Graph Algebra is, at its core, a visual method of representing social systems. So why muck about with it? We could just as easily draw a conventional flowchart. Well, flowcharts don’t give us a precise mathematical equivalence. In other words, we can translate Graph Algebraic diagrams into equations that model social systems. And that’s ballin’.
Graph Algebra works like this: Start with inputs. Inputs are kind of like independent variables. Kind of. For instance, if I wanted to model the creation of mud, I would have two inputs: water and dirt. If you add these things together, you get mud. A very basic Graph Algebraic representation would look like this:
Note we have the two inputs on the left, and the output (Mud) on the right. This is merely a convention which allows us to read a graph algebraic system in much the same way we would read a line of text, from left to right. The middle figure should be relatively self-explanatory. It’s addition! OK, not terribly impressive. As an equation, this diagram is equivalent to Rain + Dirt = Mud. Qualitatively, this makes sense. However, we can do better.
Before we do, however, a note on operators: the number of things which you can add with an addition operator is not limited to two. With this simple fact, I have told you everything you need to know in order to represent Multivariate Regression in Graph Algebra. That’s right: the tool the great Steven Levitt called the “Economists’ Favorite Trick” can be represented with the paltry amount of Graph Algebra that you, my friend, can now scrawl on the back of anything. It looks a little like this:
Cool. Now back to our mud. What if I wanted to measure the amount of mud to expect? Well, we’re going to need a recipe. Let’s say that for every cubic meter of dirt, we need 50 litres of water to generate a cubic meter of good mud. Well, the ratio is a constant. So, If I multiply the amount of water I have (in litres) by 1/50, then I can calculate exactly how much mud I’ll get, based on the amount of rainwater and mud. We denote multiplication as another value on the same path as the original value, like this:
In this case, 1/50 is what we call an operator of proportional transformation. These take units that are incompatible (litres/water, cubic metres/dirt) and transform them to compatible measures (cubic metres of mud’s worth of water, cubic meter of dirt). But frequently we don’t actually know what the operators of proportional transformation are when we are drawing our graph algebraic diagrams. So, instead of putting in the number, we would normally represent this value with a lowercase, italicized letter. I’ll call it a. Thus the equation this diagram generates is Dirt + a(Water) = Mud
To clean up the conventions a little, Dirt is actually the amount of dirt, Water is the amount of water, and Mud is the amount of mud. Better to represent these as the single letter variables D, W, and M respectively. So, our final diagram and equation,
D + aW = M
I realize this is all hardly Earth-shattering, but what if I said you could scale these techniques for far more complex processes? Next time: how feedback loops give Graph Algebraic models relevance, and why that dynamic modeling primer was important.