116 lines
4.0 KiB
Python
116 lines
4.0 KiB
Python
# 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])
|
|
|
|
|
|
@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
|