"""
Link prediction algorithms.
"""
from math import log
import networkx as nx
from networkx.utils import not_implemented_for
__all__ = [
"resource_allocation_index",
"jaccard_coefficient",
"adamic_adar_index",
"preferential_attachment",
"cn_soundarajan_hopcroft",
"ra_index_soundarajan_hopcroft",
"within_inter_cluster",
"common_neighbor_centrality",
]
def _apply_prediction(G, func, ebunch=None):
"""Applies the given function to each edge in the specified iterable
of edges.
`G` is an instance of :class:`networkx.Graph`.
`func` is a function on two inputs, each of which is a node in the
graph. The function can return anything, but it should return a
value representing a prediction of the likelihood of a "link"
joining the two nodes.
`ebunch` is an iterable of pairs of nodes. If not specified, all
non-edges in the graph `G` will be used.
"""
if ebunch is None:
ebunch = nx.non_edges(G)
return ((u, v, func(u, v)) for u, v in ebunch)
[docs]@not_implemented_for("directed")
@not_implemented_for("multigraph")
def resource_allocation_index(G, ebunch=None):
r"""Compute the resource allocation index of all node pairs in ebunch.
Resource allocation index of `u` and `v` is defined as
.. math::
\sum_{w \in \Gamma(u) \cap \Gamma(v)} \frac{1}{|\Gamma(w)|}
where $\Gamma(u)$ denotes the set of neighbors of $u$.
Parameters
----------
G : graph
A NetworkX undirected graph.
ebunch : iterable of node pairs, optional (default = None)
Resource allocation index will be computed for each pair of
nodes given in the iterable. The pairs must be given as
2-tuples (u, v) where u and v are nodes in the graph. If ebunch
is None then all non-existent edges in the graph will be used.
Default value: None.
Returns
-------
piter : iterator
An iterator of 3-tuples in the form (u, v, p) where (u, v) is a
pair of nodes and p is their resource allocation index.
Examples
--------
>>> G = nx.complete_graph(5)
>>> preds = nx.resource_allocation_index(G, [(0, 1), (2, 3)])
>>> for u, v, p in preds:
... print(f"({u}, {v}) -> {p:.8f}")
(0, 1) -> 0.75000000
(2, 3) -> 0.75000000
References
----------
.. [1] T. Zhou, L. Lu, Y.-C. Zhang.
Predicting missing links via local information.
Eur. Phys. J. B 71 (2009) 623.
https://arxiv.org/pdf/0901.0553.pdf
"""
def predict(u, v):
return sum(1 / G.degree(w) for w in nx.common_neighbors(G, u, v))
return _apply_prediction(G, predict, ebunch)
[docs]@not_implemented_for("directed")
@not_implemented_for("multigraph")
def jaccard_coefficient(G, ebunch=None):
r"""Compute the Jaccard coefficient of all node pairs in ebunch.
Jaccard coefficient of nodes `u` and `v` is defined as
.. math::
\frac{|\Gamma(u) \cap \Gamma(v)|}{|\Gamma(u) \cup \Gamma(v)|}
where $\Gamma(u)$ denotes the set of neighbors of $u$.
Parameters
----------
G : graph
A NetworkX undirected graph.
ebunch : iterable of node pairs, optional (default = None)
Jaccard coefficient will be computed for each pair of nodes
given in the iterable. The pairs must be given as 2-tuples
(u, v) where u and v are nodes in the graph. If ebunch is None
then all non-existent edges in the graph will be used.
Default value: None.
Returns
-------
piter : iterator
An iterator of 3-tuples in the form (u, v, p) where (u, v) is a
pair of nodes and p is their Jaccard coefficient.
Examples
--------
>>> G = nx.complete_graph(5)
>>> preds = nx.jaccard_coefficient(G, [(0, 1), (2, 3)])
>>> for u, v, p in preds:
... print(f"({u}, {v}) -> {p:.8f}")
(0, 1) -> 0.60000000
(2, 3) -> 0.60000000
References
----------
.. [1] D. Liben-Nowell, J. Kleinberg.
The Link Prediction Problem for Social Networks (2004).
http://www.cs.cornell.edu/home/kleinber/link-pred.pdf
"""
def predict(u, v):
union_size = len(set(G[u]) | set(G[v]))
if union_size == 0:
return 0
return len(list(nx.common_neighbors(G, u, v))) / union_size
return _apply_prediction(G, predict, ebunch)
[docs]@not_implemented_for("directed")
@not_implemented_for("multigraph")
def adamic_adar_index(G, ebunch=None):
r"""Compute the Adamic-Adar index of all node pairs in ebunch.
