I've been trying to find canonical notions of distance in hypergraphs which generalize the notion of distance in graphs. I was hoping for a distance which also encodes a metric on two subsets of the vertex set. Are there some classical/cannonical such distances on hypergraphs?
I would be thankful for any useful idea on the matter.
The most direct analogue of graph distance (at least that I can think of) is as follows.
If $u$ and $v$ are vertices of a hypergraph, and $e_1, e_2, \dots, e_k$ are hyper-edges, then we say that $P: e_1, e_2, \dots, e_k$ is a $u-v$ path of length $k$ if $u\in e_1$, $v\in e_k$, and for each consecutive pair $e_i$ and $e_{i+1}$ have at least one vertex in their intersection. In other words, a $u-v$ path is a sequence of edges, starting with one containing $u$, ending with an edge containing $v$, such that each consecutive pair of edges 'overlap'. (See the diagram for example) Then we set the distance $d(u,v)$ to be the minimum length of a $u-v$ path.
Note that any metric between vertices you find can also be adapted to a pseudo-metric between subsets (you may get two different subsets with distance 0 between them, but nothing else goes wrong).
If $A$ and $B$ are subsets of vertices, then you get the 'induced' distance $d(A,B) = \min\{d(u,v) : u\in A, v\in B \}$.