570 lines
18 KiB
Python
570 lines
18 KiB
Python
"""Algorithms to characterize the number of triangles in a graph."""
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from itertools import chain
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from itertools import combinations
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from collections import Counter
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from networkx.utils import not_implemented_for
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__all__ = [
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"triangles",
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"average_clustering",
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"clustering",
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"transitivity",
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"square_clustering",
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"generalized_degree",
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]
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@not_implemented_for("directed")
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def triangles(G, nodes=None):
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"""Compute the number of triangles.
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Finds the number of triangles that include a node as one vertex.
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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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nodes : container of nodes, optional (default= all nodes in G)
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Compute triangles for nodes in this container.
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Returns
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-------
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out : dictionary
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Number of triangles keyed by node label.
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Examples
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--------
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>>> G = nx.complete_graph(5)
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>>> print(nx.triangles(G, 0))
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6
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>>> print(nx.triangles(G))
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{0: 6, 1: 6, 2: 6, 3: 6, 4: 6}
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>>> print(list(nx.triangles(G, (0, 1)).values()))
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[6, 6]
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Notes
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-----
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When computing triangles for the entire graph each triangle is counted
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three times, once at each node. Self loops are ignored.
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"""
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# If `nodes` represents a single node in the graph, return only its number
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# of triangles.
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if nodes in G:
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return next(_triangles_and_degree_iter(G, nodes))[2] // 2
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# Otherwise, `nodes` represents an iterable of nodes, so return a
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# dictionary mapping node to number of triangles.
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return {v: t // 2 for v, d, t, _ in _triangles_and_degree_iter(G, nodes)}
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@not_implemented_for("multigraph")
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def _triangles_and_degree_iter(G, nodes=None):
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"""Return an iterator of (node, degree, triangles, generalized degree).
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This double counts triangles so you may want to divide by 2.
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See degree(), triangles() and generalized_degree() for definitions
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and details.
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"""
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if nodes is None:
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nodes_nbrs = G.adj.items()
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else:
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nodes_nbrs = ((n, G[n]) for n in G.nbunch_iter(nodes))
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for v, v_nbrs in nodes_nbrs:
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vs = set(v_nbrs) - {v}
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gen_degree = Counter(len(vs & (set(G[w]) - {w})) for w in vs)
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ntriangles = sum(k * val for k, val in gen_degree.items())
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yield (v, len(vs), ntriangles, gen_degree)
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@not_implemented_for("multigraph")
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def _weighted_triangles_and_degree_iter(G, nodes=None, weight="weight"):
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"""Return an iterator of (node, degree, weighted_triangles).
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Used for weighted clustering.
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Note: this returns the geometric average weight of edges in the triangle.
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Also, each triangle is counted twice (each direction).
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So you may want to divide by 2.
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"""
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import numpy as np
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if weight is None or G.number_of_edges() == 0:
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max_weight = 1
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else:
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max_weight = max(d.get(weight, 1) for u, v, d in G.edges(data=True))
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if nodes is None:
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nodes_nbrs = G.adj.items()
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else:
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nodes_nbrs = ((n, G[n]) for n in G.nbunch_iter(nodes))
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def wt(u, v):
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return G[u][v].get(weight, 1) / max_weight
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for i, nbrs in nodes_nbrs:
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inbrs = set(nbrs) - {i}
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weighted_triangles = 0
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seen = set()
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for j in inbrs:
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seen.add(j)
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# This avoids counting twice -- we double at the end.
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jnbrs = set(G[j]) - seen
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# Only compute the edge weight once, before the inner inner
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# loop.
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wij = wt(i, j)
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weighted_triangles += sum(
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np.cbrt([(wij * wt(j, k) * wt(k, i)) for k in inbrs & jnbrs])
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)
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yield (i, len(inbrs), 2 * weighted_triangles)
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@not_implemented_for("multigraph")
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def _directed_triangles_and_degree_iter(G, nodes=None):
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"""Return an iterator of
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(node, total_degree, reciprocal_degree, directed_triangles).
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Used for directed clustering.
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Note that unlike `_triangles_and_degree_iter()`, this function counts
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directed triangles so does not count triangles twice.