Adamic-Adar index of `u` and `v` is defined as
.. math::
\sum_{w \in \Gamma(u) \cap \Gamma(v)} \frac{1}{\log |\Gamma(w)|}
where $\Gamma(u)$ denotes the set of neighbors of $u$.
This index leads to zero-division for nodes only connected via self-loops.
It is intended to be used when no self-loops are present.
Parameters
----------
G : graph
NetworkX undirected graph.
ebunch : iterable of node pairs, optional (default = None)
Adamic-Adar index will be computed for each pair of nodes given
in the iterable. The pairs must be given as 2-tuples (u, v)
where u and v are nodes in the graph. If ebunch is None then all
non-existent edges in the graph will be used.
Default value: None.
Returns
-------
piter : iterator
An iterator of 3-tuples in the form (u, v, p) where (u, v) is a
pair of nodes and p is their Adamic-Adar index.
Examples
--------
>>> G = nx.complete_graph(5)
>>> preds = nx.adamic_adar_index(G, [(0, 1), (2, 3)])
>>> for u, v, p in preds:
... print(f"({u}, {v}) -> {p:.8f}")
(0, 1) -> 2.16404256
(2, 3) -> 2.16404256
References
----------
.. [1] D. Liben-Nowell, J. Kleinberg.
The Link Prediction Problem for Social Networks (2004).
http://www.cs.cornell.edu/home/kleinber/link-pred.pdf
"""
def predict(u, v):
return sum(1 / log(G.degree(w)) for w in nx.common_neighbors(G, u, v))
return _apply_prediction(G, predict, ebunch)
[docs]@not_implemented_for("directed")
@not_implemented_for("multigraph")
def common_neighbor_centrality(G, ebunch=None, alpha=0.8):
r"""Return the CCPA score for each pair of nodes.
Compute the Common Neighbor and Centrality based Parameterized Algorithm(CCPA)
score of all node pairs in ebunch.
CCPA score of `u` and `v` is defined as
.. math::
\alpha \cdot (|\Gamma (u){\cap }^{}\Gamma (v)|)+(1-\alpha )\cdot \frac{N}{{d}_{uv}}
where $\Gamma(u)$ denotes the set of neighbors of $u$, $\Gamma(v)$ denotes the
set of neighbors of $v$, $\alpha$ is parameter varies between [0,1], $N$ denotes
total number of nodes in the Graph and ${d}_{uv}$ denotes shortest distance
between $u$ and $v$.
This algorithm is based on two vital properties of nodes, namely the number
of common neighbors and their centrality. Common neighbor refers to the common
nodes between two nodes. Centrality refers to the prestige that a node enjoys
in a network.
.. seealso::
:func:`common_neighbors`
Parameters
----------
G : graph
NetworkX undirected graph.
ebunch : iterable of node pairs, optional (default = None)
Preferential attachment score will be computed for each pair of
nodes given in the iterable. The pairs must be given as
2-tuples (u, v) where u and v are nodes in the graph. If ebunch
is None then all non-existent edges in the graph will be used.
Default value: None.
alpha : Parameter defined for participation of Common Neighbor
and Centrality Algorithm share. Default value set to 0.8
because author found better performance at 0.8 for all the
dataset.
Default value: 0.8
Returns
-------
piter : iterator
An iterator of 3-tuples in the form (u, v, p) where (u, v) is a
pair of nodes and p is their Common Neighbor and Centrality based
Parameterized Algorithm(CCPA) score.
Examples
--------
>>> G = nx.complete_graph(5)
>>> preds = nx.common_neighbor_centrality(G, [(0, 1), (2, 3)])
>>> for u, v, p in preds:
... print(f"({u}, {v}) -> {p}")
(0, 1) -> 3.4000000000000004
(2, 3) -> 3.4000000000000004
References
----------
.. [1] Ahmad, I., Akhtar, M.U., Noor, S. et al.
Missing Link Prediction using Common Neighbor and Centrality based Parameterized Algorithm.
Sci Rep 10, 364 (2020).
https://doi.org/10.1038/s41598-019-57304-y
"""
shortest_path = nx.shortest_path(G)
def predict(u, v):
return alpha * len(list(nx.common_neighbors(G, u, v))) + (1 - alpha) * (
G.number_of_nodes() / (len(shortest_path[u][v]) - 1)
)
return _apply_prediction(G, predict, ebunch)
[docs]@not_implemented_for("directed")
@not_implemented_for("multigraph")
def preferential_attachment(G, ebunch=None):
r"""Compute the preferential attachment score of all node pairs in ebunch.
Preferential attachment score of `u` and `v` is defined as
.. math::
|\Gamma(u)| |\Gamma(v)|
where $\Gamma(u)$ denotes the set of neighbors of $u$.
