#### 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, n ^{2} 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 2^{n}. 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 n^{2}. Thus a thousand members can have a million friendships.

The magic of n^{2} 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 n^{2} (10^{2}), 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 11^{2} is 20% larger than 10^{2}). In a network economy, small efforts can lead to large results.