Let's say you have a group of people, or a network of computers, or the brain of an earthworm. Let's say a group of people: the city of Toronto. Some of the people in Toronto know each other, but most pairs of people do not. Within the city, there are groups of friends, and most of the people within these smaller groups know each other. In other words, people in Toronto cluster into smaller groups of inter-connected people. As an urban planner/sociologist/bored computer scientist, you want to know what these clusters are like. How many are there? How big are they? If you have all the people in Toronto, and know who knows who, you can search this in this information to find these clusters of dense connections.

But you're worried this approach wont work. First, Toronto is pretty big--it's hard to get that information from everyone, and hard to study it once you have it. (You could study a small part of Toronto, but you're worried that doing so will throw away long-range connections between people in different parts of the city, so it wouldn't be representative of the full city.) Second, you're not just interested in Toronto. You're interested in Boston, and New York, and in Europe. You're interested in networks of computers. So you decide not to work in Toronto, you decide to work in Proto-Toronto.

What's Proto-Toronto? Proto-Toronto is a lot like Toronto, except it's made of math. It has the same number of people, and some of the people know each other (but most don't). However, in Proto-Toronto, the only thing that determines who knows who is how close they live. People who live within a block of each other have a very high chance of knowing each other; people who live far apart probably do not. Now you ask: who are the groups of friends, the clusters, in Proto-Toronto? This you can answer! They can be big, but not too big--after all, not everyone can be friends. The bigger Proto-Toronto grows, the larger the clusters of people it can support. 

At first you're happy, but it's late at night one night and you're having doubts. The people in Proto-Toronto are not like the people in Toronto. Why should we care about Proto-Toronto? It's not Toronto, after all.  I'll tell you why. Because you can make the people in Proto-Toronto act like Torontonians much more easily than you can study Toronto itself. To figure out how a Proto-Torontonian acts, just take fifty or so real Torontonians, and figure out who they know. Take the average, and you have a proto-Torontonian.  Similarly, you can make Proto-Boston or Proto-New-York (or a proto-computer network). Even then, your Proto-villes aren't great--they aren't real cities. But they act kind of like real cities. Understanding what groups of friends look like in Proto-Toronto gives some you some intuition about what groups of friends in Toronto look like. 


This is a "human-readable" version of what I've been researching. Want more? Go read the real thing