545 lines
17 KiB
Python
545 lines
17 KiB
Python
|
"""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",
|
||
|
|
]
|
||
|
|
|
||
|
|
|
||
|
|
@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.
|
||
|
|
|
||
|
|
"""
|
||
|
|
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 prevents double counting.
|
||
|
|
jnbrs = set(G[j]) - seen
|
||
|
|
# Only compute the edge weight once, before the inner inner
|
||
|
|
# loop.
|
||
|
|
wij = wt(i, j)
|
||
|
|
weighted_triangles += sum(
|
||
|
|
(wij * wt(j, k) * wt(k, i)) ** (1 / 3) 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.
|
||
|
|
|
||
|
|
"""
|
||
|
|
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.
|
||
|
|
|
||
|
|
"""
|
||
|
|
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(
|
||
|
|
(wt(j, i) * wt(k, i) * wt(k, j)) ** (1 / 3) for k in ipreds & jpreds
|
||
|
|
)
|
||
|
|
directed_triangles += sum(
|
||
|
|
(wt(j, i) * wt(k, i) * wt(j, k)) ** (1 / 3) for k in ipreds & jsuccs
|
||
|
|
)
|
||
|
|
directed_triangles += sum(
|
||
|
|
(wt(j, i) * wt(i, k) * wt(k, j)) ** (1 / 3) for k in isuccs & jpreds
|
||
|
|
)
|
||
|
|
directed_triangles += sum(
|
||
|
|
(wt(j, i) * wt(i, k) * wt(j, k)) ** (1 / 3) for k in isuccs & jsuccs
|
||
|
|
)
|
||
|
|
|
||
|
|
for j in isuccs:
|
||
|
|
jpreds = set(G._pred[j]) - {j}
|
||
|
|
jsuccs = set(G._succ[j]) - {j}
|
||
|
|
directed_triangles += sum(
|
||
|
|
(wt(i, j) * wt(k, i) * wt(k, j)) ** (1 / 3) for k in ipreds & jpreds
|
||
|
|
)
|
||
|
|
directed_triangles += sum(
|
||
|
|
(wt(i, j) * wt(k, i) * wt(j, k)) ** (1 / 3) for k in ipreds & jsuccs
|
||
|
|
)
|
||
|
|
directed_triangles += sum(
|
||
|
|
(wt(i, j) * wt(i, k) * wt(k, j)) ** (1 / 3) for k in isuccs & jpreds
|
||
|
|
)
|
||
|
|
directed_triangles += sum(
|
||
|
|
(wt(i, j) * wt(i, k) * wt(j, k)) ** (1 / 3) for k in isuccs & jsuccs
|
||
|
|
)
|
||
|
|
|
||
|
|
dtotal = len(ipreds) + len(isuccs)
|
||
|
|
dbidirectional = len(ipreds & isuccs)
|
||
|
|
yield (i, dtotal, dbidirectional, directed_triangles)
|
||
|
|
|
||
|
|
|
||
|
|
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 v > 0]
|
||
|
|
return sum(c) / len(c)
|
||
|
|
|
||
|
|
|
||
|
|
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`.
|
||
|
|
|
||
|
|
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 [3]_.
|
||
|
|
|
||
|
|
.. math::
|
||
|
|
|
||
|
|
c_u = \frac{1}{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] 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:
|
||
|
|
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
|
||
|
|
|
||
|
|
|
||
|
|
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 = sum(t for v, d, t, _ in _triangles_and_degree_iter(G))
|
||
|
|
contri = sum(d * (d - 1) for v, d, t, _ in _triangles_and_degree_iter(G))
|
||
|
|
return 0 if triangles == 0 else triangles / contri
|
||
|
|
|
||
|
|
|
||
|
|
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.
|
||
|
|
|
||
|
|
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.
|
||
|
|
"""
|
||
|
|
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
|
||
|
|
|
||
|
|
|
||
|
|
@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)}
|