Ubiquity and Nexus
by Mark Buchanan
(Notes written in December, Year 2)
| March 30, Year 4: In these notes I expressed perplexity about how and why these "power-law" distributions and "small-world" networks come about. Recently I read Linked, a fine book that goes way beyond these two by detailing the processes. You can call it a how or a why; either way, check out the notes. |
Buchanan recounts some recent findings in science that are not too hard to understand as "how" but are devilishly tantalizing if you ask "why." In Ubiquity he brings together sandpile avalanches, revolutions, magnets, dry woodlands and the stock market, all of which have in common a behavior called "self-organized criticality" (SOC). Each system is unpredictable in the sense that the next event (grain dropped on the pile, institutional "buy" order, what have you) may cause a local rumbling, no disturbance at all, or a cataclysmic rearrangement, and there is no way to tell the outcome in advance. What's interesting is not unpredictability; dice are unpredictable without forming a major research topic in science. What's interesting is that these and many other systems are unpredictable in the same way, and it's a pretty simple way.
Take forest fires. If lightning strikes a tree tomorrow, will it kill the tree, ignite a two-acre fire, or burn down Yellowstone Park? You could try to list all the factors that lead to the answer: dryness of the tree, direction of the wind, violence of the lightning, nearness of celebrities' homes, and so on and on, maybe hundreds or thousands of items. But you still can't say what will happen if this stroke of lightning hits this tree on this future day. Look backward, though, and count how many fires covered one acre, ten acres, and so on up to a million acres. The bigger the fire, the fewer fires of that size; that's not surprising. If you make a graph of frequency (number of recorded fires) versus size (acres), the curve tilts downward. In fact, and this is the surprising thing, every time you double the size, the frequency drops by a factor of 2.48 (Ubiquity, p. 68).
Why surprising? Well, we like to think about events as being "typical" or not. How big's a typical forest fire? Because the curve in your graph always tilts downward, the world of real fires doesn't have a typical member! The closer you look, the more fires you see, but those little fires aren't typical because they are overshadowed by huge Yellowstone fires. The farther back you stand, the bigger the fires seem, but the bigger ones aren't typical either because the thousands of small fires contribute lots of acres.
In a similar way, there's no typical earthquake, and there's no typical "waiting time" between big earthquakes. There's no typical up-or-down movement of the Dow, and there's no typical number of "wave" movements between the large swings. In each system, your graph of size versus frequency slopes downward, and a given increase in size (ratio of sizes for two events) corresponds to a constant decrease in frequency (ratio of frequencies). The numbers are different but the description is the same.
The numbers are different but the description is the same . . . hey, isn't that a pretty fair description of what physics does? Yes, you look for "laws" that will operate in the same way no matter what numbers you plug into the formulas. The kind of law I've been talking about is called a power law, and it applies to a remarkable range of phenomena, all of which have the property that you can't predict them in detail. Seems anti-physics, but it still fits the frame.
The market is always ready for the placid, high-volume trading day that your broker loves, and it's always equally ready for a crash or a huge fortune-making boom. That's criticality. And getting into that critical state is somehow an inherent property of the market. "The system organizes itself so that the result of the next stress applied to the system is unpredictable" (Ubiquity, p. 119). That's self-organization, hence self-organized criticality.
Buchanan gives a fine account of how SOC plays out. The frustrating thing about this book, and for all I know it may be just as much so to the author, is the lack of any explanation of why systems act this way. While some include people, others don't, so SOC is not a specifically human quality. While some work with physical objects (magnetic domains), others don't (nothing could be less physical than the S&P 500), so it doesn't result from collisions between particles or anything like that. For once, it would be nice if physics could tell why.
Now suppose you want to lend your copy of Ubiquity to Aram Hovhanessian, an Armenian dairy-industry worker, but you don't know him or his address. How do you get the book into Aram's hands? Simple: You send it to someone you do know along with a request to forward it. When the parcel reaches Aram, how many stamps does it have on? There is a quite real chance that it has six stamps, and a rather poor chance that it has more than about ten. "By golly, it's a small world," the Hovhanessians will say, and that's the subject of Buchanan's second book, Nexus.
