Complex Spectral Analysis of Rheotrix Operators in Directed Stochastic Systems

Abstract

We investigate the spectral properties of rheotrix operators, non-symmetric flow matrices that characterize directed transitions in stochastic systems, with a focus on the role of complex eigen values in metastable state detection. Most spectral clustering methods use reversible or symmetric transition matrices, so that their eigen values are real. As a result, they cannot be used easily to many non-reversible systems with cyclical or oscillatory nature. To overcome this problem, we extend spectral clustering to work with rheotrix matrices which have complex eigen values. By splitting the complex eigenvectors into their real and imaginary parts, we can group cluster system states into meaningful metastable flow groups. With a standard cyclic rheotrix example, we are able to show that the complex spectral structure captures directional flow patterns, and that our method successfully finds coherent groups in directed stochastic dynamics. This work provides the foundation for analyzing both steady-state and transient behaviours in non-equilibrium systems by making use of spectral methods designed specifically for flow-based operators.