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collaboration_weighted_projected_graph(B, nodes)[source]

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 [R161]:

\[w_{v,u} = \sum_k \frac{\delta_{v}^{w} \delta_{w}^{k}}{k_w - 1}\]

where \(v\) and \(u\) are nodes from the same bipartite node set, and \(w\) is a node of the opposite node set. The value \(k_w\) is the degree of node \(w\) in the bipartite network and \(\delta_{v}^{w}\) is 1 if node \(v\) is linked to node \(w\) 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.


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.


[R161](1, 2) Scientific collaboration networks: II. Shortest paths, weighted networks, and centrality, M. E. J. Newman, Phys. Rev. E 64, 016132 (2001).


>>> 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])
>>> print(G.nodes())
[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})