Structure Learning via ADMM in Networks obeying Conservation Laws

Learning the edge connectivity structure of networked systems from limited data is a fundamental challenge in many critical infrastructure domains, including power, traffic, and finance. Such systems obey steady-state conservation laws: x = L*y, where x and y represent injected flows (inputs) and potentials (outputs), respectively. The sparsity pattern of the pxp Laplacian L* encodes the underlying edge structure. In a stochastic setting, the goal is to infer this sparsity pattern from zero-mean i.i.d. samples of y. Recent work by \cite{rayas2022learning} has established statistical consistency results for this learning problem by considering an $\ell_1$-regularized maximum likelihood estimator. However, their approach did not develop a scalable algorithm but relies on solving a convex program via the CVX package. To address this gap, we propose an alternating direction method of multipliers (ADMM), which is transparent and fast. A key contribution is to demonstrate the role of an algebraic matrix Riccati equation in the primal update step of ADMM. Numerical experiments on a host of synthetic and benchmark networks, including power and water systems, show the efficiency of our method.