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

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.

Figure 1

(a) Spectral density $\rho \left(\tau \right)$ for $\alpha =0.5$: the vertical line indicates the $\delta$ peak of height $1-\alpha$ at $\tau =1/\lambda$, the relaxation time in the absence of observations. (b) Spectral density $\rho (ln\tau )=\tau \rho \left(\tau \right)$ of $ln\tau$: this shifts to smaller $ln\tau$ as $\alpha$ increases, indicating shorter posterior correlation times. The spectrum also narrows and becomes concentrated around $\tau =1/\sigma \sqrt{\alpha }$ for large $\alpha$. As the posterior variance $C\propto \tau$ for each hidden space mode, the distributions of $lnC$ differ only from those of $ln\tau$ by a horizontal shift.

Figure 2

(a) Characteristic posterior relaxation time $\tau$ as a function of $\alpha$, for $\lambda =0.1$ and $\sigma =1$, defined in three different ways (see text). For $\alpha \to 0$ all three curves approach $\tau =1/\lambda =10$; asymptotically they decay as $1/\sqrt{\alpha }$. (b) Posterior power spectrum [obtained by setting $z=\text{i}\omega$ in (24)] for various $\alpha$, at $\lambda =0$. The power spectrum diverges as $\omega \to 0$ when $\alpha \le 1$. For small $\alpha$ the divergence is $\propto 1/{\omega }^2$, crossing over to $\propto 1/\omega$ as $\alpha \to 1$. Beyond $\omega \sim O\left(1\right)$ the curves for all $\alpha$ exhibit a standard Lorentzian tail $1/{\omega }^2$. See Ref. [18] for a derivation of these power laws.

Figure 3

Spectral density $\rho (ln\tau )=\tau \rho \left(\tau \right)$, of relaxation times $\tau$, for different values of $\alpha$. We plot $\rho (ln\tau )$ to make the normalization of the densities more obvious. The spectra of posterior variances $C$, which define the inference error, are identical up to a horizontal shift as $C\propto \tau$. (a) At small $\alpha$ the spectrum is broad, indicating that there is much variation in how different hidden state space directions are constrained by observations. For increasing $\alpha$ the spectrum becomes more peaked, and centered around decreasing $\tau$ or $C$: different directions become determined more strongly, and more evenly, by observations, a trend more clearly visible in (b).

Figure 4

(a) Spectral density $\rho (ln\tau )=\tau \rho \left(\tau \right)$, at $\gamma =j/\lambda =0.5$ (critical value for internal stability, with $j=0.2$ and $\lambda =0.4$) and $\alpha =0.5$ for different values of $p$: the two pieces of the spectrum at $p=0.2$ merge at $p=0.3$, giving a spectrum supported on a single interval for $p>0.3$. (b) Curve in the $(p,\gamma )$ plane for which the two pieces of the spectrum merge when coming from low $\gamma$: the black line refers to $\alpha =0.5$, the case shown in (a). The two-piece region near the origin shrinks (see curve for $\alpha =0.9$, blue dotted line) and vanishes for $\alpha \to 1$.

Figure 5

Illustration of a linear-Gaussian state space model.