In Preparation · 2026
May’s Complexity-Stability Hypothesis in Neural Networks
May’s 1972 theorem establishes that sufficiently complex random systems are generically unstable, yet evolved ecosystems systematically violate this prediction through structural mechanisms that natural selection has had billions of years to build. This project investigates whether neural networks, as another class of optimized systems, develop analogous stabilizing structure, or remain in the random-matrix regime, and what properties of the optimization process determine the difference.
In this project I study May’s complexity-stability hypothesis, the prediction from random matrix theory that sufficiently complex systems are generically unstable, in the context of neural networks. Real ecosystems violate this bound through structural mechanisms shaped by natural selection: heavy-tailed interaction-strength distributions, negative pairwise correlations, and non-random topology. The central question is whether neural networks, trained by gradient descent or by explicit selection pressure, develop analogous structure, and what properties of the optimization process, namely the objective function, selection intensity, and training dynamics, determine whether a system can escape the random-matrix regime.