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"""
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nodes_nbrs = ((n, G._pred[n], G._succ[n]) for n in G.nbunch_iter(nodes))
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for i, preds, succs in nodes_nbrs:
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ipreds = set(preds) - {i}
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isuccs = set(succs) - {i}
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directed_triangles = 0
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for j in chain(ipreds, isuccs):
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jpreds = set(G._pred[j]) - {j}
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jsuccs = set(G._succ[j]) - {j}
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directed_triangles += sum(
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1
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for k in chain(
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(ipreds & jpreds),
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(ipreds & jsuccs),
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(isuccs & jpreds),
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(isuccs & jsuccs),
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)
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)
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dtotal = len(ipreds) + len(isuccs)
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dbidirectional = len(ipreds & isuccs)
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yield (i, dtotal, dbidirectional, directed_triangles)
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@not_implemented_for("multigraph")
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def _directed_weighted_triangles_and_degree_iter(G, nodes=None, weight="weight"):
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"""Return an iterator of
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(node, total_degree, reciprocal_degree, directed_weighted_triangles).
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Used for directed weighted clustering.
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Note that unlike `_weighted_triangles_and_degree_iter()`, this function counts
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directed triangles so does not count triangles twice.
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"""
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import numpy as np
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if weight is None or G.number_of_edges() == 0:
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max_weight = 1
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else:
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max_weight = max(d.get(weight, 1) for u, v, d in G.edges(data=True))
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nodes_nbrs = ((n, G._pred[n], G._succ[n]) for n in G.nbunch_iter(nodes))
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def wt(u, v):
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return G[u][v].get(weight, 1) / max_weight
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for i, preds, succs in nodes_nbrs:
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ipreds = set(preds) - {i}
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isuccs = set(succs) - {i}
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directed_triangles = 0
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for j in ipreds:
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jpreds = set(G._pred[j]) - {j}
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jsuccs = set(G._succ[j]) - {j}
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directed_triangles += sum(
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np.cbrt([(wt(j, i) * wt(k, i) * wt(k, j)) for k in ipreds & jpreds])
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)
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directed_triangles += sum(
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np.cbrt([(wt(j, i) * wt(k, i) * wt(j, k)) for k in ipreds & jsuccs])
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)
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directed_triangles += sum(
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np.cbrt([(wt(j, i) * wt(i, k) * wt(k, j)) for k in isuccs & jpreds])
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)
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directed_triangles += sum(
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np.cbrt([(wt(j, i) * wt(i, k) * wt(j, k)) for k in isuccs & jsuccs])
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)
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for j in isuccs:
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jpreds = set(G._pred[j]) - {j}
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jsuccs = set(G._succ[j]) - {j}
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directed_triangles += sum(
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np.cbrt([(wt(i, j) * wt(k, i) * wt(k, j)) for k in ipreds & jpreds])
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)
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directed_triangles += sum(
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np.cbrt([(wt(i, j) * wt(k, i) * wt(j, k)) for k in ipreds & jsuccs])
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)
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directed_triangles += sum(
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np.cbrt([(wt(i, j) * wt(i, k) * wt(k, j)) for k in isuccs & jpreds])
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)
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directed_triangles += sum(
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np.cbrt([(wt(i, j) * wt(i, k) * wt(j, k)) for k in isuccs & jsuccs])
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)
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dtotal = len(ipreds) + len(isuccs)
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dbidirectional = len(ipreds & isuccs)
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yield (i, dtotal, dbidirectional, directed_triangles)
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def average_clustering(G, nodes=None, weight=None, count_zeros=True):
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r"""Compute the average clustering coefficient for the graph G.
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The clustering coefficient for the graph is the average,
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.. math::
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C = \frac{1}{n}\sum_{v \in G} c_v,
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where :math:`n` is the number of nodes in `G`.
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Parameters
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----------
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G : graph
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nodes : container of nodes, optional (default=all nodes in G)
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Compute average clustering for nodes in this container.
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weight : string or None, optional (default=None)
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The edge attribute that holds the numerical value used as a weight.
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If None, then each edge has weight 1.
