wehub-resource-sync
296 lines
7.9 KiB
Python
296 lines
7.9 KiB
Python
"""Algorithms for computing nodal clustering coefficients."""
|
|
|
|
import numpy as np
|
|
|
|
from easygraph.utils.exception import EasyGraphError
|
|
|
|
|
|
__all__ = [
|
|
"hypergraph_clustering_coefficient",
|
|
"hypergraph_local_clustering_coefficient",
|
|
"hypergraph_two_node_clustering_coefficient",
|
|
]
|
|
|
|
|
|
def hypergraph_clustering_coefficient(H):
|
|
r"""Return the clustering coefficients for
|
|
each node in a Hypergraph.
|
|
|
|
This clustering coefficient is defined as the
|
|
clustering coefficient of the unweighted pairwise
|
|
projection of the hypergraph, i.e.,
|
|
:math:`c = A^3_{i,i}/\binom{k}{2},`
|
|
where :math:`A` is the adjacency matrix of the network
|
|
and :math:`k` is the pairwise degree of :math:`i`.
|
|
|
|
Parameters
|
|
----------
|
|
H : Hypergraph
|
|
Hypergraph
|
|
|
|
Returns
|
|
-------
|
|
dict
|
|
nodes are keys, clustering coefficients are values.
|
|
|
|
Notes
|
|
-----
|
|
The clustering coefficient is undefined when the number of
|
|
neighbors is 0 or 1, but we set the clustering coefficient
|
|
to 0 in these cases. For more discussion, see
|
|
https://arxiv.org/abs/0802.2512
|
|
|
|
See Also
|
|
--------
|
|
local_clustering_coefficient
|
|
two_node_clustering_coefficient
|
|
|
|
References
|
|
----------
|
|
"Clustering Coefficients in Protein Interaction Hypernetworks"
|
|
by Suzanne Gallagher and Debra Goldberg.
|
|
DOI: 10.1145/2506583.2506635
|
|
|
|
Example
|
|
-------
|
|
>>> import easygraph as eg
|
|
>>> H = eg.random_hypergraph(3, [1, 1])
|
|
>>> cc = eg.clustering_coefficient(H)
|
|
>>> cc
|
|
{0: 1.0, 1: 1.0, 2: 1.0}
|
|
"""
|
|
adj = H.adjacency_matrix()
|
|
k = np.array(adj.sum(axis=1))
|
|
l = []
|
|
for i in k:
|
|
l.append(i[0])
|
|
k = np.array(l)
|
|
denom = k * (k - 1) / 2
|
|
mat = adj.dot(adj).dot(adj)
|
|
with np.errstate(divide="ignore", invalid="ignore"):
|
|
result = np.nan_to_num(0.5 * mat.diagonal() / denom)
|
|
r = {}
|
|
for i in range(0, len(H.v)):
|
|
r[i] = result[i]
|
|
return r
|
|
|
|
|
|
def hypergraph_local_clustering_coefficient(H):
|
|
"""Compute the local clustering coefficient.
|
|
|
|
This clustering coefficient is based on the
|
|
overlap of the edges connected to a given node,
|
|
normalized by the size of the node's neighborhood.
|
|
|
|
Parameters
|
|
----------
|
|
H : Hypergraph
|
|
Hypergraph
|
|
|
|
Returns
|
|
-------
|
|
dict
|
|
keys are node IDs and values are the
|
|
clustering coefficients.
|
|
|
|
Notes
|
|
-----
|
|
The clustering coefficient is undefined when the number of
|
|
neighbors is 0 or 1, but we set the clustering coefficient
|
|
to 0 in these cases. For more discussion, see
|
|
https://arxiv.org/abs/0802.2512
|
|
|
|
See Also
|
|
--------
|
|
clustering_coefficient
|
|
two_node_clustering_coefficient
|
|
|
|
References
|
|
----------
|
|
"Properties of metabolic graphs: biological organization or representation
|
|
artifacts?" by Wanding Zhou and Luay Nakhleh.
|
|
https://doi.org/10.1186/1471-2105-12-132
|
|
|
|
"Hypergraphs for predicting essential genes using multiprotein complex data"
|
|
by Florian Klimm, Charlotte M. Deane, and Gesine Reinert.
|
|
https://doi.org/10.1093/comnet/cnaa028
|
|
|
|
Example
|
|
-------
|
|
>>> import easygraph as eg
|
|
>>> H = eg.random_hypergraph(3, [1, 1])
|
|
>>> cc = eg.hypergraph_local_clustering_coefficient(H)
|
|
>>> cc
|
|
{0: 1.0, 1: 1.0, 2: 1.0}
|
|
|
|
"""
|
|
result = {}
|
|
# 节点属于哪些边
|
|
memberships = []
|
|
for n in H.v:
|
|
tmp = set()
|
|
for index, e in enumerate(H.e[0]):
|
|
if n in e:
|
|
tmp.add(index)
|
|
memberships.append(tmp)
|
|
|
|
# 每条边包含哪些节点
|
|
members = H.e[0]
|
|
for n in H.v:
|
|
ev = memberships[n]
|
|
dv = len(ev)
|
|
if dv <= 1:
|
|
result[n] = 0
|
|
else:
|
|
total_eo = 0
|
|
# go over all pairs of edges pairwise
|
|
for e1 in range(dv):
|
|
edge1 = members[e1]
|
|
for e2 in range(e1):
|
|
edge2 = members[e2]
|
|
# set differences for the hyperedges
|
|
D1 = set(edge1) - set(edge2)
|
|
D2 = set(edge2) - set(edge1)
|
|
# if edges are the same by definition the extra overlap is zero
|
|
if len(D1.union(D2)) == 0:
|
|
eo = 0
|
|
else:
