## Jul 15: What Are Dynamical Properties Anyway?

Category: Philosophy

Posted by: Aaron

One of my main research thrusts is developing methodology for complex systems, and specifically in defining new measures. New measures are necessary because existing statistical techniques were developed to report aggregate properties and trends and to recreate distributions with simple mathematical models. In complex systems we care about the relations among parts and causal process that generate behaviors at multiple levels rather than smoothed-over aggregate outcomes. Statistical techniques are refined for problems of missing, confounded, or otherwise imperfect data. Agent-based models generate data without any of these flaws, but in a tremendous quantity that requires new refinements in data mining and pattern detection. Furthermore, this new clarity and proliferation of data from simulation opens the way for new measures that would have been useless on less clean data, but are vital for understanding the complex systems we now study. One direction for new measures (and the one I’m currently most focused on) is the development of measures of dynamical properties, but people seem to have a hard time grasping what I mean by dynamical properties. I will attempt to elucidate the idea here.

The default way of capturing dynamics is to measure the differences in values of states of a system. The difference between a measure of dynamics and the changes in static measures is nuanced, and especially so because there are very few measures of dynamics to use a benchmarks. A simple (perhaps simplistic) example comes from physics. We can take the location of an object over time. Then we can measure the changes in these locations and calculate speeds between these observations...that’s a change in a static measure. Analogously we can compare the speeds calculated between different observations and calculate the differences in those...which of course is just acceleration. That technically is a property of the dynamics since speed was a dynamic. So that’s a simple example, and, as such, it’s not very illuminating on my enterprise because it’s

A more detailed and useful example of measuring dynamics comes from music. You can think of notes and chords as the static data. As time progresses you can find the tonal differences between each note for each instrument and that’s the difference in static measures. That may be helpful for some specific task, but probably not very illuminating in general. But we also have "measures" such as being: a 1-4-5 chord progression, a minor inversion of the dominant key, etc which a further categorized into things like: Irish music, punk, polka, etc. based on those dynamic properties. In some sense these dynamical properties are “emergent” since they are surely made up of the individual note changes and at the same time if all you ever measured were the individual note changes you’d never realize that such a thing as a blues chord progression existed. But it’s near impossible to find the macro dynamics from mining the microchanges, one has to be looking for the patterns in macro dynamics to find them...and that’s one version of what I’m proposing we do.

Some of the dynamical properties I've already written about are tipping points (and related phenomena), robustness (and related phenomena) and path sensitivities (and related phenomena). These are not just immediate changes in the values of variables; those features are only uncoverable through an analysis of the long-term dynamics of a system. And they describe not just changes in the changes (like acceleration) but cohesive patterns of behavior (more like chord progressions). And like chord progressions the existence of these patterns can (eventually) be used to classify systems into categories based on their behaviors regardless of domain (in the way that "chaotic" and "power law distributed" have been used to classify system behaviors).

But that's only one kind of dynamical property, and one approach to measuring them. Not all dynamics can be captured directly as changes in states. In some cases this is because we don't have access to the state variables and all we have is the need to quantify some perceivable qualitative macro behavior. Conceptually that might be the same things with just missing micro level information, but methodologically that requires a very different approach. The measures of dynamics that I've come up with so far

For the time being I have this nagging thought there are these conceptually very different dynamical properties out there, but we (including me) are just too accustomed to the old way of thinking to grasp them...for now. Hopefully by working through these first dynamical properties we will learn enough to see where they are lacking and maybe what we will find are these more purely dynamical properties. Or maybe not. For the moment we need to learn to crawl before we can fly.

The default way of capturing dynamics is to measure the differences in values of states of a system. The difference between a measure of dynamics and the changes in static measures is nuanced, and especially so because there are very few measures of dynamics to use a benchmarks. A simple (perhaps simplistic) example comes from physics. We can take the location of an object over time. Then we can measure the changes in these locations and calculate speeds between these observations...that’s a change in a static measure. Analogously we can compare the speeds calculated between different observations and calculate the differences in those...which of course is just acceleration. That technically is a property of the dynamics since speed was a dynamic. So that’s a simple example, and, as such, it’s not very illuminating on my enterprise because it’s

**too**analogous to comparing static measures (i.e. it's weird to think of acceleration as a property of speed since it's just the rate of a rate). But at least that's one conceptual difference between measures of dynamics and changes in static measures.A more detailed and useful example of measuring dynamics comes from music. You can think of notes and chords as the static data. As time progresses you can find the tonal differences between each note for each instrument and that’s the difference in static measures. That may be helpful for some specific task, but probably not very illuminating in general. But we also have "measures" such as being: a 1-4-5 chord progression, a minor inversion of the dominant key, etc which a further categorized into things like: Irish music, punk, polka, etc. based on those dynamic properties. In some sense these dynamical properties are “emergent” since they are surely made up of the individual note changes and at the same time if all you ever measured were the individual note changes you’d never realize that such a thing as a blues chord progression existed. But it’s near impossible to find the macro dynamics from mining the microchanges, one has to be looking for the patterns in macro dynamics to find them...and that’s one version of what I’m proposing we do.

Some of the dynamical properties I've already written about are tipping points (and related phenomena), robustness (and related phenomena) and path sensitivities (and related phenomena). These are not just immediate changes in the values of variables; those features are only uncoverable through an analysis of the long-term dynamics of a system. And they describe not just changes in the changes (like acceleration) but cohesive patterns of behavior (more like chord progressions). And like chord progressions the existence of these patterns can (eventually) be used to classify systems into categories based on their behaviors regardless of domain (in the way that "chaotic" and "power law distributed" have been used to classify system behaviors).

But that's only one kind of dynamical property, and one approach to measuring them. Not all dynamics can be captured directly as changes in states. In some cases this is because we don't have access to the state variables and all we have is the need to quantify some perceivable qualitative macro behavior. Conceptually that might be the same things with just missing micro level information, but methodologically that requires a very different approach. The measures of dynamics that I've come up with so far

*have*depended on them being of the weakly emergent sort, i.e. decomposable into micro changes. There is also the possibility that some dynamical properties are strongly emergent, i.e. uniquely definable at one level and not decomposable into micro phenomena, though I'm doubtful that any such phenomena are conceptually sound. But in the sense that these are patterns of behavior (or properties of patterns of behavior) then perhaps there is a sense in which reducing them to their constituent micro behaviors is simply meaningless or useless.For the time being I have this nagging thought there are these conceptually very different dynamical properties out there, but we (including me) are just too accustomed to the old way of thinking to grasp them...for now. Hopefully by working through these first dynamical properties we will learn enough to see where they are lacking and maybe what we will find are these more purely dynamical properties. Or maybe not. For the moment we need to learn to crawl before we can fly.