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
606
|
"""Graph diameter, radius, eccentricity and other properties."""
import networkx as nx
from networkx.utils import not_implemented_for
__all__ = [
"extrema_bounding",
"eccentricity",
"diameter",
"radius",
"periphery",
"center",
"barycenter",
"resistance_distance",
]
def extrema_bounding(G, compute="diameter"):
"""Compute requested extreme distance metric of undirected graph G
Computation is based on smart lower and upper bounds, and in practice
linear in the number of nodes, rather than quadratic (except for some
border cases such as complete graphs or circle shaped graphs).
Parameters
----------
G : NetworkX graph
An undirected graph
compute : string denoting the requesting metric
"diameter" for the maximal eccentricity value,
"radius" for the minimal eccentricity value,
"periphery" for the set of nodes with eccentricity equal to the diameter
"center" for the set of nodes with eccentricity equal to the radius
Returns
-------
value : value of the requested metric
int for "diameter" and "radius" or
list of nodes for "center" and "periphery"
Raises
------
NetworkXError
If the graph consists of multiple components
Notes
-----
This algorithm was proposed in the following papers:
F.W. Takes and W.A. Kosters, Determining the Diameter of Small World
Networks, in Proceedings of the 20th ACM International Conference on
Information and Knowledge Management (CIKM 2011), pp. 1191-1196, 2011.
doi: https://doi.org/10.1145/2063576.2063748
F.W. Takes and W.A. Kosters, Computing the Eccentricity Distribution of
Large Graphs, Algorithms 6(1): 100-118, 2013.
doi: https://doi.org/10.3390/a6010100
M. Borassi, P. Crescenzi, M. Habib, W.A. Kosters, A. Marino and F.W. Takes,
Fast Graph Diameter and Radius BFS-Based Computation in (Weakly Connected)
Real-World Graphs, Theoretical Computer Science 586: 59-80, 2015.
doi: https://doi.org/10.1016/j.tcs.2015.02.033
"""
# init variables
degrees = dict(G.degree()) # start with the highest degree node
minlowernode = max(degrees, key=degrees.get)
N = len(degrees) # number of nodes
# alternate between smallest lower and largest upper bound
high = False
# status variables
ecc_lower = dict.fromkeys(G, 0)
ecc_upper = dict.fromkeys(G, N)
candidates = set(G)
# (re)set bound extremes
minlower = N
maxlower = 0
minupper = N
maxupper = 0
# repeat the following until there are no more candidates
while candidates:
if high:
current = maxuppernode # select node with largest upper bound
else:
current = minlowernode # select node with smallest lower bound
high = not high
# get distances from/to current node and derive eccentricity
dist = dict(nx.single_source_shortest_path_length(G, current))
if len(dist) != N:
msg = "Cannot compute metric because graph is not connected."
raise nx.NetworkXError(msg)
current_ecc = max(dist.values())
# print status update
# print ("ecc of " + str(current) + " (" + str(ecc_lower[current]) + "/"
# + str(ecc_upper[current]) + ", deg: " + str(dist[current]) + ") is "
# + str(current_ecc))
# print(ecc_upper)
# (re)set bound extremes
maxuppernode = None
minlowernode = None
# update node bounds
for i in candidates:
# update eccentricity bounds
d = dist[i]
ecc_lower[i] = low = max(ecc_lower[i], max(d, (current_ecc - d)))
ecc_upper[i] = upp = min(ecc_upper[i], current_ecc + d)
# update min/max values of lower and upper bounds
minlower = min(ecc_lower[i], minlower)
maxlower = max(ecc_lower[i], maxlower)
minupper = min(ecc_upper[i], minupper)
maxupper = max(ecc_upper[i], maxupper)
# update candidate set
if compute == "diameter":
ruled_out = {
i
for i in candidates
if ecc_upper[i] <= maxlower and 2 * ecc_lower[i] >= maxupper
}
elif compute == "radius":
ruled_out = {
i
for i in candidates
if ecc_lower[i] >= minupper and ecc_upper[i] + 1 <= 2 * minlower
}
elif compute == "periphery":
ruled_out = {
i
for i in candidates
