Note

This documents the development version of NetworkX. Documentation for the current release can be found here.

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
BNetworkX graph

The input graph should be bipartite.

nodeslist or iterable

Nodes to project onto (the “bottom” nodes).

jaccard: Bool (default=True)
Returns
GraphNetworkX graph

A graph that is the projection onto the given nodes.

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.

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})]