Source-linked AI summary
Topological and Dynamical Complexity of Random Neural Networks
Gilles Wainrib, Jonathan Touboul
TL;DR
Estimating equilibrium complexity in random neural networks is difficult because their non-Hamiltonian dynamics lack the energy landscape and symmetry reductions used in related systems. The paper applies Kac–Rice counting near criticality and finds that equilibrium complexity grows exponentially with system size, with a rate proportional to the Lyapunov exponent.
Problem
Estimating equilibria in non-Hamiltonian random networks is technically difficult because energy-landscape and symmetry-based reductions do not apply.
Source: p. 2Method
The authors apply Kac–Rice counting to the fixed-point equations and, near criticality, use the fact that equilibria remain close to zero.
Source: p. 2Results
Near criticality, the mean number of equilibria grows exponentially with system size, with an exponential coefficient proportional to the Lyapunov exponent.
Source: p. 3Takeaways & Limitations
Circular-law universality extends the complexity result to many independent couplings, while matrices satisfying Dale’s principle may fall outside that class.
Source: p. 3For σ > 1, the lack of an energy landscape and usable symmetry reductions leaves a determinant with non-identical column variances whose spectral density is unknown.
Source: p. 2
Abstract
from arXiv · showhide
Random neural networks are dynamical descriptions of randomly interconnected neural units. These show a phase transition to chaos as a disorder parameter is increased. The microscopic mechanisms underlying this phase transition are unknown, and similarly to spin-glasses, shall be fundamentally related to the behavior of the system. In this Letter we investigate the explosion of complexity arising near that phase transition. We show that the mean number of equilibria undergoes a sharp transition from one equilibrium to a very large number scaling exponentially with the dimension on the system. Near criticality, we compute the exponential rate of divergence, called topological complexity. Strikingly, we show that it behaves exactly as the maximal Lyapunov exponent, a classical measure of dynamical complexity. This relationship unravels a microscopic mechanism leading to chaos which we further demonstrate on a simpler class of disordered systems, suggesting a deep and underexplored link between topological and dynamical complexity.