if ecc_upper[i] < maxlower
and (maxlower == maxupper or ecc_lower[i] > maxupper)
}
elif compute == "center":
ruled_out = {
i
for i in candidates
if ecc_lower[i] > minupper
and (minlower == minupper or ecc_upper[i] + 1 < 2 * minlower)
}
elif compute == "eccentricities":
ruled_out = {}
ruled_out.update(i for i in candidates if ecc_lower[i] == ecc_upper[i])
candidates -= ruled_out
# for i in ruled_out:
# print("removing %g: ecc_u: %g maxl: %g ecc_l: %g maxu: %g"%
# (i,ecc_upper[i],maxlower,ecc_lower[i],maxupper))
# print("node %g: ecc_u: %g maxl: %g ecc_l: %g maxu: %g"%
# (4,ecc_upper[4],maxlower,ecc_lower[4],maxupper))
# print("NODE 4: %g"%(ecc_upper[4] <= maxlower))
# print("NODE 4: %g"%(2 * ecc_lower[4] >= maxupper))
# print("NODE 4: %g"%(ecc_upper[4] <= maxlower
# and 2 * ecc_lower[4] >= maxupper))
# updating maxuppernode and minlowernode for selection in next round
for i in candidates:
if (
minlowernode is None
or (
ecc_lower[i] == ecc_lower[minlowernode]
and degrees[i] > degrees[minlowernode]
)
or (ecc_lower[i] < ecc_lower[minlowernode])
):
minlowernode = i
if (
maxuppernode is None
or (
ecc_upper[i] == ecc_upper[maxuppernode]
and degrees[i] > degrees[maxuppernode]
)
or (ecc_upper[i] > ecc_upper[maxuppernode])
):
maxuppernode = i
# print status update
# print (" min=" + str(minlower) + "/" + str(minupper) +
# " max=" + str(maxlower) + "/" + str(maxupper) +
# " candidates: " + str(len(candidates)))
# print("cand:",candidates)
# print("ecc_l",ecc_lower)
# print("ecc_u",ecc_upper)
# wait = input("press Enter to continue")
# return the correct value of the requested metric
if compute == "diameter":
return maxlower
elif compute == "radius":
return minupper
elif compute == "periphery":
p = [v for v in G if ecc_lower[v] == maxlower]
return p
elif compute == "center":
c = [v for v in G if ecc_upper[v] == minupper]
return c
elif compute == "eccentricities":
return ecc_lower
return None
def eccentricity(G, v=None, sp=None):
"""Returns the eccentricity of nodes in G.
The eccentricity of a node v is the maximum distance from v to
all other nodes in G.
Parameters
----------
G : NetworkX graph
A graph
v : node, optional
Return value of specified node
sp : dict of dicts, optional
All pairs shortest path lengths as a dictionary of dictionaries
Returns
-------
ecc : dictionary
A dictionary of eccentricity values keyed by node.
"""
# if v is None: # none, use entire graph
# nodes=G.nodes()
# elif v in G: # is v a single node
# nodes=[v]
# else: # assume v is a container of nodes
# nodes=v
order = G.order()
e = {}
for n in G.nbunch_iter(v):
if sp is None:
length = nx.single_source_shortest_path_length(G, n)
L = len(length)
else:
try:
length = sp[n]
L = len(length)
except TypeError as e:
raise nx.NetworkXError('Format of "sp" is invalid.') from e
if L != order:
if G.is_directed():
msg = (
"Found infinite path length because the digraph is not"
" strongly connected"
)
else:
msg = "Found infinite path length because the graph is not" " connected"
raise nx.NetworkXError(msg)
e[n] = max(length.values())
if v in G:
return e[v] # return single value
else:
return e
def diameter(G, e=None, usebounds=False):
"""Returns the diameter of the graph G.
The diameter is the maximum eccentricity.
Parameters
----------
G : NetworkX graph
A graph
e : eccentricity dictionary, optional
A precomputed dictionary of eccentricities.
Returns
-------
d : integer
Diameter of graph
See Also
--------
eccentricity
"""
if usebounds is True and e is None and not G.is_directed():
return extrema_bounding(G, compute="diameter")
if e is None:
e = eccentricity(G)
return max(e.values())
def periphery(G, e=None, usebounds=False):
"""Returns the periphery of the graph G.
The periphery is the set of nodes with eccentricity equal to the diameter.
Parameters
----------
G : NetworkX graph
A graph
e : eccentricity dictionary, optional
A precomputed dictionary of eccentricities.
