Robust reconstruction of sparse network dynamics

TL;DR

This work introduces Ergodic Basis Pursuit (EBP) to reliably reconstruct sparse directed networks from limited multivariate time series by exploiting ergodic statistics. The key idea is to adapt the candidate function library to a product-like measure via Gram–Schmidt, producing an adapted library $\mathcal{L}_{\nu}$ in which the library matrix $\Phi_{\nu}(X)$ satisfies the Restricted Isometry Property, enabling unique, sparse recovery. The authors provide rigorous conditions under which the minimum data length scales quadratically with node degree and logarithmically with network size, and they prove robustness to noise through a Quadratically constrained variant (QEBP). They validate the approach with numerical experiments on coupled logistic maps and real experimental optoelectronic networks, including a relaxing path algorithm to identify robust edges when noise levels are unknown. Overall, the work demonstrates a principled fusion of ergodic theory and compressed sensing that enables accurate, data-efficient network reconstruction in chaotic regimes with practical relevance to experimental settings.

Abstract

Reconstruction of the network interaction structure from multivariate time series is an important problem in multiple fields of science. This problem is ill-posed for large networks leading to the reconstruction of false interactions. We put forward the Ergodic Basis Pursuit (EBP) method that uses the network dynamics' statistical properties to ensure the exact reconstruction of sparse networks when a minimum length of time series is attained. We show that this minimum time series length scales quadratically with the node degree being probed and logarithmic with the network size. Our approach is robust against noise and allows us to treat the noise level as a parameter. We show the reconstruction power of the EBP in experimental multivariate time series from optoelectronic networks.

Figures (6)

• Figure 1: Ergodic Basis Pursuit performance requires only short time series. a) Illustration of a ring graph with $N = 10$. b) False positive (FP) of the reconstructed ring network with respect to the length of time series $n$ for a network size $N = 40$. c) The minimum length of time series $n_0$ for a successful reconstruction versus system size $N$. Basis pursuit (BP) and ergodic basis pursuit (EBP) are shown in (purple) squares and (green) circles, respectively. The network dynamics parameters are $a = 3.990$ and coupling strength $\alpha = 5 \times 10^{-4}$. The shaded area corresponds to the standard deviation with respect to 10 distinct initial conditions uniformly drawn in $[0, 1]^{N}$. The (black) dashed is the scaling $\ln N$ for reference. The Kernel density estimation of $\nu$ is used with bandwidth $\chi = 0.05$. The multivariate time series is generated without noise.
• Figure 2: Comparison between BP and EBP under different network structures. a) Ring graph with maximum degree $\Delta = 2$. b) The minimum length of time series $n_0$ for a successful reconstruction versus system size $N$, and similarly in d) and f). c) Lattice graph with maximum degree $\Delta = 6$. e) Star graph where the maximum degree grows with the system size. Basis pursuit (BP) and ergodic basis pursuit (EBP) are shown in (purple) squares and (green) circles, respectively. The network dynamics parameters are $a = 3.990$ and coupling strength $\alpha = 1 \times 10^{-3}/\Delta$, so the coupling term in the network dynamics is normalized as we vary $N$. The shaded area corresponds to the standard deviation with respect to 10 distinct initial conditions uniformly drawn in $[0, 1]^{N}$. The Kernel density estimation of $\nu$ is used with bandwidth $\chi = 0.05$. The multivariate time series is generated without noise.
• Figure 3: Network dynamics of experimental optoelectronic data. a) Original optoelectronic network with two groups of nodes --- dark gray node is marked for future reference. b) Return map for all nodes in the network. c) Densities function $\rho_i$ for each node $i$ (in light color) estimated using each node's time series. Clustering density estimation displays two resulting densities corresponding to two groups of nodes, in blue and red. The density estimation utilizes a Gaussian kernel with bandwidth $\chi = 0.05$.
• Figure 4: Reconstruction of the original network from experimental data. a) Relaxing path algorithm is performed in the node (in dark gray) from the left panel. There are three different relaxing parameter values, where the edges are colored accordingly: the true edges (in gray) and false positives (in orange) while the thickness is the edge weight, see \ref{['eq:weighted_edge']}. b) False positive (FP) (in orange) and false negative (FN) (in purple) of the reconstructed network versus the parameter $\epsilon$. We varied the $\epsilon$ parameter through 25 values equally spaced in the interval $\mathcal{E} = [0.20, 0.33]$. We employ ECOS convex optimization solver ECOS to solve \ref{['eq:QBP_EBP']}.
• Figure 5: Robust network reconstruction scheme using ergodic basis pursuit. The noisy data is generated from a network dynamics whose underlying measure is $\mu_{\xi}$. Using its estimated measure $\nu$, we induce an orthonormal set of basis functions $\mathcal{L}(\nu)$ representing the dynamics. Under the assumption that the network dynamics is sparse, the noisy data and $\mathcal{L}(\nu)$ are recast as a minimization problem, whose solution encodes a proxy of the network. Although the noise level may be unknown, the relaxing path algorithm searches the connections of each node varying the noise level $\epsilon$ as a parameter. The true connections remain robust over an interval of $\epsilon$.

Theorems & Definitions (51)

• Remark 1: Minimum length of time series for networks
• Definition 1: Network Library
• Definition 2: Edge via Network Library
• Definition 3: Sparse vector
• Definition 4: Sparse Network Dynamics Representation
• Proposition 1
• Proposition 2: Coherence bounds restricted isometry constant
• proof
• Theorem 1: Uniqueness of noiseless recovery CANDES_2008Foucart_mathematical_compressive_sensing
• proof