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overlap_weighted_projected_graph(B, nodes, jaccard=True)[source]

Overlap weighted projection of B onto one of its node sets.

The overlap weighted projection is the projection of the bipartite network B onto the specified nodes with weights representing the Jaccard index between the neighborhoods of the two nodes in the original bipartite network [R162]:

\[w_{v,u} = \frac{|N(u) \cap N(v)|}{|N(u) \cup N(v)|}\]

or if the parameter ‘jaccard’ is False, the fraction of common neighbors by minimum of both nodes degree in the original bipartite graph [R162]:

\[w_{v,u} = \frac{|N(u) \cap N(v)|}{min(|N(u)|,|N(v)|)}\]

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).

jaccard: Bool (default=True)

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.


[R162](1, 2, 3) 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.


>>> from networkx.algorithms import bipartite
>>> B = nx.path_graph(5)
>>> G = bipartite.overlap_weighted_projected_graph(B, [0, 2, 4])
>>> print(G.nodes())
[0, 2, 4]
>>> print(G.edges(data=True))
[(0, 2, {'weight': 0.5}), (2, 4, {'weight': 0.5})]
>>> G = bipartite.overlap_weighted_projected_graph(B, [0, 2, 4], jaccard=False)
>>> print(G.edges(data=True))
[(0, 2, {'weight': 1.0}), (2, 4, {'weight': 1.0})]