Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sparse dynamic network reconstruction through L1-regularization of a Lyapunov equation

Ian Xul Belaustegui, Marcela Ordorica Arango, Román Rossi-Pool, Naomi Ehrich Leonard, Alessio Franci

TL;DR

This work addresses reconstructing directed, weighted interaction networks from high-dimensional time-series by leveraging a covariance Lyapunov equation to generate an affine family of candidate state matrices, and then selecting a sparse solution via $L_1$-regularization formulated as a linear program over the solution space $\\mathcal{S}_{\\Gamma}$. It provides a constructive isomorphism that maps the Lyapunov-based constraints to a linear system $M v = b$, enabling efficient sparsity pursuit; the framework further allows priors on edge existence, including priors derived from Transfer Entropy (TE). TE priors are integrated into the LP through weighted $L_1$ penalties, biasing reconstructions toward plausible directed edges. Numerical experiments on random sparse Hurwitz matrices show that TE-informed priors substantially improve reconstruction accuracy over no-prior methods and compete with or approach full-prior knowledge, with consistent performance gains in weakly nonlinear settings. The approach offers a scalable, principled means to infer directed network structure with edge strengths from noisy time-series data, with potential applications in neuroscience and other domains.

Abstract

An important problem in many areas of science is that of recovering interaction networks from simultaneous time-series of many interacting dynamical processes. A common approach is to use the elements of the correlation matrix or its inverse as proxies of the interaction strengths, but the reconstructed networks are necessarily undirected. Transfer entropy methods have been proposed to reconstruct directed networks but the reconstructed network lacks information about interaction strengths. We propose a network reconstruction method that inherits the best of the two approaches by reconstructing a directed weighted network from noisy data under the assumption that the network is sparse and the dynamics are governed by a linear (or weakly-nonlinear) stochastic dynamical system. The two steps of our method are i) constructing an (infinite) family of candidate networks by solving the covariance matrix Lyapunov equation for the state matrix and ii) using L1-regularization to select a sparse solution. We further show how to use prior information on the (non)existence of a few directed edges to drastically improve the quality of the reconstruction.

Sparse dynamic network reconstruction through L1-regularization of a Lyapunov equation

TL;DR

This work addresses reconstructing directed, weighted interaction networks from high-dimensional time-series by leveraging a covariance Lyapunov equation to generate an affine family of candidate state matrices, and then selecting a sparse solution via -regularization formulated as a linear program over the solution space . It provides a constructive isomorphism that maps the Lyapunov-based constraints to a linear system , enabling efficient sparsity pursuit; the framework further allows priors on edge existence, including priors derived from Transfer Entropy (TE). TE priors are integrated into the LP through weighted penalties, biasing reconstructions toward plausible directed edges. Numerical experiments on random sparse Hurwitz matrices show that TE-informed priors substantially improve reconstruction accuracy over no-prior methods and compete with or approach full-prior knowledge, with consistent performance gains in weakly nonlinear settings. The approach offers a scalable, principled means to infer directed network structure with edge strengths from noisy time-series data, with potential applications in neuroscience and other domains.

Abstract

An important problem in many areas of science is that of recovering interaction networks from simultaneous time-series of many interacting dynamical processes. A common approach is to use the elements of the correlation matrix or its inverse as proxies of the interaction strengths, but the reconstructed networks are necessarily undirected. Transfer entropy methods have been proposed to reconstruct directed networks but the reconstructed network lacks information about interaction strengths. We propose a network reconstruction method that inherits the best of the two approaches by reconstructing a directed weighted network from noisy data under the assumption that the network is sparse and the dynamics are governed by a linear (or weakly-nonlinear) stochastic dynamical system. The two steps of our method are i) constructing an (infinite) family of candidate networks by solving the covariance matrix Lyapunov equation for the state matrix and ii) using L1-regularization to select a sparse solution. We further show how to use prior information on the (non)existence of a few directed edges to drastically improve the quality of the reconstruction.
Paper Structure (11 sections, 1 theorem, 19 equations, 4 figures)

