/usr/lib/python3/dist-packages/networkx/generators/threshold.py is in python3-networkx 1.11-1ubuntu2.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 | """
Threshold Graphs - Creation, manipulation and identification.
"""
__author__ = """Aric Hagberg (hagberg@lanl.gov)\nPieter Swart (swart@lanl.gov)\nDan Schult (dschult@colgate.edu)"""
# Copyright (C) 2004-2015 by
# Aric Hagberg <hagberg@lanl.gov>
# Dan Schult <dschult@colgate.edu>
# Pieter Swart <swart@lanl.gov>
# All rights reserved.
# BSD license.
#
__all__=[]
import random # for swap_d
from math import sqrt
import networkx
def is_threshold_graph(G):
"""
Returns True if G is a threshold graph.
"""
return is_threshold_sequence(list(G.degree().values()))
def is_threshold_sequence(degree_sequence):
"""
Returns True if the sequence is a threshold degree seqeunce.
Uses the property that a threshold graph must be constructed by
adding either dominating or isolated nodes. Thus, it can be
deconstructed iteratively by removing a node of degree zero or a
node that connects to the remaining nodes. If this deconstruction
failes then the sequence is not a threshold sequence.
"""
ds=degree_sequence[:] # get a copy so we don't destroy original
ds.sort()
while ds:
if ds[0]==0: # if isolated node
ds.pop(0) # remove it
continue
if ds[-1]!=len(ds)-1: # is the largest degree node dominating?
return False # no, not a threshold degree sequence
ds.pop() # yes, largest is the dominating node
ds=[ d-1 for d in ds ] # remove it and decrement all degrees
return True
def creation_sequence(degree_sequence,with_labels=False,compact=False):
"""
Determines the creation sequence for the given threshold degree sequence.
The creation sequence is a list of single characters 'd'
or 'i': 'd' for dominating or 'i' for isolated vertices.
Dominating vertices are connected to all vertices present when it
is added. The first node added is by convention 'd'.
This list can be converted to a string if desired using "".join(cs)
If with_labels==True:
Returns a list of 2-tuples containing the vertex number
and a character 'd' or 'i' which describes the type of vertex.
If compact==True:
Returns the creation sequence in a compact form that is the number
of 'i's and 'd's alternating.
Examples:
[1,2,2,3] represents d,i,i,d,d,i,i,i
[3,1,2] represents d,d,d,i,d,d
Notice that the first number is the first vertex to be used for
construction and so is always 'd'.
with_labels and compact cannot both be True.
Returns None if the sequence is not a threshold sequence
"""
if with_labels and compact:
raise ValueError("compact sequences cannot be labeled")
# make an indexed copy
if isinstance(degree_sequence,dict): # labeled degree seqeunce
ds = [ [degree,label] for (label,degree) in degree_sequence.items() ]
else:
ds=[ [d,i] for i,d in enumerate(degree_sequence) ]
ds.sort()
cs=[] # creation sequence
while ds:
if ds[0][0]==0: # isolated node
(d,v)=ds.pop(0)
if len(ds)>0: # make sure we start with a d
cs.insert(0,(v,'i'))
else:
cs.insert(0,(v,'d'))
continue
if ds[-1][0]!=len(ds)-1: # Not dominating node
return None # not a threshold degree sequence
(d,v)=ds.pop()
cs.insert(0,(v,'d'))
ds=[ [d[0]-1,d[1]] for d in ds ] # decrement due to removing node
if with_labels: return cs
if compact: return make_compact(cs)
return [ v[1] for v in cs ] # not labeled
def make_compact(creation_sequence):
"""
Returns the creation sequence in a compact form
that is the number of 'i's and 'd's alternating.
Examples:
[1,2,2,3] represents d,i,i,d,d,i,i,i.
[3,1,2] represents d,d,d,i,d,d.
Notice that the first number is the first vertex
to be used for construction and so is always 'd'.
Labeled creation sequences lose their labels in the
compact representation.
"""
first=creation_sequence[0]
if isinstance(first,str): # creation sequence
cs = creation_sequence[:]
elif isinstance(first,tuple): # labeled creation sequence
cs = [ s[1] for s in creation_sequence ]
elif isinstance(first,int): # compact creation sequence
return creation_sequence
else:
raise TypeError("Not a valid creation sequence type")
ccs=[]
count=1 # count the run lengths of d's or i's.
for i in range(1,len(cs)):
if cs[i]==cs[i-1]:
count+=1
else:
ccs.append(count)
count=1
ccs.append(count) # don't forget the last one
return ccs
def uncompact(creation_sequence):
"""
Converts a compact creation sequence for a threshold
graph to a standard creation sequence (unlabeled).
