In his book, At Home in the Universe, my colleague Stuart Kauffman describes a simple model of random network formation.
Imagine dumping a box of buttons (as in shirt buttons) onto your floor. Now pick up two buttons at random, tie them together with thread, and put them back down. You now have one connected component of two buttons and N-2 singleton buttons. Repeat. Over time, when you pick up a random button it will become more likely that it will lift up a small group of others with it. The interesting thing is, as the number of threads approaches one-half the number of buttons, a single, massive connected component will suddenly emerge such that when you pick up a random button, it is very likely to lift the vast majority of the other buttons. In other words, there is a critical phase transition as the system suddenly shifts from a collection of many, small, isolated groups to a single monolithic group, plus a few outliers.
To demonstrate this phenomenon, I just created a NetLogo simulation of Random Button Networks (Java applet), after following an introductory tutorial at the Complexity Workshop. Enjoy.
Random Button Network Simulation (Java applet)