Stationarity and Spectral Characterization of Random Signals on Simplicial Complexes

2026-02-03

AI Summary

This work addresses the limitations of traditional graph models in capturing non-binary higher-order relationships and the lack of statistical frameworks for random signals in existing topological signal analysis. It establishes, for the first time, a theory of stationarity for random signals defined on simplicial complexes, generalizing classical stationarity by characterizing stationary signals as outputs of white noise passed through topological filters. The paper rigorously defines the topological power spectral density (PSD) and constructs a comprehensive spectral analysis and filtering framework by integrating algebraic topology, Hodge and Dirac theory, and spectral graph methods. Experimental results demonstrate that the proposed notion of topological stationarity significantly enhances signal modeling and processing performance on both synthetic and real-world datasets.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Computational social science

Abstract

It is increasingly common for data to possess intricate structure, necessitating new models and analytical tools. Graphs, a prominent type of structure, can encode the relationships between any two entities (nodes). However, graphs neither allow connections that are not dyadic nor permit relationships between sets of nodes. We thus turn to simplicial complexes for connecting more than two nodes as well as modeling relationships between simplices, such as edges and triangles. Our data then consist of signals lying on topological spaces, represented by simplicial complexes. Much recent work explores these topological signals, albeit primarily through deterministic formulations. We propose a probabilistic framework for random signals defined on simplicial complexes. Specifically, we generalize the classical notion of stationarity. By spectral dualities of Hodge and Dirac theory, we define stationary topological signals as the outputs of topological filters given white noise. This definition naturally extends desirable properties of stationarity that hold for both time-series and graph signals. Crucially, we properly define topological power spectral density (PSD) through a clear spectral characterization. We then discuss the advantages of topological stationarity due to spectral properties via the PSD. In addition, we empirically demonstrate the practicality of these benefits through multiple synthetic and real-world simulations.

Problem

• stationarity
• simplicial complexes
• random signals
• spectral characterization
• topological signals

Innovation

• stationarity
• simplicial complexes
• topological signals
• power spectral density
• Hodge theory