517 lines
16 KiB
Python
517 lines
16 KiB
Python
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"""One-mode (unipartite) projections of bipartite graphs."""
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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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"project",
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"projected_graph",
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"weighted_projected_graph",
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"collaboration_weighted_projected_graph",
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"overlap_weighted_projected_graph",
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"generic_weighted_projected_graph",
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]
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def projected_graph(B, nodes, multigraph=False):
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r"""Returns the projection of B onto one of its node sets.
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Returns the graph G that is the projection of the bipartite graph B
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onto the specified nodes. They retain their attributes and are connected
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in G if they have a common neighbor in B.
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Parameters
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----------
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B : NetworkX graph
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The input graph should be bipartite.
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nodes : list or iterable
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Nodes to project onto (the "bottom" nodes).
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multigraph: bool (default=False)
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If True return a multigraph where the multiple edges represent multiple
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shared neighbors. They edge key in the multigraph is assigned to the
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label of the neighbor.
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Returns
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-------
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Graph : NetworkX graph or multigraph
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A graph that is the projection onto the given nodes.
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Examples
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--------
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>>> from networkx.algorithms import bipartite
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>>> B = nx.path_graph(4)
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>>> G = bipartite.projected_graph(B, [1, 3])
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>>> list(G)
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[1, 3]
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>>> list(G.edges())
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[(1, 3)]
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If nodes `a`, and `b` are connected through both nodes 1 and 2 then
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building a multigraph results in two edges in the projection onto
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[`a`, `b`]:
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>>> B = nx.Graph()
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>>> B.add_edges_from([("a", 1), ("b", 1), ("a", 2), ("b", 2)])
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>>> G = bipartite.projected_graph(B, ["a", "b"], multigraph=True)
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>>> print([sorted((u, v)) for u, v in G.edges()])
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[['a', 'b'], ['a', 'b']]
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Notes
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-----
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No attempt is made to verify that the input graph B is bipartite.
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Returns a simple graph that is the projection of the bipartite graph B
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onto the set of nodes given in list nodes. If multigraph=True then
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a multigraph is returned with an edge for every shared neighbor.
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Directed graphs are allowed as input. The output will also then
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be a directed graph with edges if there is a directed path between
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the nodes.
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The graph and node properties are (shallow) copied to the projected graph.
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See :mod:`bipartite documentation <networkx.algorithms.bipartite>`
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for further details on how bipartite graphs are handled in NetworkX.
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See Also
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--------
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is_bipartite,
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is_bipartite_node_set,
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sets,
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weighted_projected_graph,
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collaboration_weighted_projected_graph,
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overlap_weighted_projected_graph,
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generic_weighted_projected_graph
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"""
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if B.is_multigraph():
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raise nx.NetworkXError("not defined for multigraphs")
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if B.is_directed():
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directed = True
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if multigraph:
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G = nx.MultiDiGraph()
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else:
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G = nx.DiGraph()
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else:
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directed = False
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if multigraph:
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G = nx.MultiGraph()
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else:
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G = nx.Graph()
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G.graph.update(B.graph)
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G.add_nodes_from((n, B.nodes[n]) for n in nodes)
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for u in nodes:
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nbrs2 = {v for nbr in B[u] for v in B[nbr] if v != u}
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if multigraph:
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for n in nbrs2:
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if directed:
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links = set(B[u]) & set(B.pred[n])
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else:
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links = set(B[u]) & set(B[n])
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for l in links:
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if not G.has_edge(u, n, l):
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G.add_edge(u, n, key=l)
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else:
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G.add_edges_from((u, n) for n in nbrs2)
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return G
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@not_implemented_for("multigraph")
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def weighted_projected_graph(B, nodes, ratio=False):
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r"""Returns a weighted projection of B onto one of its node sets.
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The weighted projected graph is the projection of the bipartite
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network B onto the specified nodes with weights representing the
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number of shared neighbors or the ratio between actual shared
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neighbors and possible shared neighbors if ``ratio is True`` [1]_.
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The nodes retain their attributes and are connected in the resulting
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graph if they have an edge to a common node in the original graph.
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Parameters
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----------
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B : NetworkX graph
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The input graph should be bipartite.
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nodes : list or iterable
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Nodes to project onto (the "bottom" nodes).
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ratio: Bool (default=False)
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If True, edge weight is the ratio between actual shared neighbors
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and maximum possible shared neighbors (i.e., the size of the other
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node set). If False, edges weight is the number of shared neighbors.
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Returns
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-------
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Graph : NetworkX graph
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A graph that is the projection onto the given nodes.