Parameters
----------
G : graph
NetworkX undirected graph.
ebunch : iterable of node pairs, optional (default = None)
Preferential attachment score will be computed for each pair of
nodes given in the iterable. The pairs must be given as
2-tuples (u, v) where u and v are nodes in the graph. If ebunch
is None then all non-existent edges in the graph will be used.
Default value: None.
Returns
-------
piter : iterator
An iterator of 3-tuples in the form (u, v, p) where (u, v) is a
pair of nodes and p is their preferential attachment score.
Examples
--------
>>> G = nx.complete_graph(5)
>>> preds = nx.preferential_attachment(G, [(0, 1), (2, 3)])
>>> for u, v, p in preds:
... print(f"({u}, {v}) -> {p}")
(0, 1) -> 16
(2, 3) -> 16
References
----------
.. [1] D. Liben-Nowell, J. Kleinberg.
The Link Prediction Problem for Social Networks (2004).
http://www.cs.cornell.edu/home/kleinber/link-pred.pdf
"""
def predict(u, v):
return G.degree(u) * G.degree(v)
return _apply_prediction(G, predict, ebunch)
[docs]@not_implemented_for("directed")
@not_implemented_for("multigraph")
def cn_soundarajan_hopcroft(G, ebunch=None, community="community"):
r"""Count the number of common neighbors of all node pairs in ebunch
using community information.
For two nodes $u$ and $v$, this function computes the number of
common neighbors and bonus one for each common neighbor belonging to
the same community as $u$ and $v$. Mathematically,
.. math::
|\Gamma(u) \cap \Gamma(v)| + \sum_{w \in \Gamma(u) \cap \Gamma(v)} f(w)
where $f(w)$ equals 1 if $w$ belongs to the same community as $u$
and $v$ or 0 otherwise and $\Gamma(u)$ denotes the set of
neighbors of $u$.
Parameters
----------
G : graph
A NetworkX undirected graph.
ebunch : iterable of node pairs, optional (default = None)
The score will be computed for each pair of nodes given in the
iterable. The pairs must be given as 2-tuples (u, v) where u
and v are nodes in the graph. If ebunch is None then all
non-existent edges in the graph will be used.
Default value: None.
community : string, optional (default = 'community')
Nodes attribute name containing the community information.
G[u][community] identifies which community u belongs to. Each
node belongs to at most one community. Default value: 'community'.
Returns
-------
piter : iterator
An iterator of 3-tuples in the form (u, v, p) where (u, v) is a
pair of nodes and p is their score.
Examples
--------
>>> G = nx.path_graph(3)
>>> G.nodes[0]["community"] = 0
>>> G.nodes[1]["community"] = 0
>>> G.nodes[2]["community"] = 0
>>> preds = nx.cn_soundarajan_hopcroft(G, [(0, 2)])
>>> for u, v, p in preds:
... print(f"({u}, {v}) -> {p}")
(0, 2) -> 2
References
----------
.. [1] Sucheta Soundarajan and John Hopcroft.
Using community information to improve the precision of link
prediction methods.
In Proceedings of the 21st international conference companion on
World Wide Web (WWW '12 Companion). ACM, New York, NY, USA, 607-608.
http://doi.acm.org/10.1145/2187980.2188150
"""
def predict(u, v):
Cu = _community(G, u, community)
Cv = _community(G, v, community)
cnbors = list(nx.common_neighbors(G, u, v))
neighbors = (
sum(_community(G, w, community) == Cu for w in cnbors) if Cu == Cv else 0
)
return len(cnbors) + neighbors
return _apply_prediction(G, predict, ebunch)
[docs]@not_implemented_for("directed")
@not_implemented_for("multigraph")
def ra_index_soundarajan_hopcroft(G, ebunch=None, community="community"):
r"""Compute the resource allocation index of all node pairs in
ebunch using community information.
For two nodes $u$ and $v$, this function computes the resource
allocation index considering only common neighbors belonging to the
same community as $u$ and $v$. Mathematically,
.. math::
\sum_{w \in \Gamma(u) \cap \Gamma(v)} \frac{f(w)}{|\Gamma(w)|}
where $f(w)$ equals 1 if $w$ belongs to the same community as $u$
and $v$ or 0 otherwise and $\Gamma(u)$ denotes the set of
neighbors of $u$.
Parameters
----------
G : graph
A NetworkX undirected graph.
ebunch : iterable of node pairs, optional (default = None)
The score will be computed for each pair of nodes given in the
iterable. The pairs must be given as 2-tuples (u, v) where u
and v are nodes in the graph. If ebunch is None then all
non-existent edges in the graph will be used.
Default value: None.
community : string, optional (default = 'community')
Nodes attribute name containing the community information.
G[u][community] identifies which community u belongs to. Each
node belongs to at most one community. Default value: 'community'.