Is it true that "everyone in the world is connected by just six people"? It's sort of true, in a way, yes. Human society is one of the systems characterized by "small-world networks," in which most of the links are short in range (between you and your neighbors and friends) but a few make bridges between such little clusters (Jimmy and Rosalynn Carter know a mayor of Atlanta and also a West African nurse). If you can get to the Carters, the nurse is just another step away and all of her friends just two steps. You may not even need "six degrees" to get to a kid in Burkina Faso who will grow up sighted as a result of the Carters' work.
Like self-organized criticality, the small-world network appears in many kinds of systems. Consider, for example, "the food chain." You may have learned that the pond scum is at the bottom and the red-tailed hawk at the top, but it is not nearly that simple. A map of who eats who will show that while the hawk eats everybody and nobody eats the hawk, thousands of species are linked in what looks more like a web than a chain. The red-tail enters into relationships that may not be exclusive (she doesn't care if mouse or pigeon is on the table) but are quite numerous, and she may be a key linker in the whole network, not just one terminus of a chain. If the farmer shoots hawks to save his bantams, the impact can lead to all kinds of effects he doesn't foresee, such as a plague of mice uneaten by red-tails or, a few steps away in the web, mildew growing in the harmonium.
You know what I'm going to say next, the Bacon number. (An odd fact: Frances Teague has a Bacon number of 3. You could look it up.) And John Guare's play Six Degrees of Separation too. Highly trendy, this small-world business, and Buchanan gives it a pretty good exposition. Fittingly, he links the network concept with lots of very new science as well as politics, epidemiology, hydrography, the Internet, Hartsfield Airport in Atlanta, and snowflakes.
A key notion that ties Ubiquity and Nexus together is that of the scale-free organization, a system that looks "the same" close up as it does from farther away. This self-similarity appears in our size/frequency graphs and also in the details of network history. It's also a central idea in the mathematical description of the natural world: The coastline of Maine looks pretty rugged in a map of the U.S., but in a 1:400,000 topo map it looks almost equally rugged, while if you just get out and walk it you're likely to see the same picture of ruggedness. When you draw a maple tree, you use a lot of the same ratios you use in drawing a maple leaf. Computer artists have been playing with this notion for decades, tweaking a parameter in their fractal generators to change the Rockies into the Smokies. I'm going to go out on a limb and project that Buchanan's third book, for it wouldn't be much of a trilogy without a third, will address the concept of scale-free shapes in art and nature and, not at all by the way, in science.
Once again, physics doesn't tell why: What is it about networks, or more generally about systems containing links, that leads to a form of organization with this kind of economy? What causes us to hook up in such a way that I can connect to Aram Hovhanessian through a chain of people who could all sit down in my living room? There are other ways of setting up relationships (the Internet embodies one alternative). I find it wonderful that these things happen and illuminating to learn how they do, but still: Why does it do like that?
While these books are both worth reading, and neither ranks above the other in my view, I'd like to tell Mark Buchanan my heartfelt wish for the third book in the series:
May you find an editor and a proofreader this time around. When Crown Publishers brought out Ubiquity they hired good people, and the book is a pleasure to read. Then you went over to W.W. Norton & Company [you see we do not fear to Name Names] for Nexus, and the thing is dreadful. For one thing, when you wrote "principal" meaning "chief," your ignorant editor changed it to "principle." This happened not once but a few dozen times. Misspelling the foreign title Die Zeit (as Die Ziet) may be understandable, but "syphilis" is not a foreign word ("syphilus" got through), and "make do with" is not an obscure form of words (your book says "make due"). I hope you will not take these comments as gentle chiding: The bad editing and undone checking made me angry and nearly led me to throw Nexus away as a piece of trash, and I would expect the same reaction in any other educated reader. Clean it up, please.
John Guare based a play on the "six degrees" idea, and I worked on the crew for a very good community theater production. These pages have nothing to do with Mark Buchanan's books, but the pictures are pretty.
| Approved Ben Teague web site |
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Ubiquity and Nexus