Generalizations of The Collaboration Graph
Some generalizations of the collaboration graph of mathematicians have also been considered. There is a hypergraph version, where individual mathematicians are vertices and where a group of mathematicians (not necessarily just two) constitutes a hyperedge if there is a paper of which they were all co-authors. Another variation is a simple graph where two mathematicians are joined by an edge if and only if there is a paper with only two of them (and no others) as co-authors.
A multigraph version of a collaboration graph has also been considered where two mathematicians are joined by edges if they co-authored exactly papers together. Another variation is a weighted collaboration graph where with rational weights where two mathematicians are joined by an edge with weight whenever they co-authored exactly papers together. This model naturally leads to the notion of a "rational Erdős number".
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