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

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 α 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.

20 pages, 5 figures

References in corpus (8)

• Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices
• Spectra of Sparse Random Matrices
• Mean Field Theory For Non-Equilibrium Network Reconstruction
• Non-equilibrium dynamics of gene expression and the Jarzynski equality
• Inference algorithms for gene networks: a statistical mechanics analysis
• Belief-Propagation and replicas for inference and learning in a kinetic Ising model with hidden spins
• Inferring hidden states in a random kinetic Ising model: replica analysis
• Variational perturbation and extended Plefka approaches to dynamics on random networks: the case of the kinetic Ising model