I take pleasure in coincidences. One of my favorite coincidences happened just a week ago. At the anniversary weekend of my department, I met an alumna who is currently a professor at a small liberal arts college in Pennsylvania. It just so happens that one of my former students back from my Teach for America days in Brooklyn goes to this very college now. I asked the alumna if she knows my former student (then in 7th and 8th grade), and lo and behold, this student is currently in her neuroscience class! Wow – mind blown!
Then there’s the time that out of a crowd of 60,000 people at a concert in London one summer, I spotted a random guy that I had casually chatted with once at the breakfast table of a hostel in Argentina six months earlier. (Side note: incredible memory for faces apparently IS a thing). Or that time a decade ago that I randomly talked to a stranger in Fiji to learn that he (at the time) lived in my hometown AND we knew someone in common! Not to mention Facebook friend adds that result in the realization that we have random friends in common, not just locally, but globally.
I could go on and on. But I have met people that are not impressed by these stories. They have come to expect them. One such person is Steve Strogatz, a mathematician at Cornell that I had the pleasure to see speak on campus this past Friday. The topic was his co-authored work on small-world networks. By networks, we mean to say associations between individuals – be they humans, birds, or even neurons in the brain. Based on this modeling work, such coincidences as I’ve described above shouldn’t come as a surprise at all.
How do networks behave, and how do you get a small-world phenomenon to arise out a seemingly giant network (aka, the human population)? All it takes are a few key properties. One such property is the average shortest path between the nodes. If we were to play the Kevin Bacon game, this would be the smallest number of steps it takes to get from Kevin Bacon to the actor of your choice. In reality, this takes less than six degrees of separation, it’s more on the order of three or four.
Another property is the degree to which the nodes cluster. There tends to be more clustering within networks than is predicted by random chance, with nodes creating tightly-knit groups. With high connectivity between nodes and some nodes more connected than others (so called, hubs), it’s easy to see how small-world networks arise.
From neural networks in the brain to web pages on the internet, there’s growing evidence that real networks, though complex, are actually small-world networks.
So it IS a small world after all. Yet despite rationally knowing this, I know I will still be in awe of the next coincidence, no matter how probable it is.
For more from Strogatz, read his 15-part series on mathematics in the New York Times.