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count_zeros : bool
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If False include only the nodes with nonzero clustering in the average.
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Returns
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-------
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avg : float
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Average clustering
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Examples
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--------
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>>> G = nx.complete_graph(5)
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>>> print(nx.average_clustering(G))
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1.0
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Notes
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-----
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This is a space saving routine; it might be faster
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to use the clustering function to get a list and then take the average.
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Self loops are ignored.
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References
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----------
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.. [1] Generalizations of the clustering coefficient to weighted
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complex networks by J. Saramäki, M. Kivelä, J.-P. Onnela,
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K. Kaski, and J. Kertész, Physical Review E, 75 027105 (2007).
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http://jponnela.com/web_documents/a9.pdf
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.. [2] Marcus Kaiser, Mean clustering coefficients: the role of isolated
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nodes and leafs on clustering measures for small-world networks.
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https://arxiv.org/abs/0802.2512
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"""
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c = clustering(G, nodes, weight=weight).values()
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if not count_zeros:
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c = [v for v in c if abs(v) > 0]
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return sum(c) / len(c)
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def clustering(G, nodes=None, weight=None):
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r"""Compute the clustering coefficient for nodes.
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For unweighted graphs, the clustering of a node :math:`u`
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is the fraction of possible triangles through that node that exist,
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.. math::
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c_u = \frac{2 T(u)}{deg(u)(deg(u)-1)},
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where :math:`T(u)` is the number of triangles through node :math:`u` and
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:math:`deg(u)` is the degree of :math:`u`.
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For weighted graphs, there are several ways to define clustering [1]_.
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the one used here is defined
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as the geometric average of the subgraph edge weights [2]_,
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.. math::
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c_u = \frac{1}{deg(u)(deg(u)-1))}
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\sum_{vw} (\hat{w}_{uv} \hat{w}_{uw} \hat{w}_{vw})^{1/3}.
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The edge weights :math:`\hat{w}_{uv}` are normalized by the maximum weight
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in the network :math:`\hat{w}_{uv} = w_{uv}/\max(w)`.
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The value of :math:`c_u` is assigned to 0 if :math:`deg(u) < 2`.
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Additionally, this weighted definition has been generalized to support negative edge weights [3]_.
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For directed graphs, the clustering is similarly defined as the fraction
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of all possible directed triangles or geometric average of the subgraph
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edge weights for unweighted and weighted directed graph respectively [4]_.
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.. math::
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c_u = \frac{2}{deg^{tot}(u)(deg^{tot}(u)-1) - 2deg^{\leftrightarrow}(u)}
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T(u),
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where :math:`T(u)` is the number of directed triangles through node
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:math:`u`, :math:`deg^{tot}(u)` is the sum of in degree and out degree of
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:math:`u` and :math:`deg^{\leftrightarrow}(u)` is the reciprocal degree of
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:math:`u`.
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Parameters
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----------
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G : graph
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nodes : container of nodes, optional (default=all nodes in G)
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Compute clustering for nodes in this container.
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weight : string or None, optional (default=None)
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The edge attribute that holds the numerical value used as a weight.
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If None, then each edge has weight 1.
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Returns
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-------
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out : float, or dictionary
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Clustering coefficient at specified nodes
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Examples
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--------
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>>> G = nx.complete_graph(5)
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>>> print(nx.clustering(G, 0))
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1.0
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>>> print(nx.clustering(G))
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{0: 1.0, 1: 1.0, 2: 1.0, 3: 1.0, 4: 1.0}
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Notes
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-----
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Self loops are ignored.
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References
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----------
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.. [1] Generalizations of the clustering coefficient to weighted
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complex networks by J. Saramäki, M. Kivelä, J.-P. Onnela,
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K. Kaski, and J. Kertész, Physical Review E, 75 027105 (2007).
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http://jponnela.com/web_documents/a9.pdf
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.. [2] Intensity and coherence of motifs in weighted complex
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networks by J. P. Onnela, J. Saramäki, J. Kertész, and K. Kaski,
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Physical Review E, 71(6), 065103 (2005).
