Network Games Induced Prior for Graph Topology Learning

Learning the graph topology of a complex network is challenging due to limited data availability and imprecise data models. A common remedy in existing works is to incorporate priors such as sparsity or modularity which highlight on the structural property of graph topology. We depart from these approaches to develop priors that are directly inspired by complex network dynamics. Focusing on social networks with actions modeled by equilibriums of linear quadratic games, we postulate that the social network topologies are optimized with respect to a social welfare function. Utilizing this prior knowledge, we propose a network games induced regularizer to assist graph learning. We then formulate the graph topology learning problem as a bilevel program. We develop a two-timescale gradient algorithm to tackle the latter. We draw theoretical insights on the optimal graph structure of the bilevel program and show that they agree with the topology in several man-made networks. Empirically, we demonstrate the proposed formulation gives rise to reliable estimate of graph topology.

LogSpecT: Feasible Graph Learning Model from Stationary Signals with Recovery Guarantees

Graph learning from signals is a core task in Graph Signal Processing (GSP). One of the most commonly used models to learn graphs from stationary signals is SpecT. However, its practical formulation rSpecT is known to be sensitive to hyperparameter selection and, even worse, to suffer from infeasibility. In this paper, we give the first condition that guarantees the infeasibility of rSpecT and design a novel model (LogSpecT) and its practical formulation (rLogSpecT) to overcome this issue. Contrary to rSpecT, the novel practical model rLogSpecT is always feasible. Furthermore, we provide recovery guarantees of rLogSpecT, which are derived from modern optimization tools related to epi-convergence. These tools could be of independent interest and significant for various learning problems. To demonstrate the advantages of rLogSpecT in practice, a highly efficient algorithm based on the linearized alternating direction method of multipliers (L-ADMM) is proposed. The subproblems of L-ADMM admit closed-form solutions and the convergence is guaranteed. Extensive numerical results on both synthetic and real networks corroborate the stability and superiority of our proposed methods, underscoring their potential for various graph learning applications.