Network Topology Inference with Sparsity and Laplacian Constraints

We tackle the network topology inference problem by utilizing Laplacian constrained Gaussian graphical models, which recast the task as estimating a precision matrix in the form of a graph Laplacian. Recent research [1] has uncovered the limitations of the widely used ℓ
1
-norm in learning sparse graphs under this model: empirically, the number of nonzero entries in the solution grows with the regularization parameter of the ℓ
1
-norm; theoretically, a large regularization parameter leads to a fully connected (densest) graph. To overcome these challenges, we propose a graph Laplacian estimation method incorporating the ℓ
0
-norm constraint. An efficient gradient projection algorithm is developed to solve the resulting optimization problem, characterized by sparsity and Laplacian constraints. Through numerical experiments with synthetic and financial time-series datasets, we demonstrate the effectiveness of the proposed method in network topology inference.