Königsberg, Kirchhoff, the Pretzel and global village coincidences

Image Credit: Michael Taylor

Have you ever wondered how suprisingly often it happens that you know a friend of a friend’s friend and found yourself saying “it’s a small world”? You may or not be glad to know that there is good reason for such coincidences. The explanation comes from the statistical properties of social networks and has its roots in graph theory which goes all the way back to 1736 when the Swiss mathematician Leonhard Euler wrote a landmark paper on the bridges of Königsberg. During the 18th Century, the city of Königsberg (in East Prussia) was divided by the Pregel (ring a bell? read on!) river into four sections including the island of Kneiphof in the middle. Seven bridges connected these four land masses. It was said that people spent their Sundays walking around, trying to find a starting point so that they could walk about the city, cross each bridge exactly once, and then return to their starting point having completed a scenic loop. It looks really easy. But try as you might, you’ll find there’s no solution! This is exactly what Euler proved and the tools he developed to do it gave birth to graph theory as we now know it. So, you might ask, “what has all this to do with coincidences”? Well, the second half of this story begins in the mid-60s when the controversial American social psychologist Stanley Milgram started his small-world experiment. Milgram wanted to learn what the probability was that two randomly-selected individuals would know each other. He developed an ingenious procedure to count the number of links between any two people using ordinary snail-mail (post). He found that the average path length between any two people was 5.5 connections. This is where the inspiration for the 1993 film starring Donald Sutherland Six Degrees of Separation comes from. The idea is simple. In a social network, there should be at most only 6 handshakes between you and anyone else on Earth. For example, last week the university where I work gave an honorary doctorate degree to her highness Sheikha Fariha Al-Ahmad Al-Jaber Al-Sabah from Kuwait. Since I have shaken the hand of the president of the university, I am only one degree of separation from her. The president is what graph theorists would call a hub – a social connector having many incoming links. Taxi drivers and reporters are great hubs for mapping a route from you via degrees of separation to someone who you might not immediately think you have any connection to at all. Hop, hop, hop, hop, hop, hop and there you go – a connection. There is a Facebook group you can join called the Six Degrees of Facebook Separation Experiment. Perhaps you will find an old friend there. Or you can import your social connections into Digrii.com and it will produce degree of separation maps pinpointing people geographically using Google Earth!

Image Credit: Michael Taylor

But, try to connect with someone without technology and it’s a different story altogether as they need to be connected to the social network. So, dont expect to find coincidences between you and the Maasai tribes of Tanzania any time soon. Coincidences, however, are very emotional things and we unconsciously attach a lot of weight to them. For example, I have taught electrical circuits, special relativity, higher-dimensional matrices and multivariate calculus and written a degree thesis on transcendental philosophy. Should I not then be surprised to learn that the following people all came from Königsberg: Gustav Kirchhoff, Hermann Minkowski, David Hilbert, Ludwig Otto Hesse and Immanuel Kant? Possibly. But then again, maybe I just subconsciously created the coincidence? When I look at the interconnectivity graph of the Königsberg bridges, I personally see a tasty sesame-flavoured Preztel.

Image Credit: Michael Taylor

Maybe they’re named after the Pregel river. Maybe they were invented in Königsberg. Maybe I’m just hungry. See what I mean? Writing this post, I have an image in my mind of Kirchhoff, looping round and round the bridges of the East Prussian city thinking of electon flow round wire circuits. Perhaps Kant was right after all. Perhaps by “patternizing” and then checking out the facts of whatever emerges empirically, we can learn to see that why it was that Kirchhoff came up with his circuit laws in Königsberg. He may also have invented them if he lived in the bushlands of Australia but I doubt it. Yes, the internet has created a small-world, but how global it is (it’s degree of separation is currently measured to be 11) depends on how much access you have to the whole network, how interconnected its pages are, and how retrievable they are. If you live in Cuba or China your “world-wide web” will be different to if you live in Britain or America. So, if you’re a hub or know one, expect more coincidences in your life but dont let them freak you out. They are just emergent properties of the global village you are part of and which we are now constructing. Who knows what’s around the next corner as you click on another link?

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