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)
Has anyone done any work on the varying likelihood of finding cyclic strings of buttons in this way ?
Measuring the growth of cycles would be a neat extension of the current simulation. Feel free to download the NetLogo code and edit it as an exercise.
Could a button represent a biomolecule and a thread represent the relation between them that they catalyse each other’s formation (or if a “directed” thread that only one catalyses the other). Then a cycle might represent an autocatalytic network such as Kauffman mentions.
If parameters were set appropriately then the frequency of cycles appearing might have bearing on the chance of a random molecular mixture containing autocatalytic elements. This would be a useful figure to look at. I will act as you have suggested. thanks for the post.