Modeling Nonlinear Phenomena

Modeling Nonlinear Phenomena - What is "X"?

Many of us grew up hating word problems in algebra. (Some of us found them interesting, sometimes easy, and sometimes fun. We were the minority.)

For most of us, even if we understood the mathematical formulas, there was a big "gap" in our understanding and intuition when it came to applying the formulas to some real-world situation. In the problem, we'd be given a set of statements, and then told to find "something." We were supposed to turn these "starting statements" into mathematical statements of what we knew. That is, we had to say, "Let X = (something)." "X" could be the speed of a car, the distance between two cities; it could be anything among the set of known facts. Then we had to make similar "mathematical statements" about other information that was given to us. And then, we were to say, "Now I want to find Y, my 'unknown.'"

All very well and good, when we're doing algebra, and the answers are in the back of the book. (Or at least the teacher will review and correct our work.) And just slightly more difficult when we are in the "real world."

Years ago (many more years than I care to acknowledge), I made a crucial "life-decision." I knew that I was interested in the capabilities of our minds and our brains. I knew that this area was getting more and more complex, with each passing year. And, given my gift for mathematics and abstract thinking, I decided that I wanted to learn how to make mathematical models of complex situations.

I didn't know exactly how I would use such an ability, I just knew that I needed to learn the fundamentals.

So I spent the next several years happily studying quantum mechanics and statistical thermodynamics, both of which were mathematically elegant and satisfying at the soul-level. Kind of like learning mathematical analogues to poetry.

I've had many opportunities to rejoice that I took on such a disciplined formal approach when I was younger, because now that knowledge serves me well. Even more, the discipline of the approach - more than the knowledge itself - is what serves me.

Particularly when I start looking into new fields, where new methods and models are just beginning to be employed.

Such is the case in reading Beinhocker's The Origin of Wealth, which I referenced in last week's posting, and actually began some two years ago. (See initial posting on this blog; May, 2009.)

One of the more interesting chapters in The Origin of Wealth (TOW)is Chapter 7, "Networks." One of the key points of discussion here is the notion of phase transitions within a networked system. We begin by tracking the average number of connections that any given node has within the system (either a random graph or a lattice graph), and observe the change on structures "within" the system. Using an analogy first proposed by Stuart Kauffman, Beinhocker suggests that we think of nodes as "buttons," and the interconnects as "stringing the buttons together." Any button can be "strung together" with any number of other buttons; two, three, or more.

We see that when the average number of connections is relatively low, there are small clusters scattered "like little islands" (p.143). Then, as the average number of interconnects increases, "isolated clusters of connected buttons will suddenly begin to link up into giant superclusters - two fives will join to make a ten, a ten and a four will join to make a fourteen, and so on. Physicists call such a sudden change in the character of a system a phase transition." (p. 143)

After an interesting little bit on the value of random connections within a network (particularly a social network, this correlates well with what we learn about working with a system such as LinkedIn), Beinhocker moves on to Boolean networks, in which each "button" becomes a "bulb" that is either "on" or "off," or is either "black" or "white". Moving briskly through Kauffman's notion of complexity catastrophe, he arrives at one of the most salient points of the book - and, in fact, the entire crux of applying network theory and phase transitions themselves to an economic or other large-scale real-world system.

According to Beinhocker, "Kaufmann found that when each bulb in the Boolean network has between two and four connections, the system went into a highly adaptable, in-between state. In this state, the system was generally orderly, with large islands of structure, but vibrant percolating disorder around the edges of the structures. Small mutations in the switching rules of the system generally led to small changes in outcomes, but occasionally, a small change would set off larger cascades of change, which sometimes degraded the performance of the system, but sometimes led to improvements. Although this particular network was highly adaptive, Kauffman was troubled by the observation that two or four connections per node was still pretty sparsely connected, by the standards of most networks in nature or in human organizations."

A good point, and worth comment -- although in the next paragraph, Beinhocker loops back to a point he made earlier (not part of this blogpost) about hierarchical systems - which is where we begin to get some emergent structure of a defined nature. But that's for later.

For now, it's worth jumping over to a very interesting paper by Chris Langton, "Computation at the Edge of Chaos: Phase Transitions and Emergent Computation." (I'll loop back to the starting notion of this blogpost, "What is 'X'?", later. For now, just laying out some tools and useful understandings of general systems.)

These two works - Beinhocker's book and Langton's paper - each present useful models. Similarly, my previous post presenting the very classic and well-known Ising spin glass model for phase transitions in a bistate system also gives a useful model.

The real question is: When are any of these models appropriate? And, perhaps most importantly - before we go about applying any model - we need to ask and answer, "What is x?" What is it that we are modeling, and does our model make sense? Does it make gut-level, intuitive-and-logical sense?

We can use mathematics - all sorts of pretty and interesting models - to make angels dance on the head of a pin. But before we go about counting "angels," we need to ask ourselves whether or not those "angels" are really the subject of our interest, and are we really interested in modeling their dance?