2: INCREASING RETURNS
Self-Reinforcing Success
Networks have their own logic. When you connect all to all, curious things happen.
Mathematics says the sum value of a network increases as the square of the number of members. In other words, as the number of nodes in a network increases arithmetically, the value of the network increases exponentially.* Adding a few more members can dramatically increase the value for all members.
[*I use the vernacular meaning of "exponential" to mean "explosive compounded growth." Technically, n2 growth should be called polynomial, or even more precisely, a quadractic; a fixed exponent (2 in this case) is applied to a growing number n. True exponential growth in mathematics entails a fixed number (say 2) that has a growing exponent, n, as in 2n. The curves of some polynomials and exponentials look similar, except the exponential is even steeper; in common discourse the two are lumped together.]
This amazing boom is not hard to visualize. Take 4 acquaintances; there are 12 distinct one-to-one friendships among them. If we add a fifth friend to the group, the friendship network increases to 20 different relations; 6 friends makes 30 connections; 7 makes 42. As the number of members goes beyond 10, the total number of relationships among the friends escalates rapidly. When the number of people (n) involved is large, the total number of connections can be approximated as simply n X n, or n2. Thus a thousand members can have a million friendships.
The magic of n2 is that when you annex one more new member, you add many more connections; you get more value than you add. That's not true in the industrial world. Say you owned a milk factory, and you had 10 customers who bought milk once a day. If you increased your customer base by 10% by adding one new customer, you could expect an increase in milk sales of 10%. That's linear. But say, instead, you owned a telephone network with 10 customers who talked to each other once a day. Your customers would make about n2 (102), or 100 calls a day. If you added one more new customer, you increased your customer base by 10%, but you increased your calling revenue by a whopping 20% (since 112 is 20% larger than 102). In a network economy, small efforts can lead to large results.
"Mathematics says the sum value of a network increases as the square of the number of members." - this is not entirely true - mathematics only can say that the number of connections in that network grows quadratically. How the value of the network is related to the number of connections is a question outside of pure mathematics. The thesis here relies on additional assumption that all the connections have the same fixed value. This is not true for the telephone network as for everyone the value of connecting to his family is much greater than to some random stranger.
If only we had more tools to encourage networks to build long term value. I don't mean simply monetizing information flow, I'm talking about large scale collaborative web projects.
Specific web companies have positioned themselves to coordinate and profit from this type of growth (Google, YouTube, squidoo, Facebook, twitter).
I imagine a future with (open?) social media projects where incredible complex problems are broken down and solved in real time by gigantic networks of computers and people.
I was just thinking in line with Mark's comment above. The growth of the network is potentially exponential but in real practice many of the nodes fail to trigger properly.
It would be wonderful to have a network though where each node guaranteed equal involvement and participation. Making sure that the growth was truly exponential. If every business had a network that truly acted as a support and promotion group then we'd see many more small businesses become more successful.
The number of connections in a network will grow quadratically with the number of nodes if and only if the network is always approximately "compelete": that is, almost all nodes are connected to almost all other nodes. Unfortunately, complete graphs are a terrible model of social networks. Most people interact with only a tiny fraction of the other people on the planet/their telephone service/Facebook, and even among the people that they do know, they certainly don't interact equally with everyone. A better model is the small world network: http://en.wikipedia.org/wiki/Small-world_network .
Second, the distinction between quadratic growth and exponential growth is really important! The two words imply two very different kinds of underlying behaviors as the cause of the growth.