Spectral Contraction of Boundary-Weighted Filters on delta-Hyperbolic Graphs
This work addresses the problem of graph signal processing for researchers and practitioners dealing with hierarchical data, providing a novel geometric approach, though it appears incremental as it builds on existing concepts like delta-hyperbolic graphs and Busemann functions.
The paper tackled the challenge of designing graph filters for hierarchical graphs with tree-like branching patterns by introducing a boundary-weighted operator that rescales edges based on endpoint drift toward the graph's Gromov boundary. The result was a parameter-free, lightweight filter with proven stability, derived from geometric principles, offering a new analytic tool for graph signal processing on data with dense or hidden hierarchical structure.
Hierarchical graphs often exhibit tree-like branching patterns, a structural property that challenges the design of traditional graph filters. We introduce a boundary-weighted operator that rescales each edge according to how far its endpoints drift toward the graph's Gromov boundary. Using Busemann functions on delta-hyperbolic networks, we prove a closed-form upper bound on the operator's spectral norm: every signal loses a curvature-controlled fraction of its energy at each pass. The result delivers a parameter-free, lightweight filter whose stability follows directly from geometric first principles, offering a new analytic tool for graph signal processing on data with dense or hidden hierarchical structure.