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Random-Walk Laplacian for Frequency Analysis in Periodic Graphs.

Rachid Boukrab¹, Alba Pagès-Zamora¹

¹SPCOM Group, Universitat Politècnica de Catalunya-Barcelona Tech, 08034 Barcelona, Spain.

Sensors (Basel, Switzerland)

|March 6, 2021

Summary

This study introduces the random-walk normalized Laplacian matrix for graph signal processing, defining graph frequencies via eigenvalues. Replicating graphs interpolates the frequency domain, akin to zero-padding in digital signal processing.

Keywords:

frequency ordering graph Fourier transform periodic graph random-walk Laplacian

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Area of Science:

Graph Signal Processing
Linear Algebra
Digital Signal Processing

Background:

Graph signal processing extends traditional signal processing to graph-structured data.
The Laplacian matrix is a fundamental tool in graph analysis.

Purpose of the Study:

To explore the random-walk normalized Laplacian matrix as a graph-shift operator.
To define and order graph frequencies based on spectral properties.
To analyze the frequency domain behavior of periodic graphs.

Main Methods:

Utilizing the random-walk normalized Laplacian matrix and its eigenvalues to define graph frequencies.
Proposing a criterion based on Euclidean distance for frequency ordering.
Investigating the spectral properties of replicated graphs.

Main Results:

The random-walk normalized Laplacian matrix serves as an effective graph-shift operator.
A novel criterion for ordering graph frequencies is established.
Replication of basic graphs leads to interpolation in the graph frequency domain, analogous to zero-padding.

Conclusions:

The proposed graph-shift operator and frequency definition offer new insights into graph signal analysis.
The findings provide a theoretical basis for understanding spectral properties of periodic and replicated graph structures.
This work bridges concepts from graph theory and digital signal processing.