Graph Signal Processing: Vertex Multiplication

TL;DR

This paper introduces vertex multiplication, a novel operation in graph signal processing that assigns a coordinate structure to graphs, establishing a Fourier duality with differentiation and extending classical signal concepts to graph domains.

Contribution

It proposes the vertex multiplication operation for graphs, linking graph vertices to coordinate structures and demonstrating Fourier duality with differentiation.

Findings

Vertex multiplication generalizes coordinate multiplication to graphs.

The operation exhibits Fourier duality with differentiation.

Numerical examples validate the proposed approach.

Abstract

On the Euclidean domains of classical signal processing, linking of signal samples to the underlying coordinate structure is straightforward. While graph adjacency matrices totally define the quantitative associations among the underlying graph vertices, a major problem in graph signal processing is the lack of explicit association of vertices with an underlying quantitative coordinate structure. To make this link, we propose an operation, called the vertex multiplication, which is defined for graphs and can operate on graph signals. Vertex multiplication, which generalizes the coordinate multiplication operation in time series signals, can be interpreted as an operator which assigns a coordinate structure to a graph. By using the graph domain extension of differentiation and graph Fourier transform (GFT), vertex multiplication is defined such that it shows Fourier duality, which states that differentiation and coordinate multiplication operations are duals of each other under Fourier transformation (FT). The proposed definition is shown to reduce to coordinate multiplication for graphs corresponding to time series. Numerical examples are also presented.