If the creation_sequence is already standard, return it.
See creation_sequence.
"""
first=creation_sequence[0]
if isinstance(first,str): # creation sequence
return creation_sequence
elif isinstance(first,tuple): # labeled creation sequence
return creation_sequence
elif isinstance(first,int): # compact creation sequence
ccscopy=creation_sequence[:]
else:
raise TypeError("Not a valid creation sequence type")
cs = []
while ccscopy:
cs.extend(ccscopy.pop(0)*['d'])
if ccscopy:
cs.extend(ccscopy.pop(0)*['i'])
return cs
def creation_sequence_to_weights(creation_sequence):
"""
Returns a list of node weights which create the threshold
graph designated by the creation sequence. The weights
are scaled so that the threshold is 1.0. The order of the
nodes is the same as that in the creation sequence.
"""
# Turn input sequence into a labeled creation sequence
first=creation_sequence[0]
if isinstance(first,str): # creation sequence
if isinstance(creation_sequence,list):
wseq = creation_sequence[:]
else:
wseq = list(creation_sequence) # string like 'ddidid'
elif isinstance(first,tuple): # labeled creation sequence
wseq = [ v[1] for v in creation_sequence]
elif isinstance(first,int): # compact creation sequence
wseq = uncompact(creation_sequence)
else:
raise TypeError("Not a valid creation sequence type")
# pass through twice--first backwards
wseq.reverse()
w=0
prev='i'
for j,s in enumerate(wseq):
if s=='i':
wseq[j]=w
prev=s
elif prev=='i':
prev=s
w+=1
wseq.reverse() # now pass through forwards
for j,s in enumerate(wseq):
if s=='d':
wseq[j]=w
prev=s
elif prev=='d':
prev=s
w+=1
# Now scale weights
if prev=='d': w+=1
wscale=1./float(w)
return [ ww*wscale for ww in wseq]
#return wseq
def weights_to_creation_sequence(weights,threshold=1,with_labels=False,compact=False):
"""
Returns a creation sequence for a threshold graph
determined by the weights and threshold given as input.
If the sum of two node weights is greater than the
threshold value, an edge is created between these nodes.
The creation sequence is a list of single characters 'd'
or 'i': 'd' for dominating or 'i' for isolated vertices.
Dominating vertices are connected to all vertices present
when it is added. The first node added is by convention 'd'.
If with_labels==True:
Returns a list of 2-tuples containing the vertex number
and a character 'd' or 'i' which describes the type of vertex.
If compact==True:
Returns the creation sequence in a compact form that is the number
of 'i's and 'd's alternating.
Examples:
[1,2,2,3] represents d,i,i,d,d,i,i,i
[3,1,2] represents d,d,d,i,d,d
Notice that the first number is the first vertex to be used for
construction and so is always 'd'.
with_labels and compact cannot both be True.
"""
if with_labels and compact:
raise ValueError("compact sequences cannot be labeled")
# make an indexed copy
if isinstance(weights,dict): # labeled weights
wseq = [ [w,label] for (label,w) in weights.items() ]
else:
wseq = [ [w,i] for i,w in enumerate(weights) ]
wseq.sort()
cs=[] # creation sequence
cutoff=threshold-wseq[-1][0]
while wseq:
if wseq[0][0]<cutoff: # isolated node
(w,label)=wseq.pop(0)
cs.append((label,'i'))
else:
(w,label)=wseq.pop()
cs.append((label,'d'))
cutoff=threshold-wseq[-1][0]
if len(wseq)==1: # make sure we start with a d
(w,label)=wseq.pop()
cs.append((label,'d'))
# put in correct order
cs.reverse()
if with_labels: return cs
if compact: return make_compact(cs)
return [ v[1] for v in cs ] # not labeled
# Manipulating NetworkX.Graphs in context of threshold graphs
def threshold_graph(creation_sequence, create_using=None):
"""
Create a threshold graph from the creation sequence or compact
creation_sequence.
The input sequence can be a
creation sequence (e.g. ['d','i','d','d','d','i'])
labeled creation sequence (e.g. [(0,'d'),(2,'d'),(1,'i')])
compact creation sequence (e.g. [2,1,1,2,0])
Use cs=creation_sequence(degree_sequence,labeled=True)
to convert a degree sequence to a creation sequence.
