Note

This documents the development version of NetworkX. Documentation for the current release can be found here.

Source code for networkx.algorithms.triads

# See https://github.com/networkx/networkx/pull/1474
# 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>
"""Functions for analyzing triads of a graph."""

from itertools import combinations, permutations
from collections import defaultdict
from random import sample

import networkx as nx
from networkx.utils import not_implemented_for

__all__ = [
    "triadic_census",
    "is_triad",
    "all_triplets",
    "all_triads",
    "triads_by_type",
    "triad_type",
    "random_triad",
]

#: 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 type 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
def is_triad(G): """Returns True if the graph G is a triad, else False. Parameters ---------- G : graph A NetworkX Graph Returns ------- istriad : boolean Whether G is a valid triad """ if isinstance(G, nx.Graph): if G.order() == 3 and nx.is_directed(G): if not any((n, n) in G.edges() for n in G.nodes()): return True return False
[docs]@not_implemented_for("undirected") def all_triplets(G): """Returns a generator of all possible sets of 3 nodes in a DiGraph. Parameters ---------- G : digraph A NetworkX DiGraph Returns ------- triplets : generator of 3-tuples Generator of tuples of 3 nodes """ triplets = combinations(G.nodes(), 3) return triplets
[docs]@not_implemented_for("undirected") def all_triads(G): """A generator of all possible triads in G. Parameters ---------- G : digraph A NetworkX DiGraph Returns ------- all_triads : generator of DiGraphs Generator of triads (order-3 DiGraphs) """ triplets = combinations(G.nodes(), 3) for triplet in triplets: yield G.subgraph(triplet).copy()
[docs]@not_implemented_for("undirected") def triads_by_type(G): """Returns a list of all triads for each triad type in a directed graph. Parameters ---------- G : digraph A NetworkX DiGraph Returns ------- tri_by_type : dict Dictionary with triad types as keys and lists of triads as values. """ # num_triads = o * (o - 1) * (o - 2) // 6 # if num_triads > TRIAD_LIMIT: print(WARNING) all_tri = all_triads(G) tri_by_type = defaultdict(list) for triad in all_tri: name = triad_type(triad) tri_by_type[name].append(triad) return tri_by_type
[docs]@not_implemented_for("undirected") def triad_type(G): """Returns the sociological triad type for a triad. Parameters ---------- G : digraph A NetworkX DiGraph with 3 nodes Returns ------- triad_type : str A string identifying the triad type Notes ----- There can be 6 unique edges in a triad (order-3 DiGraph) (so 2^^6=64 unique triads given 3 nodes). These 64 triads each display exactly 1 of 16 topologies of triads (topologies can be permuted). These topologies are identified by the following notation: {m}{a}{n}{type} (for example: 111D, 210, 102) Here: {m} = number of mutual ties (takes 0, 1, 2, 3); a mutual tie is (0,1) AND (1,0) {a} = number of assymmetric ties (takes 0, 1, 2, 3); an assymmetric tie is (0,1) BUT NOT (1,0) or vice versa {n} = number of null ties (takes 0, 1, 2, 3); a null tie is NEITHER (0,1) NOR (1,0) {type} = a letter (takes U, D, C, T) corresponding to up, down, cyclical and transitive. This is only used for topologies that can have more than one form (eg: 021D and 021U). References ---------- .. [1] Snijders, T. (2012). "Transitivity and triads." University of Oxford. http://www.stats.ox.ac.uk/snijders/Trans_Triads_ha.pdf """ if not is_triad(G): raise nx.NetworkXAlgorithmError("G is not a triad (order-3 DiGraph)") num_edges = len(G.edges()) if num_edges == 0: return "003" elif num_edges == 1: return "012" elif num_edges == 2: e1, e2 = G.edges() if set(e1) == set(e2): return "102" elif e1[0] == e2[0]: return "021D" elif e1[1] == e2[1]: return "021U" elif e1[1] == e2[0] or e2[1] == e1[0]: return "021C" elif num_edges == 3: for (e1, e2, e3) in permutations(G.edges(), 3): if set(e1) == set(e2): if e3[0] in e1: return "111U" # e3[1] in e1: return "111D" elif set(e1).symmetric_difference(set(e2)) == set(e3): if {e1[0], e2[0], e3[0]} == {e1[0], e2[0], e3[0]} == set(G.nodes()): return "030C" # e3 == (e1[0], e2[1]) and e2 == (e1[1], e3[1]): return "030T" elif num_edges == 4: for (e1, e2, e3, e4) in permutations(G.edges(), 4): if set(e1) == set(e2): # identify pair of symmetric edges (which necessarily exists) if set(e3) == set(e4): return "201" if {e3[0]} == {e4[0]} == set(e3).intersection(set(e4)): return "120D" if {e3[1]} == {e4[1]} == set(e3).intersection(set(e4)): return "120U" if e3[1] == e4[0]: return "120C" elif num_edges == 5: return "210" elif num_edges == 6: return "300"
[docs]@not_implemented_for("undirected") def random_triad(G): """Returns a random triad from a directed graph. Parameters ---------- G : digraph A NetworkX DiGraph Returns ------- G2 : subgraph A randomly selected triad (order-3 NetworkX DiGraph) """ nodes = sample(G.nodes(), 3) G2 = G.subgraph(nodes) return G2
""" @not_implemented_for('undirected') def triadic_closures(G): '''Returns a list of order-3 subgraphs of G that are triadic closures. Parameters ---------- G : digraph A NetworkX DiGraph Returns ------- closures : list List of triads of G that are triadic closures ''' pass @not_implemented_for('undirected') def focal_closures(G, attr_name): '''Returns a list of order-3 subgraphs of G that are focally closed. Parameters ---------- G : digraph A NetworkX DiGraph attr_name : str An attribute name Returns ------- closures : list List of triads of G that are focally closed on attr_name ''' pass @not_implemented_for('undirected') def balanced_triads(G, crit_func): '''Returns a list of order-3 subgraphs of G that are stable. Parameters ---------- G : digraph A NetworkX DiGraph crit_func : function A function that determines if a triad (order-3 digraph) is stable Returns ------- triads : list List of triads in G that are stable ''' pass """