Spaces:
Running
Running
File size: 19,968 Bytes
b200bda |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 |
"""
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)
@not_implemented_for("directed")
@not_implemented_for("multigraph")
@nx._dispatch
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 nonexistent 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)
@not_implemented_for("directed")
@not_implemented_for("multigraph")
@nx._dispatch
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 nonexistent 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)
@not_implemented_for("directed")
@not_implemented_for("multigraph")
@nx._dispatch
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
nonexistent 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)
@not_implemented_for("directed")
@not_implemented_for("multigraph")
@nx._dispatch
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 nonexistent edges in the graph will be used.
Default value: None.
alpha : Parameter defined for participation of Common Neighbor
and Centrality Algorithm share. Values for alpha should
normally be between 0 and 1. 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
"""
# When alpha == 1, the CCPA score simplifies to the number of common neighbors.
if alpha == 1:
def predict(u, v):
if u == v:
raise nx.NetworkXAlgorithmError("Self links are not supported")
return sum(1 for _ in nx.common_neighbors(G, u, v))
else:
spl = dict(nx.shortest_path_length(G))
inf = float("inf")
def predict(u, v):
if u == v:
raise nx.NetworkXAlgorithmError("Self links are not supported")
path_len = spl[u].get(v, inf)
return alpha * sum(1 for _ in nx.common_neighbors(G, u, v)) + (
1 - alpha
) * (G.number_of_nodes() / path_len)
return _apply_prediction(G, predict, ebunch)
@not_implemented_for("directed")
@not_implemented_for("multigraph")
@nx._dispatch
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 nonexistent 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)
@not_implemented_for("directed")
@not_implemented_for("multigraph")
@nx._dispatch(node_attrs="community")
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
nonexistent 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)
@not_implemented_for("directed")
@not_implemented_for("multigraph")
@nx._dispatch(node_attrs="community")
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
nonexistent 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)
@not_implemented_for("directed")
@not_implemented_for("multigraph")
@nx._dispatch(node_attrs="community")
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
nonexistent 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 err:
raise nx.NetworkXAlgorithmError("No community information") from err
|