Returns None if the sequence is not valid
"""
# Turn input sequence into a labeled creation sequence
first=creation_sequence[0]
if isinstance(first,str): # creation sequence
ci = list(enumerate(creation_sequence))
elif isinstance(first,tuple): # labeled creation sequence
ci = creation_sequence[:]
elif isinstance(first,int): # compact creation sequence
cs = uncompact(creation_sequence)
ci = list(enumerate(cs))
else:
print("not a valid creation sequence type")
return None
if create_using is None:
G = networkx.Graph()
elif create_using.is_directed():
raise networkx.NetworkXError("Directed Graph not supported")
else:
G = create_using
G.clear()
G.name="Threshold Graph"
# add nodes and edges
# if type is 'i' just add nodea
# if type is a d connect to everything previous
while ci:
(v,node_type)=ci.pop(0)
if node_type=='d': # dominating type, connect to all existing nodes
for u in G.nodes():
G.add_edge(v,u)
G.add_node(v)
return G
def find_alternating_4_cycle(G):
"""
Returns False if there aren't any alternating 4 cycles.
Otherwise returns the cycle as [a,b,c,d] where (a,b)
and (c,d) are edges and (a,c) and (b,d) are not.
"""
for (u,v) in G.edges():
for w in G.nodes():
if not G.has_edge(u,w) and u!=w:
for x in G.neighbors(w):
if not G.has_edge(v,x) and v!=x:
return [u,v,w,x]
return False
def find_threshold_graph(G, create_using=None):
"""
Return a threshold subgraph that is close to largest in G.
The threshold graph will contain the largest degree node in G.
"""
return threshold_graph(find_creation_sequence(G),create_using)
def find_creation_sequence(G):
"""
Find a threshold subgraph that is close to largest in G.
Returns the labeled creation sequence of that threshold graph.
"""
cs=[]
# get a local pointer to the working part of the graph
H=G
while H.order()>0:
# get new degree sequence on subgraph
dsdict=H.degree()
ds=[ [d,v] for v,d in dsdict.items() ]
ds.sort()
# Update threshold graph nodes
if ds[-1][0]==0: # all are isolated
cs.extend( zip( dsdict, ['i']*(len(ds)-1)+['d']) )
break # Done!
# pull off isolated nodes
while ds[0][0]==0:
(d,iso)=ds.pop(0)
cs.append((iso,'i'))
# find new biggest node
(d,bigv)=ds.pop()
# add edges of star to t_g
cs.append((bigv,'d'))
# form subgraph of neighbors of big node
H=H.subgraph(H.neighbors(bigv))
cs.reverse()
return cs
### Properties of Threshold Graphs
def triangles(creation_sequence):
"""
Compute number of triangles in the threshold graph with the
given creation sequence.
"""
# shortcut algoritm that doesn't require computing number
# of triangles at each node.
cs=creation_sequence # alias
dr=cs.count("d") # number of d's in sequence
ntri=dr*(dr-1)*(dr-2)/6 # number of triangles in clique of nd d's
# now add dr choose 2 triangles for every 'i' in sequence where
# dr is the number of d's to the right of the current i
for i,typ in enumerate(cs):
if typ=="i":
ntri+=dr*(dr-1)/2
else:
dr-=1
return ntri
def triangle_sequence(creation_sequence):
"""
Return triangle sequence for the given threshold graph creation sequence.
"""
cs=creation_sequence
seq=[]
dr=cs.count("d") # number of d's to the right of the current pos
dcur=(dr-1)*(dr-2) // 2 # number of triangles through a node of clique dr
irun=0 # number of i's in the last run
drun=0 # number of d's in the last run
for i,sym in enumerate(cs):
if sym=="d":
drun+=1
tri=dcur+(dr-1)*irun # new triangles at this d
else: # cs[i]="i":
if prevsym=="d": # new string of i's
dcur+=(dr-1)*irun # accumulate shared shortest paths
irun=0 # reset i run counter
dr-=drun # reduce number of d's to right
drun=0 # reset d run counter
irun+=1
tri=dr*(dr-1) // 2 # new triangles at this i
seq.append(tri)
prevsym=sym
return seq
def cluster_sequence(creation_sequence):
"""
Return cluster sequence for the given threshold graph creation sequence.
