>> k1 = nx.generalized_degree(G, 0)
>>> sum([k * v for k, v in k1.items()]) / 2 == nx.triangles(G, 0)
True
Notes
-----
In a network of N nodes, the highest triangle multiplicty an edge can have
is N-2.
The return value does not include a `zero` entry if no edges of a
particular triangle multiplicity are present.
The number of triangles node :math:`i` is attached to can be recovered from
the generalized degree :math:`\mathbf{k}_i=(k_i^{(0)}, \dotsc,
k_i^{(N-2)})` by :math:`(k_i^{(1)}+2k_i^{(2)}+\dotsc +(N-2)k_i^{(N-2)})/2`.
References
----------
.. [1] Networks with arbitrary edge multiplicities by V. Zlatić,
D. Garlaschelli and G. Caldarelli, EPL (Europhysics Letters),
Volume 97, Number 2 (2012).
https://iopscience.iop.org/article/10.1209/0295-5075/97/28005
"""
if nodes in G:
return next(_triangles_and_degree_iter(G, nodes))[3]
return {v: gd for v, d, t, gd in _triangles_and_degree_iter(G, nodes)}