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.

翻译：先前对元胞自动机和随机布尔网络的研究表明，当内部连接的随机性（无序度）设定为中间临界值时，系统会在混沌边缘出现涌现行为。已有研究发现，最大涌现行为发生的临界值与相互连接元素的总数（即邻域大小）呈反比关系。然而，不同方程所预测的该临界值存在差异。本文通过研究一维元胞自动机（1DCA）验证了这一普适关系，但发现该临界值与邻域半径而非邻域大小存在更精确的关联。此外，研究观察到涌现状态的临界值非常接近1/e，这提示我们可能发现了涌现系统的一个基本特性。