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Source code for networkx.algorithms.triads

# triads.py - functions for analyzing triads of a graph
#
# Copyright 2015 NetworkX developers.
# Copyright 2011 Reya Group <http://www.reyagroup.com>
# Copyright 2011 Alex Levenson <alex@isnotinvain.com>
# Copyright 2011 Diederik van Liere <diederik.vanliere@rotman.utoronto.ca>
#
# This file is part of NetworkX.
#
# NetworkX is distributed under a BSD license; see LICENSE.txt for more
# information.
"""Functions for analyzing triads of a graph."""

from networkx.utils import not_implemented_for

__author__ = '\n'.join(['Alex Levenson (alex@isnontinvain.com)',
                        'Diederik van Liere (diederik.vanliere@rotman.utoronto.ca)'])

__all__ = ['triadic_census']

#: The integer codes representing each type of triad.
#:
#: Triads that are the same up to symmetry have the same code.
TRICODES = (1, 2, 2, 3, 2, 4, 6, 8, 2, 6, 5, 7, 3, 8, 7, 11, 2, 6, 4, 8, 5, 9,
            9, 13, 6, 10, 9, 14, 7, 14, 12, 15, 2, 5, 6, 7, 6, 9, 10, 14, 4, 9,
            9, 12, 8, 13, 14, 15, 3, 7, 8, 11, 7, 12, 14, 15, 8, 14, 13, 15,
            11, 15, 15, 16)

#: The names of each type of triad. The order of the elements is
#: important: it corresponds to the tricodes given in :data:`TRICODES`.
TRIAD_NAMES = ('003', '012', '102', '021D', '021U', '021C', '111D', '111U',
               '030T', '030C', '201', '120D', '120U', '120C', '210', '300')


#: A dictionary mapping triad code to triad name.
TRICODE_TO_NAME = {i: TRIAD_NAMES[code - 1] for i, code in enumerate(TRICODES)}


def _tricode(G, v, u, w):
    """Returns the integer code of the given triad.

    This is some fancy magic that comes from Batagelj and Mrvar's paper. It
    treats each edge joining a pair of `v`, `u`, and `w` as a bit in
    the binary representation of an integer.

    """
    combos = ((v, u, 1), (u, v, 2), (v, w, 4), (w, v, 8), (u, w, 16),
              (w, u, 32))
    return sum(x for u, v, x in combos if v in G[u])


[docs]@not_implemented_for('undirected') def triadic_census(G): """Determines the triadic census of a directed graph. The triadic census is a count of how many of the 16 possible types of triads are present in a directed graph. Parameters ---------- G : digraph A NetworkX DiGraph Returns ------- census : dict Dictionary with triad names as keys and number of occurrences as values. Notes ----- This algorithm has complexity $O(m)$ where $m$ is the number of edges in the graph. See also -------- triad_graph References ---------- .. [1] Vladimir Batagelj and Andrej Mrvar, A subquadratic triad census algorithm for large sparse networks with small maximum degree, University of Ljubljana, http://vlado.fmf.uni-lj.si/pub/networks/doc/triads/triads.pdf """ # Initialize the count for each triad to be zero. census = {name: 0 for name in TRIAD_NAMES} n = len(G) # m = dict(zip(G, range(n))) m = {v: i for i, v in enumerate(G)} for v in G: vnbrs = set(G.pred[v]) | set(G.succ[v]) for u in vnbrs: if m[u] <= m[v]: continue neighbors = (vnbrs | set(G.succ[u]) | set(G.pred[u])) - {u, v} # Calculate dyadic triads instead of counting them. if v in G[u] and u in G[v]: census['102'] += n - len(neighbors) - 2 else: census['012'] += n - len(neighbors) - 2 # Count connected triads. for w in neighbors: if m[u] < m[w] or (m[v] < m[w] < m[u] and v not in G.pred[w] and v not in G.succ[w]): code = _tricode(G, v, u, w) census[TRICODE_TO_NAME[code]] += 1 # null triads = total number of possible triads - all found triads # # Use integer division here, since we know this formula guarantees an # integral value. census['003'] = ((n * (n - 1) * (n - 2)) // 6) - sum(census.values()) return census