network data analysts

Abstract: Topological signal processing (TSP) processes signals on simplicial complexes with oriented boundary operators, which is the natural choice for flow signals or when the topological invariants play a role for the task at hand. However, many higher-order signals carry no orientation, and applying oriented operators to them is not well-defined since it introduces an arbitrary choice of simplex orientation. We study an unoriented TSP (UTSP) framework that replaces oriented boundaries with unoriented incidence matrices. First, we show that unoriented incidence and Laplacian matrices between arbitrary simplicial levels admit graph-like spectral properties. Second, since dropping orientation removes the Hodge decomposition, we introduce an unoriented counterpart, termed interaction-order decomposition, which quantifies how much of a higher-order signal is explained by aggregating lower-order signals. Third, we use this decomposition to derive regularizers for signal reconstruction that penalize each interaction order separately. Experiments on real-world data show that the order-aware regularizers outperform oriented baselines, with the largest gains when the signal energy is unevenly distributed across orders.

Abstract: Inferring network topology from noisy node observations is a central problem in graph signal processing. In this paper, we consider Laplacian-constrained graph estimation for Gaussian Markov random fields, focusing on the underdetermined regime in which the number of samples is smaller than the number of graph nodes. Existing approaches often formulate the problem as a sparsity-regularized maximum-likelihood estimation problem. While effective, such methods typically require iterative optimization and are often computationally demanding, particularly under Laplacian constraints. Instead, we propose a non-iterative estimator of graph Laplacians that uses effective resistance for regularization, and evaluate the method using a simple sparsification procedure. Experiments show that with some trade-off in edge and weight recovery on the considered dataset, computational cost for moderately sized graphs can be substantially reduced.