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Examples
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--------
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>>> from networkx.algorithms import bipartite
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>>> B = nx.path_graph(4)
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>>> G = bipartite.weighted_projected_graph(B, [1, 3])
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>>> list(G)
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[1, 3]
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>>> list(G.edges(data=True))
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[(1, 3, {'weight': 1})]
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>>> G = bipartite.weighted_projected_graph(B, [1, 3], ratio=True)
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>>> list(G.edges(data=True))
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[(1, 3, {'weight': 0.5})]
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Notes
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-----
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No attempt is made to verify that the input graph B is bipartite.
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The graph and node properties are (shallow) copied to the projected graph.
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See :mod:`bipartite documentation <networkx.algorithms.bipartite>`
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for further details on how bipartite graphs are handled in NetworkX.
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See Also
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--------
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is_bipartite,
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is_bipartite_node_set,
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sets,
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collaboration_weighted_projected_graph,
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overlap_weighted_projected_graph,
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generic_weighted_projected_graph
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projected_graph
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References
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----------
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.. [1] Borgatti, S.P. and Halgin, D. In press. "Analyzing Affiliation
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Networks". In Carrington, P. and Scott, J. (eds) The Sage Handbook
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of Social Network Analysis. Sage Publications.
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"""
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if B.is_directed():
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pred = B.pred
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G = nx.DiGraph()
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else:
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pred = B.adj
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G = nx.Graph()
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G.graph.update(B.graph)
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G.add_nodes_from((n, B.nodes[n]) for n in nodes)
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n_top = float(len(B) - len(nodes))
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for u in nodes:
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unbrs = set(B[u])
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nbrs2 = {n for nbr in unbrs for n in B[nbr]} - {u}
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for v in nbrs2:
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vnbrs = set(pred[v])
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common = unbrs & vnbrs
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if not ratio:
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weight = len(common)
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else:
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weight = len(common) / n_top
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G.add_edge(u, v, weight=weight)
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return G
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@not_implemented_for("multigraph")
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def collaboration_weighted_projected_graph(B, nodes):
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r"""Newman's weighted projection of B onto one of its node sets.
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The collaboration weighted projection is the projection of the
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bipartite network B onto the specified nodes with weights assigned
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using Newman's collaboration model [1]_:
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.. math::
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w_{u, v} = \sum_k \frac{\delta_{u}^{k} \delta_{v}^{k}}{d_k - 1}
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where `u` and `v` are nodes from the bottom bipartite node set,
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and `k` is a node of the top node set.
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The value `d_k` is the degree of node `k` in the bipartite
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network and `\delta_{u}^{k}` is 1 if node `u` is
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linked to node `k` in the original bipartite graph or 0 otherwise.
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The nodes retain their attributes and are connected in the resulting
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graph if have an edge to a common node in the original bipartite
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graph.
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Parameters
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----------
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B : NetworkX graph
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The input graph should be bipartite.
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nodes : list or iterable
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Nodes to project onto (the "bottom" nodes).
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Returns
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-------
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Graph : NetworkX graph
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A graph that is the projection onto the given nodes.
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Examples
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--------
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>>> from networkx.algorithms import bipartite
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>>> B = nx.path_graph(5)
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>>> B.add_edge(1, 5)
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>>> G = bipartite.collaboration_weighted_projected_graph(B, [0, 2, 4, 5])
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>>> list(G)
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[0, 2, 4, 5]
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>>> for edge in sorted(G.edges(data=True)):
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... print(edge)
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...
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(0, 2, {'weight': 0.5})
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(0, 5, {'weight': 0.5})
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(2, 4, {'weight': 1.0})
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(2, 5, {'weight': 0.5})
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Notes
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-----
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No attempt is made to verify that the input graph B is bipartite.
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The graph and node properties are (shallow) copied to the projected graph.
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See :mod:`bipartite documentation <networkx.algorithms.bipartite>`
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for further details on how bipartite graphs are handled in NetworkX.
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See Also
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--------
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is_bipartite,
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is_bipartite_node_set,
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sets,
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weighted_projected_graph,
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overlap_weighted_projected_graph,
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generic_weighted_projected_graph,
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projected_graph
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References
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----------
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.. [1] Scientific collaboration networks: II.
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Shortest paths, weighted networks, and centrality,
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M. E. J. Newman, Phys. Rev. E 64, 016132 (2001).
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"""
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if B.is_directed():
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pred = B.pred
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G = nx.DiGraph()
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else:
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pred = B.adj
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G = nx.Graph()
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G.graph.update(B.graph)
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G.add_nodes_from((n, B.nodes[n]) for n in nodes)
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for u in nodes:
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unbrs = set(B[u])
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nbrs2 = {n for nbr in unbrs for n in B[nbr] if n != u}
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for v in nbrs2:
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vnbrs = set(pred[v])
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common_degree = (len(B[n]) for n in unbrs & vnbrs)
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weight = sum(1.0 / (deg - 1) for deg in common_degree if deg > 1)
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G.add_edge(u, v, weight=weight)
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return G
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@not_implemented_for("multigraph")
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def overlap_weighted_projected_graph(B, nodes, jaccard=True):
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r"""Overlap weighted projection of B onto one of its node sets.
