I want to represent a network of one- and multiary one- and multi-valued operations (objects with zero or more inputs and zero or more outputs) that are connected so that each output is connected with an input.
An object that has zero inputs is called a source, an object that has zero outputs is called a sink.
1.) What mathematical objects/structures are there to represent such networks?
I already found directed multigraphs (e.g. network graphs, directed graphs, directed trees) and term algebra.
2.) What are the most general mathematical objects/structures that can represent nearly all different kinds of representations (graphs, knot - edge lists of a graph, matrices of a graph, tables of a graph, figures of a graph) of such networks?
I'm looking for a term that subsumes the terms "a directed multigraph" / "a relation network" / "a term of a term algebra" etc. - all kinds of representations of one network of that kind I defined above.
Can category theory for example help here?