Sometime you want to model some data from the real world using a graph, but such that edges don't just connect to a vertex; rather, they connect to some aspect of that vertex - some connection if you will.
For example, a node in a family tree would be a person, and they have a mother and father (never mind about adoptions etc.). Now, when you connect one of the parents to the child you want to connect it as "the father" or "the mother"; so a "connections graph" model for this tree would see each vertex have a set of 3 possible connections: "father, mother, child" (with the child being used to connect it to its biological children's nodes).
It's true that you can always get around really needing connections, e.g. with gadgets in their stead (say, each connection has a gadget vertex, so an original vertex is now surrounded by its connections as satellites and edges only exist between these satellites). But I'm interested in dealing with such connections explicitly.
So, connections can be represented by a set; or as an ordered sequence (perhaps even always $0...\Delta$ with $\Delta$ being the maximum degree of the graph); or the edge set could incorporate the connections apriori, i.e. an endpoint could always be a tuple of a vertex and something else. And the connection set might be shared, or per-vertex; and so on.
My question is: What are some specific formalizations of this concept which are used often (if there are such at all)?