974 lines
30 KiB
Python
974 lines
30 KiB
Python
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"""
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Threshold Graphs - Creation, manipulation and identification.
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"""
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from math import sqrt
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import networkx as nx
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from networkx.utils import py_random_state
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__all__ = ["is_threshold_graph", "find_threshold_graph"]
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def is_threshold_graph(G):
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"""
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Returns `True` if `G` is a threshold graph.
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Parameters
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----------
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G : NetworkX graph instance
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An instance of `Graph`, `DiGraph`, `MultiGraph` or `MultiDiGraph`
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Returns
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-------
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bool
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`True` if `G` is a threshold graph, `False` otherwise.
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Examples
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--------
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>>> from networkx.algorithms.threshold import is_threshold_graph
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>>> G = nx.path_graph(3)
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>>> is_threshold_graph(G)
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True
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>>> G = nx.barbell_graph(3, 3)
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>>> is_threshold_graph(G)
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False
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References
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----------
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.. [1] Threshold graphs: https://en.wikipedia.org/wiki/Threshold_graph
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"""
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return is_threshold_sequence(list(d for n, d in G.degree()))
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def is_threshold_sequence(degree_sequence):
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"""
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Returns True if the sequence is a threshold degree seqeunce.
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Uses the property that a threshold graph must be constructed by
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adding either dominating or isolated nodes. Thus, it can be
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deconstructed iteratively by removing a node of degree zero or a
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node that connects to the remaining nodes. If this deconstruction
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failes then the sequence is not a threshold sequence.
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"""
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ds = degree_sequence[:] # get a copy so we don't destroy original
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ds.sort()
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while ds:
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if ds[0] == 0: # if isolated node
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ds.pop(0) # remove it
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continue
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if ds[-1] != len(ds) - 1: # is the largest degree node dominating?
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return False # no, not a threshold degree sequence
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ds.pop() # yes, largest is the dominating node
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ds = [d - 1 for d in ds] # remove it and decrement all degrees
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return True
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def creation_sequence(degree_sequence, with_labels=False, compact=False):
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"""
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Determines the creation sequence for the given threshold degree sequence.
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The creation sequence is a list of single characters 'd'
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or 'i': 'd' for dominating or 'i' for isolated vertices.
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Dominating vertices are connected to all vertices present when it
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is added. The first node added is by convention 'd'.
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This list can be converted to a string if desired using "".join(cs)
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If with_labels==True:
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Returns a list of 2-tuples containing the vertex number
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and a character 'd' or 'i' which describes the type of vertex.
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If compact==True:
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Returns the creation sequence in a compact form that is the number
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of 'i's and 'd's alternating.
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Examples:
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[1,2,2,3] represents d,i,i,d,d,i,i,i
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[3,1,2] represents d,d,d,i,d,d
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Notice that the first number is the first vertex to be used for
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construction and so is always 'd'.
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with_labels and compact cannot both be True.
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Returns None if the sequence is not a threshold sequence
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"""
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if with_labels and compact:
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raise ValueError("compact sequences cannot be labeled")
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# make an indexed copy
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if isinstance(degree_sequence, dict): # labeled degree seqeunce
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ds = [[degree, label] for (label, degree) in degree_sequence.items()]
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else:
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ds = [[d, i] for i, d in enumerate(degree_sequence)]
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ds.sort()
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cs = [] # creation sequence
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while ds:
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if ds[0][0] == 0: # isolated node
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(d, v) = ds.pop(0)
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if len(ds) > 0: # make sure we start with a d
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cs.insert(0, (v, "i"))
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else:
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cs.insert(0, (v, "d"))
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continue
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if ds[-1][0] != len(ds) - 1: # Not dominating node
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return None # not a threshold degree sequence
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(d, v) = ds.pop()
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cs.insert(0, (v, "d"))
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ds = [[d[0] - 1, d[1]] for d in ds] # decrement due to removing node
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if with_labels:
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return cs
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if compact:
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return make_compact(cs)
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return [v[1] for v in cs] # not labeled
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def make_compact(creation_sequence):
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"""
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Returns the creation sequence in a compact form
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that is the number of 'i's and 'd's alternating.
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Examples
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--------
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>>> from networkx.algorithms.threshold import make_compact
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>>> make_compact(["d", "i", "i", "d", "d", "i", "i", "i"])
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[1, 2, 2, 3]
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>>> make_compact(["d", "d", "d", "i", "d", "d"])
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[3, 1, 2]
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Notice that the first number is the first vertex
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to be used for construction and so is always 'd'.
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Labeled creation sequences lose their labels in the
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compact representation.
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>>> make_compact([3, 1, 2])
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[3, 1, 2]
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"""
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first = creation_sequence[0]
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if isinstance(first, str): # creation sequence
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cs = creation_sequence[:]
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elif isinstance(first, tuple): # labeled creation sequence
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cs = [s[1] for s in creation_sequence]
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elif isinstance(first, int): # compact creation sequence
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return creation_sequence
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else:
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raise TypeError("Not a valid creation sequence type")
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ccs = []
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count = 1 # count the run lengths of d's or i's.
