What is the name of mathematical object similar to a graph, but with different kind of edges?

Question

While modeling biochemical networks in the cell using Lisp programming language, I recognized that I use a certain model for this task. This model resembles a directed graph, with edges pointed not only to vertices but sometimes to other edges.

I wonder is there any distinct name for this kind of model in mathematics?

Here is a simple example of the model pictured in a graph-like way.

Answer 1

Mathematically spoken a (directed) graph is a set of vertices, say, $V=\{a,b,c\}$ and a set of (ordered) pairs of this vertices, say, $\{(a,b),(a,c),(c,b)\}$, The pairs are called edges (or in the directed graph, arcs), and are sometimes abbreviated by (as in your case) $e1$, etc.

But it is not prohibitted, that the set of vertices contains semantically different things, like "simple" vertices, and ordered pairs of simple vertices, and pairs of ordered pairs and so on. Everything can be a vertex.

In your case the vertex set would be $V=\{v1, v2, v3, v4, (v1,v2), (v2, v3), (v4, (v1,v2))\}$, where I indicate an ordered pair by $(.. , ..)$. You see I avoided totally the notion $e3$ etc., since these would be only abbreviations for ordered pairs. The arc set for your example is then $E=\{(v1,v2), (v4, (v1,v2)), (v2,v3), ((v2,v3),(v4,(v1,v2)))\}$. You invented already a graphical representation for this kind of graphs.

Answer 2

The paper linked from comments describes 3 types of components:

1. Entities (represented by v nodes in your chart)
2. Statements (e1 & e2)
3. Influences (e3 & e4)

One way to represent this is to define a graph tower: level one graph $G_1$ is a graph with vertices $V_1$ and edges $E_1$. Level $n$ graph $G_n$ is a graph with vertices $V_n=V_{n-1}\cup E_{n-1}$ and edges $E_n$.

In your case the tower has just two levels:

1. E/S level 1 graph with nodes=entities and edges=statements
2. ES/I level 2 graph with nodes=elements of the E/S graph and edges=influences

PS. The reason to define the the graph tower in terms of traditional graphs is that those have been studied for long time, there are many theorems and algorithms about them, and they can be built upon in your study/manipulation of graph tower.