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projections.py
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projections.py
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import networkx as nx
from networkx.algorithms import bipartite
def projected_graph(B, nodes, multigraph=False):
"""Returns the projection of B onto one of its node sets.
Returns the graph G that is the projection of the bipartite graph B
onto the specified nodes. They retain their attributes and are connected
in G if they have a common neighbor in B.
Parameters
----------
B : NetworkX graph
The input graph should be bipartite.
nodes : list or iterable
Nodes to project onto (the "bottom" nodes).
multigraph: bool (default=False)
If True return a multigraph where the multiple edges represent multiple
shared neighbors. They edge key in the multigraph is assigned to the
label of the neighbor.
Returns
-------
Graph : NetworkX graph or multigraph
A graph that is the projection onto the given nodes.
Examples
--------
>>> from networkx.algorithms import bipartite
>>> B = nx.path_graph(4)
>>> G = bipartite.projected_graph(B, [1, 3])
>>> list(G)
[1, 3]
>>> list(G.edges())
[(1, 3)]
If nodes `a`, and `b` are connected through both nodes 1 and 2 then
building a multigraph results in two edges in the projection onto
[`a`, `b`]:
>>> B = nx.Graph()
>>> B.add_edges_from([('a', 1), ('b', 1), ('a', 2), ('b', 2)])
>>> G = bipartite.projected_graph(B, ['a', 'b'], multigraph=True)
>>> print([sorted((u, v)) for u, v in G.edges()])
[['a', 'b'], ['a', 'b']]
Notes
-----
No attempt is made to verify that the input graph B is bipartite.
Returns a simple graph that is the projection of the bipartite graph B
onto the set of nodes given in list nodes. If multigraph=True then
a multigraph is returned with an edge for every shared neighbor.
Directed graphs are allowed as input. The output will also then
be a directed graph with edges if there is a directed path between
the nodes.
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,
generic_weighted_projected_graph
"""
if B.is_multigraph():
raise nx.NetworkXError("not defined for multigraphs")
if B.is_directed():
directed = True
if multigraph:
G = nx.MultiDiGraph()
else:
G = nx.DiGraph()
else:
directed = False
if multigraph:
G = nx.MultiGraph()
else:
G = nx.Graph()
G.graph.update(B.graph)
G.add_nodes_from((n, {**B.nodes[n], **{'start': min([x[2] for x in B.edges(n, data='start')])}}) for n in nodes)
for u in nodes:
nbrs2 = set(v for nbr in B[u] for v in B[nbr] if v != u)
if multigraph:
for n in nbrs2:
if directed:
links = set(B[u]) & set(B.pred[n])
else:
links = set(B[u]) & set(B[n])
for l in links:
if not G.has_edge(u, n, l):
G.add_edge(u, n, key=l)
else:
G.add_edges_from((u, n, {'weight': 1, 'start': start_conversation(B.edges(u, data='start'), B.edges(n, data='start'))}) for n in nbrs2)
return G
# We need the weight to be type float instead of int so we slightly adjust the
# networkx weighted_projected_graph function.
def weighted_projected_graph(B, nodes, ratio=False):
"""Returns a weighted projection of B onto one of its node sets.
The weighted projected graph is the projection of the bipartite
network B onto the specified nodes with weights representing the
number of shared neighbors or the ratio between actual shared
neighbors and possible shared neighbors if ``ratio is True`` [1]_.
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).
ratio: Bool (default=False)
If True, edge weight is the ratio between actual shared neighbors
and possible shared neighbors. If False, edges weight is 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
>>> B = nx.path_graph(4)
>>> G = bipartite.weighted_projected_graph(B, [1, 3])
>>> list(G)
[1, 3]
>>> list(G.edges(data=True))
[(1, 3, {'weight': 1})]
>>> G = bipartite.weighted_projected_graph(B, [1, 3], ratio=True)
>>> list(G.edges(data=True))
[(1, 3, {'weight': 0.5})]
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,
collaboration_weighted_projected_graph,
overlap_weighted_projected_graph,
generic_weighted_projected_graph
projected_graph
References
----------
.. [1] Borgatti, S.P. and Halgin, D. In press. "Analyzing Affiliation
Networks". In Carrington, P. and Scott, J. (eds) The Sage Handbook
of Social Network Analysis. Sage Publications.
