Topology Learning of unknown Networked Linear Dynamical System excited by Cyclostationary inputs

TL;DR

The paper addresses topology learning for networks of linear dynamical systems driven by cyclostationary inputs, extending prior work to complex-valued interdependencies and lifting the problem to $T$-dimensional vector processes. It develops a principled framework that connects inverse power spectral density and multivariate Wiener filtering, employing group-sparse regularization to reliably identify network edges. For fully observed networks, the approach uses phase-constant criteria on eigenvalues and a sequence of relaxations to prune spurious edges, culminating in a regularized Wiener filter estimator that supports both time- and frequency-domain implementations. For partially observed (latent-node) trees, the method reconstructs the observed topology and iteratively infers latent nodes and their connections, leveraging a matrix-inversion-based decomposition of $\Phi_X^{-1}$ and graphical-separation arguments. The authors validate their theory on simulated data and real-world climate SST data, demonstrating accurate topology recovery and latent-node inference, with practical implications for infrastructure and climate networks where cyclostationary excitation is common.

Abstract

Topology learning of networked dynamical systems is an important problem with implications to optimal control, decision-making over networks, cybersecurity and safety. The majority of prior work in consistent topology estimation relies on dynamical systems excited by temporally uncorrelated processes. In this article, we present a novel algorithm for guaranteed topology learning of networks that are excited by temporally (colored) cyclostationary processes, which encompasses a wide range of temporal correlation including wide-sense stationarity. Furthermore, unlike prior work, the framework applies to linear dynamic system with complex valued dependencies, and leverages group lasso regularization for effective learning of the network structure. In the second part of the article, we analyze conditions for consistent topology learning for bidirected tree networks when a subset of the network is unobserved. Here, the full topology along with unobserved nodes are recovered from observed node's time-series alone. Our theoretical contributions are validated on simulated data as well as on real-world climate data.

Figures (5)

• Figure 1: Current reconstruction using frequency-domain regression presented in Subsection \ref{['subsec:RWFE']}. Reconstructed currents are: gulf stream: $15 \rightarrow 14 \rightarrow 12 \rightarrow 10 \rightarrow 8 \rightarrow 17,$ Benguela current: $7 \rightarrow 4 ,$ Braziliean current: $1 \rightarrow 3,$ Malvinas current: $6 \rightarrow 5,$ Canary current: $9 \rightarrow 11 \rightarrow 13 \rightarrow 16.$
• Figure 2: (a) Generative graph topology $\mathcal{G_T}(\mathcal{V}, \mathcal{E_T}):$ Under full observability, green colored edges are included in $\mathcal{G_T}$. Under partial observability, red colored nodes are latent, and green colored edges are excluded from $\mathcal{G_T}.$ (b) Reconstruction under full observability: color map of $H_{\infty}[ \boldsymbol{\hat{\mathsf{W}}}_{ji} ]+H_{\infty}[ \boldsymbol{\hat{\mathsf{W}}}_{ij}]$ for $i,j \in \{1,\cdots,50 \}.$ Some of the spurious edges have very low color intensity. All edges in $\mathcal{G_T}$ are identified correctly.
• Figure 3: Partial observability: Left figure is the magnitude of model-based Wiener filter (exact values based on generative model). Right figure is the magnitude of regularized Wiener filter obtained by solving (\ref{['eqn:vec_opt_sparse']}) with $\gamma=0.07$ and $L_*=3$.
• Figure 4: Phase plot of eigenvalues (model-based) for various edges $j-i.$ Here, $\epsilon_{ji}$ ( $\epsilon_{ij}$) is the largest component of the vector $\mathcal{E}_{ji}$ ($\mathcal{E}_{ij}$).
• Figure 5: Reconstruction without regularization: Left figure is the magnitude of model-based Wiener filter (exact values based on generative model). Right figure is the magnitude of Wiener filter estimate without regularizer ($\gamma=0$) and $L_*=3$, computed from 628400 samples.