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.. [3] Generalization of Clustering Coefficients to Signed Correlation Networks
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by G. Costantini and M. Perugini, PloS one, 9(2), e88669 (2014).
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.. [4] Clustering in complex directed networks by G. Fagiolo,
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Physical Review E, 76(2), 026107 (2007).
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"""
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if G.is_directed():
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if weight is not None:
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td_iter = _directed_weighted_triangles_and_degree_iter(G, nodes, weight)
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clusterc = {
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v: 0 if t == 0 else t / ((dt * (dt - 1) - 2 * db) * 2)
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for v, dt, db, t in td_iter
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}
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else:
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td_iter = _directed_triangles_and_degree_iter(G, nodes)
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clusterc = {
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v: 0 if t == 0 else t / ((dt * (dt - 1) - 2 * db) * 2)
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for v, dt, db, t in td_iter
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}
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else:
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# The formula 2*T/(d*(d-1)) from docs is t/(d*(d-1)) here b/c t==2*T
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if weight is not None:
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td_iter = _weighted_triangles_and_degree_iter(G, nodes, weight)
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clusterc = {v: 0 if t == 0 else t / (d * (d - 1)) for v, d, t in td_iter}
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else:
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td_iter = _triangles_and_degree_iter(G, nodes)
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clusterc = {v: 0 if t == 0 else t / (d * (d - 1)) for v, d, t, _ in td_iter}
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if nodes in G:
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# Return the value of the sole entry in the dictionary.
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return clusterc[nodes]
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return clusterc
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def transitivity(G):
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r"""Compute graph transitivity, the fraction of all possible triangles
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present in G.
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Possible triangles are identified by the number of "triads"
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(two edges with a shared vertex).
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The transitivity is
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.. math::
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T = 3\frac{\#triangles}{\#triads}.
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Parameters
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----------
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G : graph
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Returns
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-------
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out : float
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Transitivity
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Examples
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--------
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>>> G = nx.complete_graph(5)
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>>> print(nx.transitivity(G))
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1.0
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"""
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triangles_contri = [
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(t, d * (d - 1)) for v, d, t, _ in _triangles_and_degree_iter(G)
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]
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# If the graph is empty
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if len(triangles_contri) == 0:
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return 0
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triangles, contri = map(sum, zip(*triangles_contri))
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return 0 if triangles == 0 else triangles / contri
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def square_clustering(G, nodes=None):
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r"""Compute the squares clustering coefficient for nodes.
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For each node return the fraction of possible squares that exist at
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the node [1]_
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.. math::
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C_4(v) = \frac{ \sum_{u=1}^{k_v}
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\sum_{w=u+1}^{k_v} q_v(u,w) }{ \sum_{u=1}^{k_v}
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\sum_{w=u+1}^{k_v} [a_v(u,w) + q_v(u,w)]},
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where :math:`q_v(u,w)` are the number of common neighbors of :math:`u` and
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:math:`w` other than :math:`v` (ie squares), and :math:`a_v(u,w) = (k_u -
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(1+q_v(u,w)+\theta_{uv})) + (k_w - (1+q_v(u,w)+\theta_{uw}))`, where
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:math:`\theta_{uw} = 1` if :math:`u` and :math:`w` are connected and 0
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otherwise. [2]_
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Parameters
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----------
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G : graph
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nodes : container of nodes, optional (default=all nodes in G)
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Compute clustering for nodes in this container.
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Returns
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-------
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c4 : dictionary
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A dictionary keyed by node with the square clustering coefficient value.
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Examples
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--------
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>>> G = nx.complete_graph(5)
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>>> print(nx.square_clustering(G, 0))
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1.0
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>>> print(nx.square_clustering(G))
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{0: 1.0, 1: 1.0, 2: 1.0, 3: 1.0, 4: 1.0}
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Notes
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-----
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While :math:`C_3(v)` (triangle clustering) gives the probability that
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two neighbors of node v are connected with each other, :math:`C_4(v)` is
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the probability that two neighbors of node v share a common
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neighbor different from v. This algorithm can be applied to both
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bipartite and unipartite networks.
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References
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----------
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.. [1] Pedro G. Lind, Marta C. González, and Hans J. Herrmann. 2005
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Cycles and clustering in bipartite networks.