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The overlap weighted projection is the projection of the bipartite
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network B onto the specified nodes with weights representing
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the Jaccard index between the neighborhoods of the two nodes in the
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original bipartite network [1]_:
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.. math::
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w_{v, u} = \frac{|N(u) \cap N(v)|}{|N(u) \cup N(v)|}
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or if the parameter 'jaccard' is False, the fraction of common
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neighbors by minimum of both nodes degree in the original
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bipartite graph [1]_:
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.. math::
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w_{v, u} = \frac{|N(u) \cap N(v)|}{min(|N(u)|, |N(v)|)}
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The nodes retain their attributes and are connected in the resulting
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graph if have an edge to a common node in the original bipartite graph.
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Parameters
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----------
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B : NetworkX graph
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The input graph should be bipartite.
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nodes : list or iterable
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Nodes to project onto (the "bottom" nodes).
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jaccard: Bool (default=True)
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Returns
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-------
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Graph : NetworkX graph
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A graph that is the projection onto the given nodes.
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Examples
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--------
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>>> from networkx.algorithms import bipartite
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>>> B = nx.path_graph(5)
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>>> nodes = [0, 2, 4]
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>>> G = bipartite.overlap_weighted_projected_graph(B, nodes)
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>>> list(G)
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[0, 2, 4]
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>>> list(G.edges(data=True))
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[(0, 2, {'weight': 0.5}), (2, 4, {'weight': 0.5})]
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>>> G = bipartite.overlap_weighted_projected_graph(B, nodes, jaccard=False)
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>>> list(G.edges(data=True))
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[(0, 2, {'weight': 1.0}), (2, 4, {'weight': 1.0})]
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Notes
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-----
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No attempt is made to verify that the input graph B is bipartite.
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The graph and node properties are (shallow) copied to the projected graph.
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See :mod:`bipartite documentation <networkx.algorithms.bipartite>`
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for further details on how bipartite graphs are handled in NetworkX.
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See Also
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--------
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is_bipartite,
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is_bipartite_node_set,
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sets,
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weighted_projected_graph,
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collaboration_weighted_projected_graph,
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generic_weighted_projected_graph,
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projected_graph
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References
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----------
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.. [1] Borgatti, S.P. and Halgin, D. In press. Analyzing Affiliation
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Networks. In Carrington, P. and Scott, J. (eds) The Sage Handbook
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of Social Network Analysis. Sage Publications.
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"""
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if B.is_directed():
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pred = B.pred
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G = nx.DiGraph()
|
||
|
|
else:
|
||
|
|
pred = B.adj
|
||
|
|
G = nx.Graph()
|
||
|
|
G.graph.update(B.graph)
|
||
|
|
G.add_nodes_from((n, B.nodes[n]) for n in nodes)
|
||
|
|
for u in nodes:
|
||
|
|
unbrs = set(B[u])
|
||
|
|
nbrs2 = {n for nbr in unbrs for n in B[nbr]} - {u}
|
||
|
|
for v in nbrs2:
|
||
|
|
vnbrs = set(pred[v])
|
||
|
|
if jaccard:
|
||
|
|
wt = float(len(unbrs & vnbrs)) / len(unbrs | vnbrs)
|
||
|
|
else:
|
||
|
|
wt = float(len(unbrs & vnbrs)) / min(len(unbrs), len(vnbrs))
|
||
|
|
G.add_edge(u, v, weight=wt)
|
||
|
|
return G
|
||
|
|
|
||
|
|
|
||
|
|
@not_implemented_for("multigraph")
|
||
|
|
def generic_weighted_projected_graph(B, nodes, weight_function=None):
|
||
|
|
r"""Weighted projection of B with a user-specified weight function.
|
||
|
|
|
||
|
|
The bipartite network B is projected on to the specified nodes
|
||
|
|
with weights computed by a user-specified function. This function
|
||
|
|
must accept as a parameter the neighborhood sets of two nodes and
|
||
|
|
return an integer or a float.
|
||
|
|
|
||
|
|
The nodes retain their attributes and are connected in the resulting graph
|
||
|
|
if they have an edge to a common node in the original graph.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
B : NetworkX graph
|
||
|
|
The input graph should be bipartite.
|
||
|
|
|
||
|
|
nodes : list or iterable
|
||
|
|
Nodes to project onto (the "bottom" nodes).
