116 lines
4.0 KiB
Python
116 lines
4.0 KiB
Python
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# triads.py - functions for analyzing triads of a graph
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#
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# Copyright 2015 NetworkX developers.
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# Copyright 2011 Reya Group <http://www.reyagroup.com>
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# Copyright 2011 Alex Levenson <alex@isnotinvain.com>
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# Copyright 2011 Diederik van Liere <diederik.vanliere@rotman.utoronto.ca>
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#
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# This file is part of NetworkX.
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#
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# NetworkX is distributed under a BSD license; see LICENSE.txt for more
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# information.
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"""Functions for analyzing triads of a graph."""
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from networkx.utils import not_implemented_for
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__author__ = '\n'.join(['Alex Levenson (alex@isnontinvain.com)',
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'Diederik van Liere (diederik.vanliere@rotman.utoronto.ca)'])
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__all__ = ['triadic_census']
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#: The integer codes representing each type of triad.
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#:
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#: Triads that are the same up to symmetry have the same code.
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TRICODES = (1, 2, 2, 3, 2, 4, 6, 8, 2, 6, 5, 7, 3, 8, 7, 11, 2, 6, 4, 8, 5, 9,
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9, 13, 6, 10, 9, 14, 7, 14, 12, 15, 2, 5, 6, 7, 6, 9, 10, 14, 4, 9,
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9, 12, 8, 13, 14, 15, 3, 7, 8, 11, 7, 12, 14, 15, 8, 14, 13, 15,
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11, 15, 15, 16)
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#: The names of each type of triad. The order of the elements is
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#: important: it corresponds to the tricodes given in :data:`TRICODES`.
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TRIAD_NAMES = ('003', '012', '102', '021D', '021U', '021C', '111D', '111U',
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'030T', '030C', '201', '120D', '120U', '120C', '210', '300')
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#: A dictionary mapping triad code to triad name.
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TRICODE_TO_NAME = {i: TRIAD_NAMES[code - 1] for i, code in enumerate(TRICODES)}
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def _tricode(G, v, u, w):
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"""Returns the integer code of the given triad.
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This is some fancy magic that comes from Batagelj and Mrvar's paper. It
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treats each edge joining a pair of `v`, `u`, and `w` as a bit in
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the binary representation of an integer.
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"""
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combos = ((v, u, 1), (u, v, 2), (v, w, 4), (w, v, 8), (u, w, 16),
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(w, u, 32))
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return sum(x for u, v, x in combos if v in G[u])
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@not_implemented_for('undirected')
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def triadic_census(G):
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"""Determines the triadic census of a directed graph.
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The triadic census is a count of how many of the 16 possible types of
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triads are present in a directed graph.
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Parameters
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----------
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G : digraph
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A NetworkX DiGraph
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Returns
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-------
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census : dict
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Dictionary with triad names as keys and number of occurrences as values.
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Notes
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-----
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This algorithm has complexity $O(m)$ where $m$ is the number of edges in
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the graph.
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See also
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--------
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triad_graph
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References
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----------
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.. [1] Vladimir Batagelj and Andrej Mrvar, A subquadratic triad census
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algorithm for large sparse networks with small maximum degree,
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University of Ljubljana,
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http://vlado.fmf.uni-lj.si/pub/networks/doc/triads/triads.pdf
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"""
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# Initialize the count for each triad to be zero.
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census = {name: 0 for name in TRIAD_NAMES}
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n = len(G)
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# m = dict(zip(G, range(n)))
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m = {v: i for i, v in enumerate(G)}
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for v in G:
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vnbrs = set(G.pred[v]) | set(G.succ[v])
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for u in vnbrs:
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if m[u] <= m[v]:
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continue
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neighbors = (vnbrs | set(G.succ[u]) | set(G.pred[u])) - {u, v}
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# Calculate dyadic triads instead of counting them.
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if v in G[u] and u in G[v]:
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census['102'] += n - len(neighbors) - 2
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else:
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census['012'] += n - len(neighbors) - 2
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# Count connected triads.
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for w in neighbors:
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if m[u] < m[w] or (m[v] < m[w] < m[u] and
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v not in G.pred[w] and
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v not in G.succ[w]):
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code = _tricode(G, v, u, w)
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census[TRICODE_TO_NAME[code]] += 1
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# null triads = total number of possible triads - all found triads
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#
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# Use integer division here, since we know this formula guarantees an
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# integral value.
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census['003'] = ((n * (n - 1) * (n - 2)) // 6) - sum(census.values())
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return census
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