"""
triseq=triangle_sequence(creation_sequence)
degseq=degree_sequence(creation_sequence)
cseq=[]
for i,deg in enumerate(degseq):
tri=triseq[i]
if deg <= 1: # isolated vertex or single pair gets cc 0
cseq.append(0)
continue
max_size=(deg*(deg-1)) // 2
cseq.append(float(tri)/float(max_size))
return cseq
def degree_sequence(creation_sequence):
"""
Return degree sequence for the threshold graph with the given
creation sequence
"""
cs=creation_sequence # alias
seq=[]
rd=cs.count("d") # number of d to the right
for i,sym in enumerate(cs):
if sym=="d":
rd-=1
seq.append(rd+i)
else:
seq.append(rd)
return seq
def density(creation_sequence):
"""
Return the density of the graph with this creation_sequence.
The density is the fraction of possible edges present.
"""
N=len(creation_sequence)
two_size=sum(degree_sequence(creation_sequence))
two_possible=N*(N-1)
den=two_size/float(two_possible)
return den
def degree_correlation(creation_sequence):
"""
Return the degree-degree correlation over all edges.
"""
cs=creation_sequence
s1=0 # deg_i*deg_j
s2=0 # deg_i^2+deg_j^2
s3=0 # deg_i+deg_j
m=0 # number of edges
rd=cs.count("d") # number of d nodes to the right
rdi=[ i for i,sym in enumerate(cs) if sym=="d"] # index of "d"s
ds=degree_sequence(cs)
for i,sym in enumerate(cs):
if sym=="d":
if i!=rdi[0]:
print("Logic error in degree_correlation",i,rdi)
raise ValueError
rdi.pop(0)
degi=ds[i]
for dj in rdi:
degj=ds[dj]
s1+=degj*degi
s2+=degi**2+degj**2
s3+=degi+degj
m+=1
denom=(2*m*s2-s3*s3)
numer=(4*m*s1-s3*s3)
if denom==0:
if numer==0:
return 1
raise ValueError("Zero Denominator but Numerator is %s"%numer)
return numer/float(denom)
def shortest_path(creation_sequence,u,v):
"""
Find the shortest path between u and v in a
threshold graph G with the given creation_sequence.
For an unlabeled creation_sequence, the vertices
u and v must be integers in (0,len(sequence)) refering
to the position of the desired vertices in the sequence.
For a labeled creation_sequence, u and v are labels of veritices.
Use cs=creation_sequence(degree_sequence,with_labels=True)
to convert a degree sequence to a creation sequence.
Returns a list of vertices from u to v.
Example: if they are neighbors, it returns [u,v]
"""
# Turn input sequence into a labeled creation sequence
first=creation_sequence[0]
if isinstance(first,str): # creation sequence
cs = [(i,creation_sequence[i]) for i in range(len(creation_sequence))]
elif isinstance(first,tuple): # labeled creation sequence
cs = creation_sequence[:]
elif isinstance(first,int): # compact creation sequence
ci = uncompact(creation_sequence)
cs = [(i,ci[i]) for i in range(len(ci))]
else:
raise TypeError("Not a valid creation sequence type")
verts=[ s[0] for s in cs ]
if v not in verts:
raise ValueError("Vertex %s not in graph from creation_sequence"%v)
if u not in verts:
raise ValueError("Vertex %s not in graph from creation_sequence"%u)
# Done checking
if u==v: return [u]
uindex=verts.index(u)
vindex=verts.index(v)
bigind=max(uindex,vindex)
if cs[bigind][1]=='d':
return [u,v]
# must be that cs[bigind][1]=='i'
cs=cs[bigind:]
while cs:
vert=cs.pop()
if vert[1]=='d':
return [u,vert[0],v]
# All after u are type 'i' so no connection
return -1
def shortest_path_length(creation_sequence,i):
"""
Return the shortest path length from indicated node to
every other node for the threshold graph with the given
creation sequence.
Node is indicated by index i in creation_sequence unless
creation_sequence is labeled in which case, i is taken to
be the label of the node.
Paths lengths in threshold graphs are at most 2.
Length to unreachable nodes is set to -1.