Returns
-------
p : list
List of nodes in periphery
See Also
--------
barycenter
center
"""
if usebounds is True and e is None and not G.is_directed():
return extrema_bounding(G, compute="periphery")
if e is None:
e = eccentricity(G)
diameter = max(e.values())
p = [v for v in e if e[v] == diameter]
return p
def radius(G, e=None, usebounds=False):
"""Returns the radius of the graph G.
The radius is the minimum eccentricity.
Parameters
----------
G : NetworkX graph
A graph
e : eccentricity dictionary, optional
A precomputed dictionary of eccentricities.
Returns
-------
r : integer
Radius of graph
"""
if usebounds is True and e is None and not G.is_directed():
return extrema_bounding(G, compute="radius")
if e is None:
e = eccentricity(G)
return min(e.values())
def center(G, e=None, usebounds=False):
"""Returns the center of the graph G.
The center is the set of nodes with eccentricity equal to radius.
Parameters
----------
G : NetworkX graph
A graph
e : eccentricity dictionary, optional
A precomputed dictionary of eccentricities.
Returns
-------
c : list
List of nodes in center
See Also
--------
barycenter
periphery
"""
if usebounds is True and e is None and not G.is_directed():
return extrema_bounding(G, compute="center")
if e is None:
e = eccentricity(G)
radius = min(e.values())
p = [v for v in e if e[v] == radius]
return p
def barycenter(G, weight=None, attr=None, sp=None):
r"""Calculate barycenter of a connected graph, optionally with edge weights.
The :dfn:`barycenter` a
:func:`connected <networkx.algorithms.components.is_connected>` graph
:math:`G` is the subgraph induced by the set of its nodes :math:`v`
minimizing the objective function
.. math::
\sum_{u \in V(G)} d_G(u, v),
where :math:`d_G` is the (possibly weighted) :func:`path length
<networkx.algorithms.shortest_paths.generic.shortest_path_length>`.
The barycenter is also called the :dfn:`median`. See [West01]_, p. 78.
Parameters
----------
G : :class:`networkx.Graph`
The connected graph :math:`G`.
weight : :class:`str`, optional
Passed through to
:func:`~networkx.algorithms.shortest_paths.generic.shortest_path_length`.
attr : :class:`str`, optional
If given, write the value of the objective function to each node's
`attr` attribute. Otherwise do not store the value.
sp : dict of dicts, optional
All pairs shortest path lengths as a dictionary of dictionaries
Returns
-------
list
Nodes of `G` that induce the barycenter of `G`.
Raises
------
NetworkXNoPath
If `G` is disconnected. `G` may appear disconnected to
:func:`barycenter` if `sp` is given but is missing shortest path
lengths for any pairs.
ValueError
If `sp` and `weight` are both given.
See Also
--------
center
periphery
"""
if sp is None:
sp = nx.shortest_path_length(G, weight=weight)
else:
sp = sp.items()
if weight is not None:
raise ValueError("Cannot use both sp, weight arguments together")
smallest, barycenter_vertices, n = float("inf"), [], len(G)
for v, dists in sp:
if len(dists) < n:
raise nx.NetworkXNoPath(
f"Input graph {G} is disconnected, so every induced subgraph "
"has infinite barycentricity."
)
barycentricity = sum(dists.values())
if attr is not None:
G.nodes[v][attr] = barycentricity
if barycentricity < smallest:
smallest = barycentricity
barycenter_vertices = [v]
elif barycentricity == smallest:
barycenter_vertices.append(v)
return barycenter_vertices
def _laplacian_submatrix(node, mat, node_list):
"""Removes row/col from a sparse matrix and returns the submatrix"""
j = node_list.index(node)
n = list(range(len(node_list)))
n.pop(j)
if mat.shape[0] != mat.shape[1]:
raise nx.NetworkXError("Matrix must be square")
elif len(node_list) != mat.shape[0]:
msg = "Node list length does not match matrix dimentions"
raise nx.NetworkXError(msg)
mat = mat.tocsr()
mat = mat[n, :]
mat = mat.tocsc()
mat = mat[:, n]
node_list.pop(j)
return mat, node_list
def _count_lu_permutations(perm_array):
"""Counts the number of permutations in SuperLU perm_c or perm_r"""
perm_cnt = 0
arr = perm_array.tolist()
for i in range(len(arr)):
if i != arr[i]:
perm_cnt += 1
n = arr.index(i)
arr[n] = arr[i]
arr[i] = i
return perm_cnt
@not_implemented_for("directed")
def resistance_distance(G, nodeA, nodeB, weight=None, invert_weight=True):
"""Returns the resistance distance between node A and node B on graph G.
