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networkx.algorithms.bipartite.projection.overlap_weighted_projected_graph¶

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 [1]:

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

or if the parameter ‘jaccard’ is False, the fraction of common neighbors by minimum of both nodes degree in the original bipartite graph [1]:

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

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)) Graph – A graph that is the projection onto the given nodes. NetworkX graph

Examples

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


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 bipartite documentation for further details on how bipartite graphs are handled in NetworkX.

References

 [1] (1, 2) 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.