Returns
-------
piter : iterator
An iterator of 3-tuples in the form (u, v, p) where (u, v) is a
pair of nodes and p is their score.
Examples
--------
>>> G = nx.Graph()
>>> G.add_edges_from([(0, 1), (0, 2), (1, 3), (2, 3)])
>>> G.nodes[0]["community"] = 0
>>> G.nodes[1]["community"] = 0
>>> G.nodes[2]["community"] = 1
>>> G.nodes[3]["community"] = 0
>>> preds = nx.ra_index_soundarajan_hopcroft(G, [(0, 3)])
>>> for u, v, p in preds:
... print(f"({u}, {v}) -> {p:.8f}")
(0, 3) -> 0.50000000
References
----------
.. [1] Sucheta Soundarajan and John Hopcroft.
Using community information to improve the precision of link
prediction methods.
In Proceedings of the 21st international conference companion on
World Wide Web (WWW '12 Companion). ACM, New York, NY, USA, 607-608.
http://doi.acm.org/10.1145/2187980.2188150
"""
def predict(u, v):
Cu = _community(G, u, community)
Cv = _community(G, v, community)
if Cu != Cv:
return 0
cnbors = nx.common_neighbors(G, u, v)
return sum(1 / G.degree(w) for w in cnbors if _community(G, w, community) == Cu)
return _apply_prediction(G, predict, ebunch)
[docs]@not_implemented_for("directed")
@not_implemented_for("multigraph")
def within_inter_cluster(G, ebunch=None, delta=0.001, community="community"):
"""Compute the ratio of within- and inter-cluster common neighbors
of all node pairs in ebunch.
For two nodes `u` and `v`, if a common neighbor `w` belongs to the
same community as them, `w` is considered as within-cluster common
neighbor of `u` and `v`. Otherwise, it is considered as
inter-cluster common neighbor of `u` and `v`. The ratio between the
size of the set of within- and inter-cluster common neighbors is
defined as the WIC measure. [1]_
Parameters
----------
G : graph
A NetworkX undirected graph.
ebunch : iterable of node pairs, optional (default = None)
The WIC measure will be computed for each pair of nodes given in
the iterable. The pairs must be given as 2-tuples (u, v) where
u and v are nodes in the graph. If ebunch is None then all
non-existent edges in the graph will be used.
Default value: None.
delta : float, optional (default = 0.001)
Value to prevent division by zero in case there is no
inter-cluster common neighbor between two nodes. See [1]_ for
details. Default value: 0.001.
community : string, optional (default = 'community')
Nodes attribute name containing the community information.
G[u][community] identifies which community u belongs to. Each
node belongs to at most one community. Default value: 'community'.
Returns
-------
piter : iterator
An iterator of 3-tuples in the form (u, v, p) where (u, v) is a
pair of nodes and p is their WIC measure.
Examples
--------
>>> G = nx.Graph()
>>> G.add_edges_from([(0, 1), (0, 2), (0, 3), (1, 4), (2, 4), (3, 4)])
>>> G.nodes[0]["community"] = 0
>>> G.nodes[1]["community"] = 1
>>> G.nodes[2]["community"] = 0
>>> G.nodes[3]["community"] = 0
>>> G.nodes[4]["community"] = 0
>>> preds = nx.within_inter_cluster(G, [(0, 4)])
>>> for u, v, p in preds:
... print(f"({u}, {v}) -> {p:.8f}")
(0, 4) -> 1.99800200
>>> preds = nx.within_inter_cluster(G, [(0, 4)], delta=0.5)
>>> for u, v, p in preds:
... print(f"({u}, {v}) -> {p:.8f}")
(0, 4) -> 1.33333333
References
----------
.. [1] Jorge Carlos Valverde-Rebaza and Alneu de Andrade Lopes.
Link prediction in complex networks based on cluster information.
In Proceedings of the 21st Brazilian conference on Advances in
Artificial Intelligence (SBIA'12)
https://doi.org/10.1007/978-3-642-34459-6_10
"""
if delta <= 0:
raise nx.NetworkXAlgorithmError("Delta must be greater than zero")
def predict(u, v):
Cu = _community(G, u, community)
Cv = _community(G, v, community)
if Cu != Cv:
return 0
cnbors = set(nx.common_neighbors(G, u, v))
within = {w for w in cnbors if _community(G, w, community) == Cu}
inter = cnbors - within
return len(within) / (len(inter) + delta)
return _apply_prediction(G, predict, ebunch)
def _community(G, u, community):
"""Get the community of the given node."""
node_u = G.nodes[u]
try:
return node_u[community]
except KeyError as e:
raise nx.NetworkXAlgorithmError("No community information") from e