This paper contains 11 sections, 1 theorem, 19 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Model formulation
  4. Solving the covariance Lyapunov equation for the state matrix
  5. $L_1$-regularization on $\mathcal{S}_{\Gamma}$ and sparse network reconstruction
  6. Including priors on edge existence
  7. Edge priors from Transfer Entropy
  8. Numerical validation
  9. Implementation
  10. Results and discussion
  11. Conclusions and perspectives

Key Result

Proposition IV.1

Given a covariance matrix $\Gamma=UCU^T$, define a matrix $\mathbf{M}$ of dimensions $\frac{n(n+1)}{2}\times n^2$ and a vector $\mathbf b$ of dimension $\frac{n(n+1)}{2}$ with entries, respectively, given by and where $\iota_1:\{(i,j) \mid i,j\in [n], i \leq j\}\to[\frac{n(n+1)}{2}]$ and $\iota_2:[n]^2\to[n^2]$ are one-to-one functions that give the index of the vectorizations of the upper trian

Figures (4)

  • Figure 1: Heatmaps for three examples of weighted connectivity matrices of dimension $8\times 8$. The leftmost heatmap is for a sparse matrix $A\in \mathbb{R}^{8\times 8}$ generated randomly using the algorithm in Section \ref{['sec:Val']}. The other two matrices were selected randomly from the solution space $\mathcal{S}_\Gamma$ for the matrix $A$. This means the solution to the system in (\ref{['eq:lin_sys']}) has the same covariance matrix for all three matrices.
  • Figure 2: Diagram illustrating the way the optimization problem (\ref{['eq:l1_prob']}) can be encoded as a linear programming problem. The top $\frac{n(n+1)}{2}\times 2n^2$ sub-matrix is used as an equality constraint, which restricts the search to matrices on $\mathcal{S}_\Gamma$. The $2n^2\times 2n^2$ sub-matrix bellow is used as an inequality constraint which ensures that for each $\ell, k$ we have $\lvert v_{(\ell, k)} \rvert \leq u_{(\ell, k)}$. An example of this encoding for the case $n=5$ is show on the right.
  • Figure 3: Three examples of reconstructed connected networks using our method with edge priors inferred using TE (middle), and no info (right), applied to time-series generated from the weakly-nonlinear system in (\ref{['eq:sat_sys']}). The ground truth connectivity matrix $A$ is show on the left. The top row shows examples of good reconstruction (high alignment between reconstructed matrix and the ground truth), the middle row shows reconstruction with median alignment, and the bottom row shows a bad reconstruction (low alignment). The alignments for the reconstructions with TE Info were $0.99$, $0.90$ and $0.31$ for the good, median, and bad respectively. For the No Info the alignments were $0.43$, $0.42$, and $0.53$ for the good, median, and bad respectively. Examples were selected from $600$ simulated dynamics of matrices with $20$ edges and different $\varepsilon$ values by sorting according to alignment of TE Info reconstruction and selecting the best, median, and worst reconstructions.
  • Figure 4: Plots showing the performance of our method for $100$ samples of random Hurwitz matrices of size $10\times 10$ generated with different values of $\varepsilon$, and with a)$10$, b)$20$, c)$30$ edges. The $L_1$ optimization methods compared are: Full Info (blue), TE Info (red), No Info (green) on edge existence. Performance of precision and correlation matrices were added as reference in purple and orange respectively. Darker colors correspond to reconstructions for data simulated using the linear system in (\ref{['eq:lin_sys']}), and lighter colors for the weakly-nonlinear system in (\ref{['eq:sat_sys']}). Notches in boxes represent median values and dotted lines mean values. Boxes encompass $50\%$ of the data points. Isolated points are outliers.

Theorems & Definitions (2)

  • Proposition IV.1
  • proof