Strogatz @ MIT

Two weeks ago, Steven Strogatz gave a series of three lectures as part of the Simons Lecture Series. (The pure math half of the series was given last week by Manjul Bhargava.)

The first lecture focused on coupled oscillators as a self-organizing system. My favorite example of this phenomenon relates to the behavior of Hungarian audiences: at the end of (for example) an opera in Budapest, the audience members initially begin applauding as in the US, each beginning with his or her own frequency and phase. However, after a moment or two, the entire audience synchronizes on a common frequency and phase. This remarkable phenomenon happens far more quickly than would be possible if everyone had to get together and agree on a common frequency; instead, it’s the result of everyone subtly (and subconciously) adjusting their own rate and phase in response to the collective responses of the rest of the audience. It turns out that with some reasonable assumptions and simplifications (e.g., involving mean field theory — I link that article partly in the hope that someone will improve it) one can write down and solve a system of differential equations that model this situation, and this allows one to say a lot about the overall behavior. In particular, given a distribution of natural individual frequencies, there is some value for sensitivity to the behavior to others below which synchronization does not occur, followed by an abrupt phase-change at the critical value.Here’s one natural question that I thought was raised by the talk: suppose we live in a regime below but near the critical value, but that a group of the oscillators collaborate to try to generate synchrony. (One might imagine, for example, a large group of Hungarians at the Boston Symphony, or a group of determined graduate students at a math colloquium.) How large or forceful does this group need to be to succeed? Experimental evidence suggests that six grad students in an audience of 100 or so is insufficient. (Actually, I think that the mean field assumption isn’t ideal for modeling applause in an auditorium because perceived volume decreases very quickly with distance, but whatever.)

The second talk was about another kind of self-organizing system. In this case, the model is a social network of some fixed number of people, each of whom is either friends or enemies with every other. In other words, we’re doing graph theory. We call a graph balanced if among every three nodes, the number of edges (= pairs who like each other) is odd. Motivation: if one person has two friends who hate each other, this is awkward, and social forces pull towards either the dissolution of one of the friendships or the resolution of the dispute between the two who hate each other. If three people mutually hate each other, there is incentive for two of them to team up against the third. The other situations are stable. It’s a theorem of Harary that the balanced graphs are exactly the complements of complete bipartite graphs. (The proof is cute, and I leave it to you to reconstruct.) So, the statics question “what sort of balanced configurations are possible” has been answered, but that still leaves the dynamics question “if I’m in an unbalanced configuration, how do I get to a balanced one?” A natural guess of a good strategy is to take whatever configuration you’re in and choose an edge; if flipping the edge (from friendship to hatred or vice-versa) increases the number of balanced triangles (those with an odd number of friend-pairs), then do it, and otherwise don’t. Unfortunately, this strategy doesn’t work in general: a simple counter-example is provided by three tribes (Strogatz called them “Lakers fans”, “Celtics fans” and “Knicks fans”) such that everyone within the trible likes everyone else within the tribe but everyone hates everyone from other tribes. As long as the tribes are large enough and roughly similar sizes (e.g., three members each does the trick) this graph is not balanced but adding or removing any edge makes it worse. More exotic examples are provided by Paley graphs, a family of graphs with some beautiful properties coming from elementary number theoretic considerations. Near the end of his talk, Strogatz described a different approach, using a dynamical system: the vertex i has an opinion (now a real number) X_{i, j} about every other vertex j, which it changes according to

\displaystyle \frac{d X_{i, j}}{dt} = \sum_k X_{i, k} X_{k, j}.

In essence, each person consults everyone else in the network when deciding how to update his or her opinions. If you run this system, people’s opinions of each other change for a while (the X_{i, j} may change sign) before stabilizing in a balanced configuration; after it stabilizes, the whole system blows up in a finite amount of time. Nifty!

The third talk was non-mathematical, and was instead about the experience of blogging for the New York Times. Rather than turn an already long post in to a ridiculously long one, I’ll just provide a link to the series. I personally read and enjoyed the first three (I particularly recommend the embedded Sesame Street video in the first piece) when he was writing them, then never got around to the other 12. (I was also surprised to see what a large percentage of the audience hadn’t read any of them.) Still on my to-do list!

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