"""
# Turn input sequence into a labeled creation sequence
first=creation_sequence[0]
if isinstance(first,str): # creation sequence
if isinstance(creation_sequence,list):
cs = creation_sequence[:]
else:
cs = list(creation_sequence)
elif isinstance(first,tuple): # labeled creation sequence
cs = [ v[1] for v in creation_sequence]
i = [v[0] for v in creation_sequence].index(i)
elif isinstance(first,int): # compact creation sequence
cs = uncompact(creation_sequence)
else:
raise TypeError("Not a valid creation sequence type")
# Compute
N=len(cs)
spl=[2]*N # length 2 to every node
spl[i]=0 # except self which is 0
# 1 for all d's to the right
for j in range(i+1,N):
if cs[j]=="d":
spl[j]=1
if cs[i]=='d': # 1 for all nodes to the left
for j in range(i):
spl[j]=1
# and -1 for any trailing i to indicate unreachable
for j in range(N-1,0,-1):
if cs[j]=="d":
break
spl[j]=-1
return spl
def betweenness_sequence(creation_sequence,normalized=True):
"""
Return betweenness for the threshold graph with the given creation
sequence. The result is unscaled. To scale the values
to the iterval [0,1] divide by (n-1)*(n-2).
"""
cs=creation_sequence
seq=[] # betweenness
lastchar='d' # first node is always a 'd'
dr=float(cs.count("d")) # number of d's to the right of curren pos
irun=0 # number of i's in the last run
drun=0 # number of d's in the last run
dlast=0.0 # betweenness of last d
for i,c in enumerate(cs):
if c=='d': #cs[i]=="d":
# betweennees = amt shared with eariler d's and i's
# + new isolated nodes covered
# + new paths to all previous nodes
b=dlast + (irun-1)*irun/dr + 2*irun*(i-drun-irun)/dr
drun+=1 # update counter
else: # cs[i]="i":
if lastchar=='d': # if this is a new run of i's
dlast=b # accumulate betweenness
dr-=drun # update number of d's to the right
drun=0 # reset d counter
irun=0 # reset i counter
b=0 # isolated nodes have zero betweenness
irun+=1 # add another i to the run
seq.append(float(b))
lastchar=c
# normalize by the number of possible shortest paths
if normalized:
order=len(cs)
scale=1.0/((order-1)*(order-2))
seq=[ s*scale for s in seq ]
return seq
def eigenvectors(creation_sequence):
"""
Return a 2-tuple of Laplacian eigenvalues and eigenvectors
for the threshold network with creation_sequence.
The first value is a list of eigenvalues.
The second value is a list of eigenvectors.
The lists are in the same order so corresponding eigenvectors
and eigenvalues are in the same position in the two lists.
Notice that the order of the eigenvalues returned by eigenvalues(cs)
may not correspond to the order of these eigenvectors.
"""
ccs=make_compact(creation_sequence)
N=sum(ccs)
vec=[0]*N
val=vec[:]
# get number of type d nodes to the right (all for first node)
dr=sum(ccs[::2])
nn=ccs[0]
vec[0]=[1./sqrt(N)]*N
val[0]=0
e=dr
dr-=nn
type_d=True
i=1
dd=1
while dd<nn:
scale=1./sqrt(dd*dd+i)
vec[i]=i*[-scale]+[dd*scale]+[0]*(N-i-1)
val[i]=e
i+=1
dd+=1
if len(ccs)==1: return (val,vec)
for nn in ccs[1:]:
scale=1./sqrt(nn*i*(i+nn))
vec[i]=i*[-nn*scale]+nn*[i*scale]+[0]*(N-i-nn)
# find eigenvalue
type_d=not type_d
if type_d:
e=i+dr
dr-=nn
else:
e=dr
val[i]=e
st=i
i+=1
dd=1
while dd<nn:
scale=1./sqrt(i-st+dd*dd)
vec[i]=[0]*st+(i-st)*[-scale]+[dd*scale]+[0]*(N-i-1)
val[i]=e
i+=1
dd+=1
return (val,vec)
def spectral_projection(u,eigenpairs):
"""
Returns the coefficients of each eigenvector
in a projection of the vector u onto the normalized
eigenvectors which are contained in eigenpairs.
eigenpairs should be a list of two objects. The
first is a list of eigenvalues and the second a list
of eigenvectors. The eigenvectors should be lists.
There's not a lot of error checking on lengths of
arrays, etc. so be careful.
"""
coeff=[]
evect=eigenpairs[1]
for ev in evect:
c=sum([ evv*uv for (evv,uv) in zip(ev,u)])
coeff.append(c)
return coeff
def eigenvalues(creation_sequence):
"""
Return sequence of eigenvalues of the Laplacian of the threshold
graph for the given creation_sequence.
Based on the Ferrer's diagram method. The spectrum is integral
and is the conjugate of the degree sequence.
