Inferring hidden states in Langevin dynamics on large networks: Average case performance

TL;DR

This paper analyzes the average performance of dynamical inference in large networks with hidden nodes, using Langevin dynamics and random Gaussian couplings, providing insights into inference error and relaxation times.

Contribution

It introduces two methods to characterize the inference error in Langevin dynamics on large networks, combining Random Matrix Theory and dynamical functionals.

Findings

Inference error depends on system parameters and hidden-to-observed node ratio.

Spectral properties of the inference error are characterized via Random Matrix Theory.

Posterior relaxation times are derived from the analysis of the effective drift.

Abstract

We present average performance results for dynamical inference problems in large networks, where a set of nodes is hidden while the time trajectories of the others are observed. Examples of this scenario can occur in signal transduction and gene regulation networks. We focus on the linear stochastic dynamics of continuous variables interacting via random Gaussian couplings of generic symmetry. We analyze the inference error, given by the variance of the posterior distribution over hidden paths, in the thermodynamic limit and as a function of the system parameters and the ratio {\alpha} between the number of hidden and observed nodes. By applying Kalman filter recursions we find that the posterior dynamics is governed by an "effective" drift that incorporates the effect of the observations. We present two approaches for characterizing the posterior variance that allow us to tackle, respectively, equilibrium and nonequilibrium dynamics. The first appeals to Random Matrix Theory and reveals average spectral properties of the inference error and typical posterior relaxation times, the second is based on dynamical functionals and yields the inference error as the solution of an algebraic equation.