From Covariance to Response: What Is Identifiable, and What Should We Measure?

Equilibrium measurements describe how variables fluctuate together, whereas a perturbation experiment asks how a change propagates from one variable to another. When can covariance and a list of decay rates answer this response question, and what should be measured when they cannot? We study stable Ornstein–Uhlenbeck networks with a known acyclic graph and unknown diagonal diffusion. For complete two-layer networks, we establish a sharp identification boundary using the rate sum and show how the full anonymous spectrum can resolve the remaining rate assignments. For a sparse family, an odd-cycle criterion identifies the missing information and guides targeted temporal measurements. Matching bounds on stars quantify the value of target self-lags; multiroot results connect response risk to graph geometry and amplitude calibration. On the star, with unknown independent sensor variances, tracking source and target attains the four-coordinate risk order. A local risk calculation predicts when temporal inversion improves on a prior-center prediction. All sixteen known-amplitude comparisons in independent synthetic experiments support the predicted directions, alongside matched-readout and likelihood-based comparisons. These results connect response identification to the choice and budget of temporal measurements.