Almost 400 years ago, John Donne penned his famous line about no man being an island, and that sentiment has never been truer. Think about all the available ways to be connected today: phone systems, broadcast media, social networks (both physical and virtual), the internet in general. Human beings are tangled up in a web of connections.
But not all networks are created equal; some are more valuable than others. Depending on your point of view, there are lots of competing factors that determine how much a particular network is "worth," but one simple measure is how much connectivity it offers. In other words, if a network is valuable because of its ability to connect people, then shouldn't networks with the capacity for more connections be considered more valuable?
If that's true, we can start to ask questions about how we might calculate the values of differently-sized networks, or, equivalently, how the value of a network changes as it grows. Certainly a network with more people has more possible connections, but how many more? Let's start with a few examples.
As you might expect, a network with one person isn't particularly valuable since it offers precisely zero connections. A network with two people is a little better, because at least they can connect with each each other. A three-person network has three connections; a four-person network has six; a five-person network has 10. A table might help.
For small networks we might be able to directly count the number of connections, but it would be helpful to have a general rule for a network of any given size. There are a few lovely ways to do that, but here's a good one:
In a network with n people, a person can be connected to at most n - 1 others (everybody but herself). But that's true of everyone in the network, so there are n people with at most (n - 1) connections each, for a potential total of n(n - 1) connections overall. Except that's a bit too high. That calculation includes, for example, Alice's connection with Bob, but it also includes Bob's connection with Alice, so each connection is getting counted twice. That means the actual number of possible connections in a network with n people is n(n - 1)/2. Go ahead, check the table.
As an important aside, notice that, when n is very large, the difference between n and n - 1 is negligible, so a network's "value" is approximately proportional to n2. This observation that network value grows quadratically is known as Metcalfe's Law.
So far we've been talking about a network's value from a perspective of global potential, but what about its value to individual users? Take, for instance, Facebook, which is by any reasonable standard a very large network. Even though it has a billion users, and therefore something like 5 × 1o17 possible connections, the average person only utilizes fewer than 200 of them. That's a staggeringly tiny percentage.
So maybe Facebook's value is asymmetrical: the company gets a lot of value by having lots of connections --- each one, after all, a potential conduit for advertising revenue via "likes" --- but human beings can only manage a relatively small number of relationships. So huge networks aren't necessarily as valuable to individual members, there being little practical difference to any one person between a network of 30 million people and a network of a billion people. Maybe the number of connections isn't actually a meaningful metric at the user level.
Enter Path, a social network founded in 2010 that limits users to 150 connections and makes its money not by advertising, but by selling products directly to users. A larger network still means more value for Path, but maybe only proportional to n instead of n2 now, and the value to users might be proportional to something even less than n.
All of this is to say that a network's value might not be so easy to calculate, and maybe the potential to connect people just isn't enough. We somehow need more than that.
Teachers: want to have this conversation with your students? Check out the lesson materials on our website.
 Maybe more like Metcalfe's Conjecture, since lots of smart people think it's wrong.