# 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})]