Humanity forms a non-Euclidean network
Last week we talked about some of the features of complex systems: emergence and self-organization. The discussion got me thinking about the complex system created by human beings. If we accept that the human network is non-Euclidean, what does that mean for the mathematics of our interactions?
Euclidean Network Geometry
When I think of networks, I tend to imagine two-dimensional diagrams of nodes and connections, the stereotypical centralized, decentralized, and distributed versions:
When you look at the above networks, you can see immediately that they are finite: there’s a border to each one. And, yes, they are simplified and simplistic, but take another look at the image I posted last week, ostensibly a representation of the Internet:
This image looks spherical, but I’ve noticed something important: you’d actually have to cover its outer nodes with a membrane to make it form a sphere. It’s the difference between a soccer ball and a pom-pom.
Non-Euclidean Network Geometry
So here’s something else to think about: what about a network shaped like a soccer ball? One that doesn’t need a membrane to cover it and that doesn’t have any terminal nodes?
It may sound like an innocuous question, but consider this: Euclidean geometry is what we all learned in school; it includes some of the oldest known mathematics. But it took non-Euclidean geometry?not accepted until the 19th century?to allow for navigation of the globe and Einstein’s Special Theory of Relativity.
The basic difference here is straightforward: Euclidean geometry is based on a flat plane, which allows for things like parallel lines and right triangles. Non-Euclidean geometry, specifically the elliptical geometry that refers to spheres, does not allow for parallel lines and allows for triangles with two 90-degree angles.
So think for a moment about the network constructs above. Imagine that each node were a person, and each line a connection between people. Now imagine a network consisting of all the people on our planet. A simplistic model of it might look like this:
So here’s the ultimate question: what are the mathematical implications of a non-Euclidean human network?
I invite your ideas on the matter…
(Updated July 4 to correct a grammatical error I had made.)
July 3rd, 2008 at 6:07 pm
Some implications are found in: http://www.ditext.com/hempel/geo-frame.html Real question might be: “Can we get there from here?” The reductionistic mind might not grasp the math.
July 4th, 2008 at 4:03 am
Thanks, Sam!
My mind might certainly not grasp the math, but let’s not allow that to stop us from trying
July 5th, 2008 at 6:11 am
I’ve thought - with no proof nor argument to back it up - that “if it’s not simple, it won’t happen here”.
As if to say we are too near, too compressed, too tightly near the center of a star to grow grand filigree and roundabout twining. We are too whelped into a tight frame, not a Big Bang, but gravitation is real enough to restrain us. Our truths here may not ever be infinite loops and complicated things.
This is a silly postulate, I know that, but it keeps me away from excessive theory and triumphant revelation hiding in labyrinthine claims. Salt matters.
If I cannot see it or say it, then is it?
Of course I always defer to bright folks, inspired folks, studious folks… we haven’t begun to honor our brilliant people.
Yet from Marx to Maslow there’s so much gumming and still little we march for. Even if wrapped in party talk or accrued intellect, the great duty is to bring it to the mundane.
I hope wisdom isn’t a bumper sticker.