I have just a little stub today. There are some other things you can do with Graph Algebra, like measure things over time. People who are familiar with modeling in the social sciences will readily recognize the distinction between discrete and continuous time, and these terms are no different for Graph Algebra.
Discrete time, E^-1
In discrete time, events are evenly spaced. This is not to say that events occur on an inflexible schedule, but that you may be measuring things that happen periodically, and measuring them at the time of occurrence. The ideal example is elections. In America, they occur every two years. However, in the United Kingdom, elections are called periodically but not on a fixed schedule. You can still use the discrete time operator for British elections because you’re just measuring results or popularity or whatever at the time of the election and not measuring anything about the time in between elections. Just to illustrate, elections in which a President is not being elected tend to be unpopular. There’s an interesting dynamic model to show that.
From the points on this graph we can see voter turnout at various election years. The oscillating line comes from the model we generate, based on code from the earlier posts on dynamic modeling. Now, the voter turnout between elections is zero, but we don’t need to show that because it’s obvious. If there’s nothing to vote on, no one will show up to vote. That’s why this is discrete time. We simply jump from one period to the next.
In graph algebra, we have specially designated operators called time operator to deal with the passage of time. The discrete time operator looks like a capital E raised to the negative first. In short, it means that we step back one unit of time in order for the model to work.
What if we wanted to measure something that doesn’t jump from one point to the next? For instance, what if we wanted to describe Presidential Approval ratings? People judge the president every time they hear anything about him, so approval ratings can fluctuate literally from moment to moment. Thus we measure Presidential approval in continuous time, and infer that in the times in between our measurement, Presidential approval fell somewhere in between the two values, or nearby. This is continuous time.
Continuous time Δ^-1
Continuous time is a little more complicated. There’s calculus involved. I won’t go into it, but for the calc-inclined, think of continuous time as the limit as the distance between points closes to zero. We represent differences (as most sciences do) with the Greek letter delta, Δ, like so:
ΔY(t) = Y(t) – Y(t-1)
However, in order to really sink our teeth into continuous time models, we need a very different operator. If Δ is a difference operator, then we need Δ^-1 (reads “delta inverse”). This is sometimes called a summation operator, because it undoes the difference operator, exactly the way addition can undo subtraction.
At this point, I would normally introduce some model to demonstrate the use of these tools, but there’s something I haven’t really explained yet, and in order to make this stuff meaningful, we’ll really need to know about it. It’s called feedback. Stay tuned.