Learning Networks from Wide-Sense Stationary Stochastic Processes.

Area of Science:

• Network Science
• Statistical Inference
• Systems Engineering

Background:

• Complex networked systems with latent inputs are prevalent across neuroscience, finance, and engineering.
• A critical challenge is inferring network edge connectivity from observed node potentials.
• Systems governed by steady-state linear conservation laws are frequently encountered.

Purpose of the Study:

• To develop a method for learning edge connectivity in complex networked systems from node potentials.
• To address the challenge of network inference in high-dimensional settings where network size exceeds sample size.
• To provide theoretical guarantees for the accuracy of the learned network structure.

Main Methods:

• Utilizing an $\ell_1$-regularized Whittle's maximum likelihood estimator (MLE) on temporally correlated node potential samples.
• Assuming latent inputs follow a wide-sense stationary stochastic process with a known spectral density matrix.
• Leveraging the sparsity pattern of the Laplacian matrix to encode network structure.

Main Results:

• The MLE problem is shown to be strictly convex, ensuring a unique solution.
• Under a novel mutual incoherence condition and specific sample-size constraints, the ML estimate accurately recovers the network's sparsity pattern with high probability.
• Recovery guarantees are provided for the Laplacian matrix in element-wise maximum, Frobenius, and operator norms.

Conclusions:

• The proposed $\ell_1$-regularized MLE method effectively infers network connectivity in complex systems.
• The method demonstrates strong performance in high-dimensional settings and provides robust theoretical guarantees.
• The approach is validated through simulations on engineered and real-world neural network datasets.