Graph Signal Processing of Indefinite and Complex Graphs using Directed Variation

TL;DR

This paper extends graph signal processing techniques to indefinite and complex-valued directed graphs, introducing a new graph Fourier transform and demonstrating its effectiveness through simulations and a biological case study.

Contribution

It develops a generalized graph Fourier transform for indefinite and complex graphs, addressing challenges posed by asymmetric and complex weights in directed graphs.

Findings

Effective GFT for indefinite and complex graphs demonstrated

Simulation results show promising signal analysis capabilities

Case study on fruit fly connectome validates approach

Abstract

In the field of graph signal processing (GSP), directed graphs present a particular challenge for the "standard approaches" of GSP to due to their asymmetric nature. The presence of negative- or complex-weight directed edges, a graphical structure used in fields such as neuroscience, critical infrastructure, and robot coordination, further complicates the issue. Recent results generalized the total variation of a graph signal to that of directed variation as a motivating principle for developing a graphical Fourier transform (GFT). Here, we extend these techniques to concepts of signal variation appropriate for indefinite and complex-valued graphs and use them to define a GFT for these classes of graph. Simulation results on random graphs are presented, as well as a case study of a portion of the fruit fly connectome.