459 lines
11 KiB
Python
459 lines
11 KiB
Python
# See https://github.com/networkx/networkx/pull/1474
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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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"""Functions for analyzing triads of a graph."""
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from itertools import combinations, permutations
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from collections import defaultdict
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from random import sample
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import networkx as nx
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from networkx.utils import not_implemented_for
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__all__ = [
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"triadic_census",
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"is_triad",
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"all_triplets",
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"all_triads",
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"triads_by_type",
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"triad_type",
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"random_triad",
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]
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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 = (
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1,
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2,
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2,
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3,
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2,
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4,
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6,
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8,
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2,
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6,
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5,
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7,
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3,
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8,
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7,
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11,
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2,
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6,
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4,
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8,
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5,
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9,
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9,
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13,
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6,
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10,
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9,
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14,
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7,
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14,
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12,
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15,
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2,
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5,
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6,
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7,
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6,
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9,
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10,
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14,
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4,
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9,
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9,
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12,
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8,
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13,
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14,
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15,
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3,
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7,
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8,
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11,
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7,
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12,
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14,
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15,
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8,
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14,
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13,
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15,
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11,
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15,
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15,
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16,
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)
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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 = (
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"003",
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"012",
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"102",
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"021D",
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"021U",
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"021C",
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"111D",
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"111U",
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"030T",
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"030C",
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"201",
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"120D",
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"120U",
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"120C",
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"210",
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"300",
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)
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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), (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, nodelist=None):
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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. If a list of nodes is passed, then
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only those triads are taken into account which have elements of nodelist in them.
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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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nodelist : list
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List of nodes for which you want to calculate triadic census
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Returns
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-------
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census : dict
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Dictionary with triad type 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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if nodelist is None:
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nodelist = list(G.nodes())
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for v in nodelist:
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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 (
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m[v] < m[w] < m[u] and v not in G.pred[w] and v not in G.succ[w]
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):
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code = _tricode(G, v, u, w)
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census[TRICODE_TO_NAME[code]] += 1
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if len(nodelist) != len(G):
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census["003"] = 0
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for v in nodelist:
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vnbrs = set(G.pred[v]) | set(G.succ[v])
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not_vnbrs = G.nodes() - vnbrs - set(v)
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triad_003_count = 0
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for u in not_vnbrs:
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unbrs = set(set(G.succ[u]) | set(G.pred[u])) - vnbrs
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triad_003_count += len(not_vnbrs - unbrs) - 1
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triad_003_count //= 2
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census["003"] += triad_003_count
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else:
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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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def is_triad(G):
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"""Returns True if the graph G is a triad, else False.
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Parameters
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----------
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G : graph
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A NetworkX Graph
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Returns
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-------
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istriad : boolean
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Whether G is a valid triad
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"""
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if isinstance(G, nx.Graph):
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if G.order() == 3 and nx.is_directed(G):
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if not any((n, n) in G.edges() for n in G.nodes()):
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return True
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return False
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@not_implemented_for("undirected")
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def all_triplets(G):
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"""Returns a generator of all possible sets of 3 nodes in a DiGraph.
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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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triplets : generator of 3-tuples
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Generator of tuples of 3 nodes
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"""
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triplets = combinations(G.nodes(), 3)
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return triplets
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@not_implemented_for("undirected")
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def all_triads(G):
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"""A generator of all possible triads in G.
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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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all_triads : generator of DiGraphs
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Generator of triads (order-3 DiGraphs)
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"""
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triplets = combinations(G.nodes(), 3)
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for triplet in triplets:
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yield G.subgraph(triplet).copy()
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@not_implemented_for("undirected")
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def triads_by_type(G):
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"""Returns a list of all triads for each triad type 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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tri_by_type : dict
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Dictionary with triad types as keys and lists of triads as values.
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"""
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# num_triads = o * (o - 1) * (o - 2) // 6
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# if num_triads > TRIAD_LIMIT: print(WARNING)
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all_tri = all_triads(G)
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tri_by_type = defaultdict(list)
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for triad in all_tri:
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name = triad_type(triad)
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tri_by_type[name].append(triad)
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return tri_by_type
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@not_implemented_for("undirected")
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def triad_type(G):
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"""Returns the sociological triad type for a triad.
