Unified Network Spectroscopy: Inferring Reduced Biological Dynamics from Fluctuations under Structured Noise and Broken Ergodicity

Abstract

Mechanistic models of biological networks predict but cannot be calibrated at scale, while learned representations describe but expose no dynamical operator. Both lack an inverse step mapping data onto the few dynamical quantities the data determine. We develop Unified Network Spectroscopy (UNS) as that operator and establish when it is well posed. Reading fluctuations about a physiological setpoint through the Lyapunov relation of the linear noise approximation, UNS returns dominant timescales and frequencies, the subspace on which they act, and the projected noise. The inversion itself is established; our contribution is architectural. Three results govern its success. First, biochemical noise is generically anisotropic, and extrinsic modulation makes the dynamics non-normal. Assuming isotropy can invert mode rankings and destroy precision-matrix topology recovery; we give the exact, checkable condition under which recovery survives. Second, equal-time covariance identifies neither the dynamics nor their spectrum: the residual freedom is precisely the probability current that non-equilibrium living systems generically carry. Third, population snapshots replace time series only under ergodicity; heterogeneity biases inferred relaxation rates downward, making systems appear closer to bifurcation, whatever the sample size. Lagged covariance or per-cell velocity restores identifiability and turns noise structure into a recovered, falsifiable quantity. We provide a three-regime ergodicity taxonomy with diagnostics and show that the variational operator remains well posed when samples are fewer than variables. The output is a typed interface contract for hierarchical model composition, carrying its uncertainty, attainment and validity domain, with controllability inheritance as a falsification test.