"""
if B.is_directed():
pred = B.pred
G = nx.DiGraph()
else:
pred = B.adj
G = nx.Graph()
G.graph.update(B.graph)
# print([(n, {**B.nodes[n], **{'start': min([x[2] for x in B.edges(n, data='start')])}}) for n in nodes])
G.add_nodes_from((n, {**B.nodes[n], **{'start': min([x[2] for x in B.edges(n, data='start')])}}) for n in nodes)
n_top = float(len(B) - len(nodes))
for u in nodes:
unbrs = set(B[u])
nbrs2 = set((n for nbr in unbrs for n in B[nbr])) - set([u])
for v in nbrs2:
vnbrs = set(pred[v])
common = unbrs & vnbrs
if not ratio:
weight = float(len(common))
else:
weight = float(len(common) / n_top)
G.add_edge(u, v, weight=weight, start=start_conversation(B.edges(u, data='start'), B.edges(v, data='start')))
return G
def collaboration_weighted_projected_graph(B, nodes):
"""Newman's weighted projection of B onto one of its node sets.
The collaboration weighted projection is the projection of the
bipartite network B onto the specified nodes with weights assigned
using Newman's collaboration model [1]_:
.. math::
w_{u, v} = \sum_k \frac{\delta_{u}^{k} \delta_{v}^{k}}{d_k - 1}
where `u` and `v` are nodes from the bottom bipartite node set,
and `k` is a node of the top node set.
The value `d_k` is the degree of node `k` in the bipartite
network and `\delta_{u}^{k}` is 1 if node `u` is
linked to node `k` in the original bipartite graph or 0 otherwise.
The nodes retain their attributes and are connected in the resulting
graph if have an edge to a common node in the original bipartite
graph.
Parameters
----------
B : NetworkX graph
The input graph should be bipartite.
nodes : list or iterable
Nodes to project onto (the "bottom" nodes).
Returns
-------
Graph : NetworkX graph
A graph that is the projection onto the given nodes.
Examples
--------
>>> from networkx.algorithms import bipartite
>>> B = nx.path_graph(5)
>>> B.add_edge(1, 5)
>>> G = bipartite.collaboration_weighted_projected_graph(B, [0, 2, 4, 5])
>>> list(G)
[0, 2, 4, 5]
>>> for edge in G.edges(data=True): print(edge)
...
(0, 2, {'weight': 0.5})
(0, 5, {'weight': 0.5})
(2, 4, {'weight': 1.0})
(2, 5, {'weight': 0.5})
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,
overlap_weighted_projected_graph,
generic_weighted_projected_graph,
projected_graph
References
----------
.. [1] Scientific collaboration networks: II.
Shortest paths, weighted networks, and centrality,
M. E. J. Newman, Phys. Rev. E 64, 016132 (2001).
"""
if B.is_directed():
pred = B.pred
G = nx.DiGraph()
else:
pred = B.adj
G = nx.Graph()
G.graph.update(B.graph)
G.add_nodes_from((n, {**B.nodes[n], **{'start': min([x[2] for x in B.edges(n, data='start')])}}) for n in nodes)
for u in nodes:
unbrs = set(B[u])
nbrs2 = set(n for nbr in unbrs for n in B[nbr] if n != u)
for v in nbrs2:
vnbrs = set(pred[v])
common_degree = (len(B[n]) for n in unbrs & vnbrs)
weight = sum(1.0 / (deg - 1) for deg in common_degree if deg > 1)
G.add_edge(u, v, weight=weight, start=start_conversation(B.edges(u, data='start'), B.edges(v, data='start')))
return G
def start_conversation(adj_u, adj_v):
u_starts = [x[2] for x in adj_u]
v_starts = [x[2] for x in adj_v]
return max(min(u_starts), min(v_starts))
mapped_projections = {
1: projected_graph,
2: weighted_projected_graph,
3: collaboration_weighted_projected_graph
}