I have part of a physical simulation which I've realised can be modelled using a directed graph where each node is a polynomial. I then calculate this graph by functional composition and summing to calculate a "function flow" as shown in the example below.
e.g. three polynomials $f(x)$, $g(x)$, $h(x)$ are connected as follows: $f$ -> $g$ -> $h$
We start at $f(x)$ because it only has an outgoing connection. Because $f(x)$ has an outgoing edge to $g(x)$ we combine them to make $g(f(x))$. Because $g(x)$ has an outgoing edge to $h(x)$ we combine them to make $h(g(f(x)))$. Because $h(x)$ has no outgoing edges we stop.
If there were multiple incoming edges to a node, the incoming polynomials would be summed before the composition. Also loops are allowed as the compositions will converge in my case.
The question is, is this something that has already been studied somewhere? I imagine there are a lot of physical processes like this. It also looks very Category-like.