|
||
|
|
|
||
|
|
weight_function : function
|
||
|
|
This function must accept as parameters the same input graph
|
||
|
|
that this function, and two nodes; and return an integer or a float.
|
||
|
|
The default function computes the number of shared neighbors.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
Graph : NetworkX graph
|
||
|
|
A graph that is the projection onto the given nodes.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> from networkx.algorithms import bipartite
|
||
|
|
>>> # Define some custom weight functions
|
||
|
|
>>> def jaccard(G, u, v):
|
||
|
|
... unbrs = set(G[u])
|
||
|
|
... vnbrs = set(G[v])
|
||
|
|
... return float(len(unbrs & vnbrs)) / len(unbrs | vnbrs)
|
||
|
|
...
|
||
|
|
>>> def my_weight(G, u, v, weight="weight"):
|
||
|
|
... w = 0
|
||
|
|
... for nbr in set(G[u]) & set(G[v]):
|
||
|
|
... w += G[u][nbr].get(weight, 1) + G[v][nbr].get(weight, 1)
|
||
|
|
... return w
|
||
|
|
...
|
||
|
|
>>> # A complete bipartite graph with 4 nodes and 4 edges
|
||
|
|
>>> B = nx.complete_bipartite_graph(2, 2)
|
||
|
|
>>> # Add some arbitrary weight to the edges
|
||
|
|
>>> for i, (u, v) in enumerate(B.edges()):
|
||
|
|
... B.edges[u, v]["weight"] = i + 1
|
||
|
|
...
|
||
|
|
>>> for edge in B.edges(data=True):
|
||
|
|
... print(edge)
|
||
|
|
...
|
||
|
|
(0, 2, {'weight': 1})
|
||
|
|
(0, 3, {'weight': 2})
|
||
|
|
(1, 2, {'weight': 3})
|
||
|
|
(1, 3, {'weight': 4})
|
||
|
|
>>> # By default, the weight is the number of shared neighbors
|
||
|
|
>>> G = bipartite.generic_weighted_projected_graph(B, [0, 1])
|
||
|
|
>>> print(list(G.edges(data=True)))
|
||
|
|
[(0, 1, {'weight': 2})]
|
||
|
|
>>> # To specify a custom weight function use the weight_function parameter
|
||
|
|
>>> G = bipartite.generic_weighted_projected_graph(
|
||
|
|
... B, [0, 1], weight_function=jaccard
|
||
|
|
... )
|
||
|
|
>>> print(list(G.edges(data=True)))
|
||
|
|
[(0, 1, {'weight': 1.0})]
|
||
|
|
>>> G = bipartite.generic_weighted_projected_graph(
|
||
|
|
... B, [0, 1], weight_function=my_weight
|
||
|
|
... )
|
||
|
|
>>> print(list(G.edges(data=True)))
|
||
|
|
[(0, 1, {'weight': 10})]
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
No attempt is made to verify that the input graph B is bipartite.
|
||
|
|
The graph and node properties are (shallow) copied to the projected graph.
|
||
|
|
|
||
|
|
See :mod:`bipartite documentation <networkx.algorithms.bipartite>`
|
||
|
|
for further details on how bipartite graphs are handled in NetworkX.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
is_bipartite,
|
||
|
|
is_bipartite_node_set,
|
||
|
|
sets,
|
||
|
|
weighted_projected_graph,
|
||
|
|
collaboration_weighted_projected_graph,
|
||
|
|
overlap_weighted_projected_graph,
|
||
|
|
projected_graph
|
||
|
|
|
||
|
|
"""
|
||
|
|
if B.is_directed():
|
||
|
|
pred = B.pred
|
||
|
|
G = nx.DiGraph()
|
||
|
|
else:
|
||
|
|
pred = B.adj
|
||
|
|
G = nx.Graph()
|
||
|
|
if weight_function is None:
|
||
|
|
|
||
|
|
def weight_function(G, u, v):
|
||
|
|
# Notice that we use set(pred[v]) for handling the directed case.
|
||
|
|
return len(set(G[u]) & set(pred[v]))
|
||
|
|
|
||
|
|
G.graph.update(B.graph)
|
||
|
|
G.add_nodes_from((n, B.nodes[n]) for n in nodes)
|
||
|
|
for u in nodes:
|
||
|
|
nbrs2 = {n for nbr in set(B[u]) for n in B[nbr]} - {u}
|
||
|
|
for v in nbrs2:
|
||
|
|
weight = weight_function(B, u, v)
|
||
|
|
G.add_edge(u, v, weight=weight)
|
||
|
|
return G
|
||
|
|
|
||
|
|
|
||
|
|
def project(B, nodes, create_using=None):
|
||
|
|
return projected_graph(B, nodes)
|