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for i in range(1, len(cs)):
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if cs[i] == cs[i - 1]:
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count += 1
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else:
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ccs.append(count)
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count = 1
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ccs.append(count) # don't forget the last one
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return ccs
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def uncompact(creation_sequence):
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"""
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Converts a compact creation sequence for a threshold
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graph to a standard creation sequence (unlabeled).
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If the creation_sequence is already standard, return it.
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See creation_sequence.
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"""
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first = creation_sequence[0]
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if isinstance(first, str): # creation sequence
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return creation_sequence
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elif isinstance(first, tuple): # labeled creation sequence
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return creation_sequence
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elif isinstance(first, int): # compact creation sequence
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ccscopy = creation_sequence[:]
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else:
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raise TypeError("Not a valid creation sequence type")
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cs = []
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while ccscopy:
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cs.extend(ccscopy.pop(0) * ["d"])
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if ccscopy:
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cs.extend(ccscopy.pop(0) * ["i"])
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return cs
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def creation_sequence_to_weights(creation_sequence):
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"""
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Returns a list of node weights which create the threshold
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graph designated by the creation sequence. The weights
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are scaled so that the threshold is 1.0. The order of the
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nodes is the same as that in the creation sequence.
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"""
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# Turn input sequence into a labeled creation sequence
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first = creation_sequence[0]
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if isinstance(first, str): # creation sequence
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if isinstance(creation_sequence, list):
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wseq = creation_sequence[:]
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else:
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wseq = list(creation_sequence) # string like 'ddidid'
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elif isinstance(first, tuple): # labeled creation sequence
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wseq = [v[1] for v in creation_sequence]
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elif isinstance(first, int): # compact creation sequence
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wseq = uncompact(creation_sequence)
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else:
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raise TypeError("Not a valid creation sequence type")
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# pass through twice--first backwards
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wseq.reverse()
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w = 0
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prev = "i"
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for j, s in enumerate(wseq):
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if s == "i":
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wseq[j] = w
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prev = s
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elif prev == "i":
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prev = s
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w += 1
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wseq.reverse() # now pass through forwards
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for j, s in enumerate(wseq):
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if s == "d":
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wseq[j] = w
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prev = s
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elif prev == "d":
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prev = s
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w += 1
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# Now scale weights
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if prev == "d":
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w += 1
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wscale = 1.0 / float(w)
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return [ww * wscale for ww in wseq]
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# return wseq
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def weights_to_creation_sequence(
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weights, threshold=1, with_labels=False, compact=False
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):
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"""
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Returns a creation sequence for a threshold graph
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determined by the weights and threshold given as input.
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If the sum of two node weights is greater than the
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threshold value, an edge is created between these nodes.
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The creation sequence is a list of single characters 'd'
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or 'i': 'd' for dominating or 'i' for isolated vertices.
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Dominating vertices are connected to all vertices present
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when it is added. The first node added is by convention 'd'.
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If with_labels==True:
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Returns a list of 2-tuples containing the vertex number
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and a character 'd' or 'i' which describes the type of vertex.
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If compact==True:
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Returns the creation sequence in a compact form that is the number
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of 'i's and 'd's alternating.
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Examples:
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[1,2,2,3] represents d,i,i,d,d,i,i,i
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[3,1,2] represents d,d,d,i,d,d
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Notice that the first number is the first vertex to be used for
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construction and so is always 'd'.
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with_labels and compact cannot both be True.
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"""
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if with_labels and compact:
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raise ValueError("compact sequences cannot be labeled")
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# make an indexed copy
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if isinstance(weights, dict): # labeled weights
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wseq = [[w, label] for (label, w) in weights.items()]
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else:
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wseq = [[w, i] for i, w in enumerate(weights)]
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wseq.sort()
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cs = [] # creation sequence
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cutoff = threshold - wseq[-1][0]
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while wseq:
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if wseq[0][0] < cutoff: # isolated node
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(w, label) = wseq.pop(0)
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cs.append((label, "i"))
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else:
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(w, label) = wseq.pop()
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cs.append((label, "d"))
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cutoff = threshold - wseq[-1][0]
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if len(wseq) == 1: # make sure we start with a d
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(w, label) = wseq.pop()
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cs.append((label, "d"))
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# put in correct order
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cs.reverse()
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if with_labels:
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return cs
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if compact:
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return make_compact(cs)
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return [v[1] for v in cs] # not labeled
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# Manipulating NetworkX.Graphs in context of threshold graphs
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def threshold_graph(creation_sequence, create_using=None):
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"""
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Create a threshold graph from the creation sequence or compact
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creation_sequence.
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The input sequence can be a
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creation sequence (e.g. ['d','i','d','d','d','i'])
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labeled creation sequence (e.g. [(0,'d'),(2,'d'),(1,'i')])
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compact creation sequence (e.g. [2,1,1,2,0])
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Use cs=creation_sequence(degree_sequence,labeled=True)
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to convert a degree sequence to a creation sequence.
