Non-Hermitian Gain–Loss Diagnostics for Directed Process Networks: Spectral Stability, Irreversibility, and Escape Routes

Introduction: Many process systems are open, irreversible, and structurally asymmetric: material, energy, information, or value may be amplified along some pathways while being dissipated, trapped, recycled, or lost along others. Conventional balances, transfer functions, and multivariate monitoring methods characterize process behavior effectively, but they do not directly expose how gain, loss, recirculation, and directed topology combine to control dynamical stability.

Methods: We formulate a diagnostic framework for directed process networks using non-Hermitian operator methods for open systems. A process is represented by a state-transition operator whose asymmetric couplings encode transfer, gain, loss, recirculation, dissipation, and forcing. Spectral and pathway diagnostics characterize dominant amplification or decay modes, escape routes, irreversible concentration of flow, and sensitivity to structural perturbations. Quantitative comparison is illustrated with an eight-node linear-Gaussian recirculating network. One recycle coupling is tripled while all other coefficients and stochastic forcing are held fixed. The proposed spectral diagnostic is compared with a six-component PCA squared-prediction-error (SPE) monitor explaining 93.2% of baseline variance and with linear-Gaussian pairwise transfer entropy.

Results: The recycle perturbation increases the dominant spectral radius from 0.850 to 0.960 and reduces the associated decay width, defined as −ln(ρ), from 0.163 to 0.041, a 74.6% loss of decay margin. Transfer entropy across the perturbed recycle edge increases from 0.0087 to 0.0845 nats, approximately 9.7-fold, while expected PCA-SPE increases by 1.2%. The methods are therefore complementary: PCA-SPE measures state-space deviation, transfer entropy measures directed statistical dependence, and the non-Hermitian spectrum links network topology and gain-loss structure directly to proximity to dynamical instability. Full dense eigendecomposition scales as O(N³) with O(N²) storage, whereas sparse leading-mode Krylov/Arnoldi calculations require storage proportional to network connectivity and approximately linear edge-wise work per iteration.

Conclusion: Non-Hermitian gain-loss diagnostics provide an interpretable additional layer for monitoring open directed processes, particularly where recirculation, amplification, dissipation, and topology are central to process behavior.