Where is the Edge of Chaos?

Previous study of cellular automata and random Boolean networks has shown emergent behavior occurring at the edge of chaos where the randomness (disorder) of internal connections is set to an intermediate critical value. The value at which maximal emergent behavior occurs has been observed to be inversely related to the total number of interconnected elements, the neighborhood size. However, different equations predict different values. This paper presents a study of one-dimensional cellular automata (1DCA) verifying the general relationship but finding a more precise correlation with the radius of the neighborhood rather than neighborhood size. Furthermore, the critical value of the emergent regime is observed to be very close to 1/e hinting at the discovery of a fundamental characteristic of emergent systems.