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Returns None if the sequence is not valid
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"""
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# Turn input sequence into a labeled creation sequence
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first = creation_sequence[0]
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if isinstance(first, str): # creation sequence
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ci = list(enumerate(creation_sequence))
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elif isinstance(first, tuple): # labeled creation sequence
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ci = creation_sequence[:]
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elif isinstance(first, int): # compact creation sequence
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cs = uncompact(creation_sequence)
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ci = list(enumerate(cs))
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else:
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print("not a valid creation sequence type")
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return None
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G = nx.empty_graph(0, create_using)
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if G.is_directed():
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raise nx.NetworkXError("Directed Graph not supported")
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G.name = "Threshold Graph"
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# add nodes and edges
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# if type is 'i' just add nodea
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# if type is a d connect to everything previous
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while ci:
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(v, node_type) = ci.pop(0)
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if node_type == "d": # dominating type, connect to all existing nodes
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# We use `for u in list(G):` instead of
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# `for u in G:` because we edit the graph `G` in
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# the loop. Hence using an iterator will result in
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# `RuntimeError: dictionary changed size during iteration`
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for u in list(G):
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G.add_edge(v, u)
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G.add_node(v)
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return G
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def find_alternating_4_cycle(G):
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"""
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Returns False if there aren't any alternating 4 cycles.
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Otherwise returns the cycle as [a,b,c,d] where (a,b)
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and (c,d) are edges and (a,c) and (b,d) are not.
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"""
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for (u, v) in G.edges():
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for w in G.nodes():
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if not G.has_edge(u, w) and u != w:
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for x in G.neighbors(w):
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if not G.has_edge(v, x) and v != x:
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return [u, v, w, x]
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return False
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def find_threshold_graph(G, create_using=None):
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"""
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Returns a threshold subgraph that is close to largest in `G`.
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||
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The threshold graph will contain the largest degree node in G.
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Parameters
|
||
|
|
----------
|
||
|
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G : NetworkX graph instance
|
||
|
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An instance of `Graph`, or `MultiDiGraph`
|
||
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create_using : NetworkX graph class or `None` (default), optional
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||
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Type of graph to use when constructing the threshold graph.
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||
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If `None`, infer the appropriate graph type from the input.
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|
|
Returns
|
||
|
|
-------
|
||
|
|
graph :
|
||
|
|
A graph instance representing the threshold graph
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> from networkx.algorithms.threshold import find_threshold_graph
|
||
|
|
>>> G = nx.barbell_graph(3, 3)
|
||
|
|
>>> T = find_threshold_graph(G)
|
||
|
|
>>> T.nodes # may vary
|
||
|
|
NodeView((7, 8, 5, 6))
|
||
|
|
|
||
|
|
References
|
||
|
|
----------
|
||
|
|
.. [1] Threshold graphs: https://en.wikipedia.org/wiki/Threshold_graph
|
||
|
|
"""
|
||
|
|
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 = dict(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 algorithm 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(f"Zero Denominator but Numerator is {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)) referring
|
||
|
|
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(f"Vertex {v} not in graph from creation_sequence")
|
||
|
|
if u not in verts:
|
||
|
|
raise ValueError(f"Vertex {u} not in graph from creation_sequence")
|
||
|
|
# 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] = [1.0 / sqrt(N)] * N
|
||
|
|
val[0] = 0
|
||
|
|
e = dr
|
||
|
|
dr -= nn
|
||
|
|
type_d = True
|
||
|
|
i = 1
|
||
|
|
dd = 1
|
||
|
|
while dd < nn:
|
||
|
|
scale = 1.0 / 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 = 1.0 / 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 = 1.0 / 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
|
||
|
|
|
||
|
|
|
||
|
|
@py_random_state(2)
|
||
|
|
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)
|
||
|
|
|
||
|
|
seed : integer, random_state, or None (default)
|
||
|
|
Indicator of random number generation state.
|
||
|
|
See :ref:`Randomness<randomness>`.
|
||
|
|
"""
|
||
|
|
if not (0 <= p <= 1):
|
||
|
|
raise ValueError("p must be in [0,1]")
|
||
|
|
|
||
|
|
cs = ["d"] # threshold sequences always start with a d
|
||
|
|
for i in range(1, n):
|
||
|
|
if seed.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
|
||
|
|
|
||
|
|
|
||
|
|
@py_random_state(3)
|
||
|
|
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.
|
||
|
|
|
||
|
|
seed : integer, random_state, or None (default)
|
||
|
|
Indicator of random number generation state.
|
||
|
|
See :ref:`Randomness<randomness>`.
|
||
|
|
"""
|
||
|
|
# preprocess the creation sequence
|
||
|
|
dlist = [i for (i, node_type) in enumerate(cs[1:-1]) if node_type == "d"]
|
||
|
|
# split
|
||
|
|
if seed.random() < p_split:
|
||
|
|
choice = seed.choice(dlist)
|
||
|
|
split_to = seed.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 seed.random() < p_combine and dlist:
|
||
|
|
first_choice = seed.choice(dlist)
|
||
|
|
second_choice = seed.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
|