See::
@Article{degree-merris-1994,
author = {Russel Merris},
title = {Degree maximal graphs are Laplacian integral},
journal = {Linear Algebra Appl.},
year = {1994},
volume = {199},
pages = {381--389},
}
"""
degseq=degree_sequence(creation_sequence)
degseq.sort()
eiglist=[] # zero is always one eigenvalue
eig=0
row=len(degseq)
bigdeg=degseq.pop()
while row:
if bigdeg<row:
eiglist.append(eig)
row-=1
else:
eig+=1
if degseq:
bigdeg=degseq.pop()
else:
bigdeg=0
return eiglist
### Threshold graph creation routines
def random_threshold_sequence(n,p,seed=None):
"""
Create a random threshold sequence of size n.
A creation sequence is built by randomly choosing d's with
probabiliy p and i's with probability 1-p.
s=nx.random_threshold_sequence(10,0.5)
returns a threshold sequence of length 10 with equal
probably of an i or a d at each position.
A "random" threshold graph can be built with
G=nx.threshold_graph(s)
"""
if not seed is None:
random.seed(seed)
if not (p<=1 and p>=0):
raise ValueError("p must be in [0,1]")
cs=['d'] # threshold sequences always start with a d
for i in range(1,n):
if random.random() < p:
cs.append('d')
else:
cs.append('i')
return cs
# maybe *_d_threshold_sequence routines should
# be (or be called from) a single routine with a more descriptive name
# and a keyword parameter?
def right_d_threshold_sequence(n,m):
"""
Create a skewed threshold graph with a given number
of vertices (n) and a given number of edges (m).
The routine returns an unlabeled creation sequence
for the threshold graph.
FIXME: describe algorithm
"""
cs=['d']+['i']*(n-1) # create sequence with n insolated nodes
# m <n : not enough edges, make disconnected
if m < n:
cs[m]='d'
return cs
# too many edges
if m > n*(n-1)/2:
raise ValueError("Too many edges for this many nodes.")
# connected case m >n-1
ind=n-1
sum=n-1
while sum<m:
cs[ind]='d'
ind -= 1
sum += ind
ind=m-(sum-ind)
cs[ind]='d'
return cs
def left_d_threshold_sequence(n,m):
"""
Create a skewed threshold graph with a given number
of vertices (n) and a given number of edges (m).
The routine returns an unlabeled creation sequence
for the threshold graph.
FIXME: describe algorithm
"""
cs=['d']+['i']*(n-1) # create sequence with n insolated nodes
# m <n : not enough edges, make disconnected
if m < n:
cs[m]='d'
return cs
# too many edges
if m > n*(n-1)/2:
raise ValueError("Too many edges for this many nodes.")
# Connected case when M>N-1
cs[n-1]='d'
sum=n-1
ind=1
while sum<m:
cs[ind]='d'
sum += ind
ind += 1
if sum>m: # be sure not to change the first vertex
cs[sum-m]='i'
return cs
def swap_d(cs,p_split=1.0,p_combine=1.0,seed=None):
"""
Perform a "swap" operation on a threshold sequence.
The swap preserves the number of nodes and edges
in the graph for the given sequence.
The resulting sequence is still a threshold sequence.
Perform one split and one combine operation on the
'd's of a creation sequence for a threshold graph.
This operation maintains the number of nodes and edges
in the graph, but shifts the edges from node to node
maintaining the threshold quality of the graph.
"""
if not seed is None:
random.seed(seed)
# preprocess the creation sequence
dlist= [ i for (i,node_type) in enumerate(cs[1:-1]) if node_type=='d' ]
# split
if random.random()<p_split:
choice=random.choice(dlist)
split_to=random.choice(range(choice))
flip_side=choice-split_to
if split_to!=flip_side and cs[split_to]=='i' and cs[flip_side]=='i':
cs[choice]='i'
cs[split_to]='d'
cs[flip_side]='d'
dlist.remove(choice)
# don't add or combine may reverse this action
# dlist.extend([split_to,flip_side])
# print >>sys.stderr,"split at %s to %s and %s"%(choice,split_to,flip_side)
# combine
if random.random()<p_combine and dlist:
first_choice= random.choice(dlist)
second_choice=random.choice(dlist)
target=first_choice+second_choice
if target >= len(cs) or cs[target]=='d' or first_choice==second_choice:
return cs
# OK to combine
cs[first_choice]='i'
cs[second_choice]='i'
cs[target]='d'
# print >>sys.stderr,"combine %s and %s to make %s."%(first_choice,second_choice,target)
return cs
|