00001 """ Threshold Graphs - Creation, manipulation and identification. """ __author__ = """Aric Hagberg (hagberg@lanl.gov)\nPieter Swart (swart@lanl.gov)\nDan Schult (dschult@colgate.edu)""" __date__ = "$Date: 2005-06-17 08:06:22 -0600 (Fri, 17 Jun 2005) $" __credits__ = """""" __version__ = "$Revision: 1049 $" # $Header$ # Copyright (C) 2004,2005 by # Aric Hagberg <hagberg@lanl.gov> # Dan Schult <dschult@colgate.edu> # Pieter Swart <swart@lanl.gov> # Distributed under the terms of the GNU Lesser General Public License # http://www.gnu.org/copyleft/lesser.html # import random # for swap_d from math import sqrt import networkx # try numpy first and Numeric, second. Fail if neither is available. try: import numpy as N except ImportError: try: import Numeric as N except ImportError: raise ImportError,"Neither Numeric nor numpy can be imported." 00030 def is_threshold_graph(G): """ Returns True if G is a threshold graph. """ return is_threshold_sequence(G.degree()) 00036 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 00059 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 00115 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 00149 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 00172 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 00217 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 00276 def threshold_graph(creation_sequence): """ 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 G=networkx.Graph() 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) G.add_node(v) if node_type=='d': # dominating type, connect to all existing nodes for u in G.nodes(): G.add_edge(v,u) # will silently ignore self loop return G 00321 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 00337 def find_threshold_graph(G): """ 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)) 00346 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(with_labels=True) ds=[ [d,v] for v,d in dsdict.iteritems() ] 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 00379 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 00399 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 00426 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 00443 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 00459 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 00470 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) 00504 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 xrange(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 xrange(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 00555 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 00601 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 00641 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) 00697 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. FIXME--allow Numeric/numpy arrays for eigenvectors. 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 00720 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 00761 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=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=threshold_graph(random_threshold_sequence(10,0.5)) """ 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? 00798 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 00831 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 00865 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 def _test_suite(): import doctest suite = doctest.DocFileSuite('tests/threshold.txt', package='networkx') return suite if __name__ == "__main__": import os import sys import unittest if sys.version_info[:2] < (2, 4): print "Python version 2.4 or later required for tests (%d.%d detected)." % sys.version_info[:2] sys.exit(-1) # directory of networkx package (relative to this) nxbase=sys.path[0]+os.sep+os.pardir sys.path.insert(0,nxbase) # prepend to search path unittest.TextTestRunner().run(_test_suite())

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