|
|
# otherwise we have to look at their neighbors
|
|
# the neighbors of D1 and D2, respectively.
|
|
neighD1 = {i for d in D1 for i in H.neighbor_of_node(d)}
|
|
neighD2 = {i for d in D2 for i in H.neighbor_of_node(d)}
|
|
# compute extra overlap [len() is used for cardinality of edges]
|
|
eo = (
|
|
len(neighD1.intersection(D2))
|
|
+ len(neighD2.intersection(D1))
|
|
) / len(
|
|
D1.union(D2)
|
|
) # add it up
|
|
# add it up
|
|
total_eo = total_eo + eo
|
|
|
|
# include normalization by degree k*(k-1)/2
|
|
result[n] = 2 * total_eo / (dv * (dv - 1))
|
|
return result
|
|
|
|
|
|
def hypergraph_two_node_clustering_coefficient(H, kind="union"):
|
|
"""Return the clustering coefficients for
|
|
each node in a Hypergraph.
|
|
|
|
This definition averages over all of the
|
|
two-node clustering coefficients involving the node.
|
|
|
|
Parameters
|
|
----------
|
|
H : Hypergraph
|
|
Hypergraph
|
|
kind : string, optional
|
|
The type of two node clustering coefficient. Options
|
|
are "union", "max", and "min". By default, "union".
|
|
|
|
Returns
|
|
-------
|
|
dict
|
|
nodes are keys, clustering coefficients are values.
|
|
|
|
Notes
|
|
-----
|
|
The clustering coefficient is undefined when the number of
|
|
neighbors is 0 or 1, but we set the clustering coefficient
|
|
to 0 in these cases. For more discussion, see
|
|
https://arxiv.org/abs/0802.2512
|
|
|
|
See Also
|
|
--------
|
|
clustering_coefficient
|
|
local_clustering_coefficient
|
|
|
|
References
|
|
----------
|
|
"Clustering Coefficients in Protein Interaction Hypernetworks"
|
|
by Suzanne Gallagher and Debra Goldberg.
|
|
DOI: 10.1145/2506583.2506635
|
|
|
|
Example
|
|
-------
|
|
>>> import easygraph as eg
|
|
>>> H = eg.random_hypergraph(3, [1, 1])
|
|
>>> cc = eg.two_node_clustering_coefficient(H, kind="union")
|
|
>>> cc
|
|
{0: 0.5, 1: 0.5, 2: 0.5}
|
|
"""
|
|
result = {}
|
|
memberships = {}
|
|
for n in H.v:
|
|
tmp = set()
|
|
for index, e in enumerate(H.e[0]):
|
|
if n in e:
|
|
tmp.add(index)
|
|
memberships[n] = tmp
|
|
|
|
for n in H.v:
|
|
neighbors = H.neighbor_of_node(n)
|
|
result[n] = 0.0
|
|
for v in neighbors:
|
|
result[n] += _uv_cc(n, v, memberships, kind=kind) / len(neighbors)
|
|
return result
|
|
|
|
|
|
def _uv_cc(u, v, memberships, kind="union"):
|
|
"""Helper function to compute the two-node
|
|
clustering coefficient.
|
|
|
|
Parameters
|
|
----------
|
|
u : hashable
|
|
First node
|
|
v : hashable
|
|
Second node
|
|
memberships : dict
|
|
node IDs are keys, edge IDs to which they belong
|
|
are values.
|
|
kind : str, optional
|
|
Type of clustering coefficient to compute, by default "union".
|
|
Options:
|
|
|
|
- "union"
|
|
- "max"
|
|
- "min"
|
|
|
|
Returns
|
|
-------
|
|
float
|
|
The clustering coefficient
|
|
|
|
Raises
|
|
------
|
|
EasyGraphError
|
|
If an invalid clustering coefficient kind
|
|
is specified.
|
|
|
|
References
|
|
----------
|
|
"Clustering Coefficients in Protein Interaction Hypernetworks"
|
|
by Suzanne Gallagher and Debra Goldberg.
|
|
DOI: 10.1145/2506583.2506635
|
|
"""
|
|
m_u = memberships[u]
|
|
m_v = memberships[v]
|
|
|
|
num = len(m_u.intersection(m_v))
|
|
|
|
if kind == "union":
|
|
denom = len(m_u.union(m_v))
|
|
elif kind == "min":
|
|
denom = min(len(m_u), len(m_v))
|
|
elif kind == "max":
|
|
denom = max(len(m_u), len(m_v))
|
|
else:
|
|
raise EasyGraphError("Invalid kind of clustering.")
|
|
|
|
if denom == 0:
|
|
return np.nan
|
|
|
|
return num / denom
|