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Physical Review E (72) 056127.
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.. [2] Zhang, Peng et al. Clustering Coefficient and Community Structure of
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Bipartite Networks. Physica A: Statistical Mechanics and its Applications 387.27 (2008): 6869–6875.
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https://arxiv.org/abs/0710.0117v1
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"""
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if nodes is None:
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node_iter = G
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else:
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node_iter = G.nbunch_iter(nodes)
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clustering = {}
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for v in node_iter:
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clustering[v] = 0
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potential = 0
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for u, w in combinations(G[v], 2):
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squares = len((set(G[u]) & set(G[w])) - {v})
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clustering[v] += squares
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degm = squares + 1
|
||
if w in G[u]:
|
||
degm += 1
|
||
potential += (len(G[u]) - degm) + (len(G[w]) - degm) + squares
|
||
if potential > 0:
|
||
clustering[v] /= potential
|
||
if nodes in G:
|
||
# Return the value of the sole entry in the dictionary.
|
||
return clustering[nodes]
|
||
return clustering
|
||
|
||
|
||
@not_implemented_for("directed")
|
||
def generalized_degree(G, nodes=None):
|
||
r"""Compute the generalized degree for nodes.
|
||
|
||
For each node, the generalized degree shows how many edges of given
|
||
triangle multiplicity the node is connected to. The triangle multiplicity
|
||
of an edge is the number of triangles an edge participates in. The
|
||
generalized degree of node :math:`i` can be written as a vector
|
||
:math:`\mathbf{k}_i=(k_i^{(0)}, \dotsc, k_i^{(N-2)})` where
|
||
:math:`k_i^{(j)}` is the number of edges attached to node :math:`i` that
|
||
participate in :math:`j` triangles.
|
||
|
||
Parameters
|
||
----------
|
||
G : graph
|
||
|
||
nodes : container of nodes, optional (default=all nodes in G)
|
||
Compute the generalized degree for nodes in this container.
|
||
|
||
Returns
|
||
-------
|
||
out : Counter, or dictionary of Counters
|
||
Generalized degree of specified nodes. The Counter is keyed by edge
|
||
triangle multiplicity.
|
||
|
||
Examples
|
||
--------
|
||
>>> G = nx.complete_graph(5)
|
||
>>> print(nx.generalized_degree(G, 0))
|
||
Counter({3: 4})
|
||
>>> print(nx.generalized_degree(G))
|
||
{0: Counter({3: 4}), 1: Counter({3: 4}), 2: Counter({3: 4}), 3: Counter({3: 4}), 4: Counter({3: 4})}
|
||
|
||
To recover the number of triangles attached to a node:
|
||
|
||
>>> k1 = nx.generalized_degree(G, 0)
|
||
>>> sum([k * v for k, v in k1.items()]) / 2 == nx.triangles(G, 0)
|
||
True
|
||
|
||
Notes
|
||
-----
|
||
In a network of N nodes, the highest triangle multiplicty an edge can have
|
||
is N-2.
|
||
|
||
The return value does not include a `zero` entry if no edges of a
|
||
particular triangle multiplicity are present.
|
||
|
||
The number of triangles node :math:`i` is attached to can be recovered from
|
||
the generalized degree :math:`\mathbf{k}_i=(k_i^{(0)}, \dotsc,
|
||
k_i^{(N-2)})` by :math:`(k_i^{(1)}+2k_i^{(2)}+\dotsc +(N-2)k_i^{(N-2)})/2`.
|
||
|
||
References
|
||
----------
|
||
.. [1] Networks with arbitrary edge multiplicities by V. Zlatić,
|
||
D. Garlaschelli and G. Caldarelli, EPL (Europhysics Letters),
|
||
Volume 97, Number 2 (2012).
|
||
https://iopscience.iop.org/article/10.1209/0295-5075/97/28005
|
||
"""
|
||
if nodes in G:
|
||
return next(_triangles_and_degree_iter(G, nodes))[3]
|
||
return {v: gd for v, d, t, gd in _triangles_and_degree_iter(G, nodes)}
|