The resistance distance between two nodes of a graph is akin to treating
the graph as a grid of resistorses with a resistance equal to the provided
weight.
If weight is not provided, then a weight of 1 is used for all edges.
Parameters
----------
G : NetworkX graph
A graph
nodeA : node
A node within graph G.
nodeB : node
A node within graph G, exclusive of Node A.
weight : string or None, optional (default=None)
The edge data key used to compute the resistance distance.
If None, then each edge has weight 1.
invert_weight : boolean (default=True)
Proper calculation of resistance distance requires building the
Laplacian matrix with the reciprocal of the weight. Not required
if the weight is already inverted. Weight cannot be zero.
Returns
-------
rd : float
Value of effective resistance distance
Notes
-----
Overview discussion:
* https://en.wikipedia.org/wiki/Resistance_distance
* http://mathworld.wolfram.com/ResistanceDistance.html
Additional details:
Vaya Sapobi Samui Vos, “Methods for determining the effective resistance,” M.S.,
Mathematisch Instituut, Universiteit Leiden, Leiden, Netherlands, 2016
Available: `Link to thesis <https://www.universiteitleiden.nl/binaries/content/assets/science/mi/scripties/master/vos_vaya_master.pdf>`_
"""
import numpy as np
import scipy as sp
import scipy.sparse.linalg # call as sp.sparse.linalg
if not nx.is_connected(G):
msg = "Graph G must be strongly connected."
raise nx.NetworkXError(msg)
elif nodeA not in G:
msg = "Node A is not in graph G."
raise nx.NetworkXError(msg)
elif nodeB not in G:
msg = "Node B is not in graph G."
raise nx.NetworkXError(msg)
elif nodeA == nodeB:
msg = "Node A and Node B cannot be the same."
raise nx.NetworkXError(msg)
G = G.copy()
node_list = list(G)
if invert_weight and weight is not None:
if G.is_multigraph():
for (u, v, k, d) in G.edges(keys=True, data=True):
d[weight] = 1 / d[weight]
else:
for (u, v, d) in G.edges(data=True):
d[weight] = 1 / d[weight]
# Replace with collapsing topology or approximated zero?
# Using determinants to compute the effective resistance is more memory
# efficent than directly calculating the psuedo-inverse
L = nx.laplacian_matrix(G, node_list, weight=weight)
Lsub_a, node_list_a = _laplacian_submatrix(nodeA, L.copy(), node_list[:])
Lsub_ab, node_list_ab = _laplacian_submatrix(nodeB, Lsub_a.copy(), node_list_a[:])
# Factorize Laplacian submatrixes and extract diagonals
# Order the diagonals to minimize the likelihood over overflows
# during computing the determinant
lu_a = sp.sparse.linalg.splu(Lsub_a, options=dict(SymmetricMode=True))
LdiagA = lu_a.U.diagonal()
LdiagA_s = np.product(np.sign(LdiagA)) * np.product(lu_a.L.diagonal())
LdiagA_s *= (-1) ** _count_lu_permutations(lu_a.perm_r)
LdiagA_s *= (-1) ** _count_lu_permutations(lu_a.perm_c)
LdiagA = np.absolute(LdiagA)
LdiagA = np.sort(LdiagA)
lu_ab = sp.sparse.linalg.splu(Lsub_ab, options=dict(SymmetricMode=True))
LdiagAB = lu_ab.U.diagonal()
LdiagAB_s = np.product(np.sign(LdiagAB)) * np.product(lu_ab.L.diagonal())
LdiagAB_s *= (-1) ** _count_lu_permutations(lu_ab.perm_r)
LdiagAB_s *= (-1) ** _count_lu_permutations(lu_ab.perm_c)
LdiagAB = np.absolute(LdiagAB)
LdiagAB = np.sort(LdiagAB)
# Calculate the ratio of determinant, rd = det(Lsub_ab)/det(Lsub_a)
Ldet = np.product(np.divide(np.append(LdiagAB, [1]), LdiagA))
rd = Ldet * LdiagAB_s / LdiagA_s
return rd
|