Spectral Contraction of Boundary-Weighted Filters on delta-Hyperbolic Graphs

TL;DR

This paper introduces a boundary-weighted operator for delta-hyperbolic graphs that provides a stable, curvature-controlled filtering method based on geometric principles, suitable for hierarchical network data.

Contribution

It presents a novel boundary-weighted filter with a spectral norm bound derived from hyperbolic geometry, enabling stable graph signal processing on hierarchical structures.

Findings

Operator's spectral norm is bounded by a curvature-dependent factor.

Signals lose a predictable fraction of energy per filter pass.

The filter is parameter-free and grounded in geometric principles.

Abstract

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.