Typed Graphs

Why

We want to visualize function composition.

Definition

Let $G$ be a graph (directed or undirected) on $\set{1, \dots , n}$ and let $A = (A_1, \dots , A_n)$ be a list of sets. We call the ordered pair $(G, A)$ a typed graph. We call $A_i$ the $i$th domain. For $S \subset \set{1, \dots , n}$, we denote the product $\prod_{s \in S} A_s$ by $A_S$.

If $G$ is directed, we call a source vertex exogenous and otherwise we call a vertex endogenous.

Let $\bar{G} = (G, A)$ be a typed graph where $G$ is directed. Let $f_i: A_{\pa_{i}} \to A_i$ for $i = 1, \dots , n$ so that $f$ is a sequence of functions. We call the ordered pair $(\bar{G}, f)$ a function graph¹ (or function diagram).

1. This sheet is not to be confused with the graph of a function (see Functions). ↩︎