One of the questions that historians may ask about our era is why technology became so ubiquitous, and so central to our lives. The important idea is not technology, of course, but the way we behave as a result of the tools that technology has provided. Mostly, the historian’s answers will focus on new forms of group dynamics, for these provide the underpinning for nearly all of our digital success stories.

(Many of the ideas in this article, and in several articles that follow, were sparked by the brilliant NYU professor Clay Shirky. You should buy his book right now. It is entitled Here Comes Everybody. Stop reading this blog, get your credit card, click here, then c’mon back to finish reading.)

(Welcome back.) While you were away, thirty six of us formed a big circle. And because you were away for a while, we were struggling to pass the time, and the woman next to me proposed a wager. She was willing to bet $50 that no two people in the circle shared a birthday. Nobody took the bet–it seemed like an easy way to lose money.

Shirky: “With 36 people and 365 possible birthdays, it seems like there would be about a one-in-ten chance of a match, leaving you a 90 percent chance of losing fifty dollars. In fact, you should take the bet, since you have better than an 80 percent chance of winning fifty dollars… Most people get the odds of a birthday match wrong… First, in situations involving many people, they think about themselves rather than the group…instead of counting people, you need to count the links between people.”

When counting connections, 1 plus 1 equals 1, but 1 times 4 equals 6. If I’ve done my math correctly, each of the 36 people in the circle has 35 connections, so the equation would be 36*35 or 1,260. If we were calculating unique connections–so we don’t double count both your connection to me and my connection to you, then we would divide by 2, and the number of unique connections would be 630, still a number far larger than 35, the number most people would choose in the bet.

Why does this matter? Consider LinkedIn, an Internet company whose entire operating theory is based in Internet connections. If you are reading this blog, you are likely to be one of my 500+ primary connections (that is, we are directly connected), but you are more likely to be two or three steps away–that is, you may be connected to one of the tens of thousands of people who are connected to my 500+ and even more likely to be connected to the hundreds of thousands (millions?) of people who are connected to the tens of thousands who are two steps away from me.

And why does that matter? It matters because I want to maintain my network, but it is nearly impossible to productively make use of such a large network–the connections are too diffuse, too unreliable, too far out of reach. Instead, my network, and your network, consists of a few dozen people, perhaps as many as a hundred or two hundred. And as long as at least a few of those people–the dozens or hundreds–remain connected to one another, my network remains viable. If, however, I lose contact with a few important connectors, the size and resilience of my network may dissolve.

Shirky again:

“A group’s complexity grows faster than its size…You can see this phenomenon even in small situations, such as when people clink glasses during toast. In a small group, everyone can clink with everyone else; in a larger one, people clink glasses only with those near them.”

And here’s why that matters. If you are trying to accomplish anything meaningful on the Internet that involves connections or interactions between people, you need to understand small world networks. And with that, Shirky closes us out:

“In 1998, Duncan Watts and Steve Strogatz published their research on a pattern they dubbed the “Small World Network.” Small World networks have two characteristics that, when balanced properly, let messages move through the network effectively. The first is that small groups are densely connected. In a small group, the best pattern of connection is that everyone connects with everyone else. The second characteristic of Small World networks is that large groups are sparsely connected. As the size of your your network grew, your small group pattern, where everyone connected to everyone, would become first impractical, then unbuildable. By the time you wanted to connect five thousand people, you would need a half million connections.”

” So what do you do? You adopt both strategies–dense and sparse communities– at different scales…As long as a couple of people in each small group know a couple of people in other groups, you get the advantages of tight connection at the small scale and loose connection at the large scale. The network will be sparse but efficient and robust.”

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Thanks to Emil for working out the basic mathematical formula that calculates connections (time prevented us from completing it for all cases). That formula is:

((n)*(n-1)/2 where n = the number of people in the group. Example: ((6)*5)/2 = 15

Since the connection between Person A and Person B is the same connection as Person B and Person A, division by 2 eliminates the double counting.

The formula starts working with 6 people. If anybody knows why the formula falls apart with groups of 5 or fewer group members, please comment below. And, how do we deal with fractions (half a connection divided by two)?