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Parameters
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----------
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G : digraph
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A NetworkX DiGraph with 3 nodes
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Returns
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-------
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triad_type : str
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A string identifying the triad type
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Notes
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-----
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There can be 6 unique edges in a triad (order-3 DiGraph) (so 2^^6=64 unique
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triads given 3 nodes). These 64 triads each display exactly 1 of 16
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topologies of triads (topologies can be permuted). These topologies are
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identified by the following notation:
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{m}{a}{n}{type} (for example: 111D, 210, 102)
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Here:
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{m} = number of mutual ties (takes 0, 1, 2, 3); a mutual tie is (0,1)
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AND (1,0)
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{a} = number of assymmetric ties (takes 0, 1, 2, 3); an assymmetric tie
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is (0,1) BUT NOT (1,0) or vice versa
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{n} = number of null ties (takes 0, 1, 2, 3); a null tie is NEITHER
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(0,1) NOR (1,0)
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{type} = a letter (takes U, D, C, T) corresponding to up, down, cyclical
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and transitive. This is only used for topologies that can have
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more than one form (eg: 021D and 021U).
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References
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----------
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.. [1] Snijders, T. (2012). "Transitivity and triads." University of
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Oxford.
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https://web.archive.org/web/20170830032057/http://www.stats.ox.ac.uk/~snijders/Trans_Triads_ha.pdf
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"""
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if not is_triad(G):
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raise nx.NetworkXAlgorithmError("G is not a triad (order-3 DiGraph)")
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num_edges = len(G.edges())
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if num_edges == 0:
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return "003"
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elif num_edges == 1:
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return "012"
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elif num_edges == 2:
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e1, e2 = G.edges()
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if set(e1) == set(e2):
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return "102"
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elif e1[0] == e2[0]:
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return "021D"
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elif e1[1] == e2[1]:
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return "021U"
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elif e1[1] == e2[0] or e2[1] == e1[0]:
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return "021C"
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elif num_edges == 3:
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for (e1, e2, e3) in permutations(G.edges(), 3):
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if set(e1) == set(e2):
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if e3[0] in e1:
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return "111U"
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# e3[1] in e1:
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return "111D"
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elif set(e1).symmetric_difference(set(e2)) == set(e3):
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if {e1[0], e2[0], e3[0]} == {e1[0], e2[0], e3[0]} == set(G.nodes()):
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return "030C"
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# e3 == (e1[0], e2[1]) and e2 == (e1[1], e3[1]):
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return "030T"
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elif num_edges == 4:
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for (e1, e2, e3, e4) in permutations(G.edges(), 4):
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if set(e1) == set(e2):
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# identify pair of symmetric edges (which necessarily exists)
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if set(e3) == set(e4):
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return "201"
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if {e3[0]} == {e4[0]} == set(e3).intersection(set(e4)):
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return "120D"
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if {e3[1]} == {e4[1]} == set(e3).intersection(set(e4)):
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return "120U"
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if e3[1] == e4[0]:
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return "120C"
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elif num_edges == 5:
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return "210"
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elif num_edges == 6:
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return "300"
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@not_implemented_for("undirected")
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def random_triad(G):
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"""Returns a random triad from 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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G2 : subgraph
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A randomly selected triad (order-3 NetworkX DiGraph)
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"""
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nodes = sample(list(G.nodes()), 3)
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G2 = G.subgraph(nodes)
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return G2
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"""
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@not_implemented_for('undirected')
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def triadic_closures(G):
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'''Returns a list of order-3 subgraphs of G that are triadic closures.
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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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closures : list
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List of triads of G that are triadic closures
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'''
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pass
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@not_implemented_for('undirected')
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def focal_closures(G, attr_name):
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'''Returns a list of order-3 subgraphs of G that are focally closed.
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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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attr_name : str
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An attribute name
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Returns
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-------
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closures : list
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List of triads of G that are focally closed on attr_name
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'''
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pass
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@not_implemented_for('undirected')
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def balanced_triads(G, crit_func):
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'''Returns a list of order-3 subgraphs of G that are stable.
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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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crit_func : function
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A function that determines if a triad (order-3 digraph) is stable
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Returns
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-------
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triads : list
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List of triads in G that are stable
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'''
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pass
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"""
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