Harmonic Analysis for Directed Networks

This paper addresses the challenges of analyzing diffusion processes on directed networks, where the standard tools of spectral graph theory (which rely on symmetry) are not directly applicable. It introduces a Biorthogonal Graph Fourier Transform (BGFT) using biorthogonal eigenvectors to handle the non-self-adjoint nature of the Markov transition operator in directed graphs. The paper's significance lies in providing a framework for understanding stability and signal processing in these complex systems, going beyond the limitations of traditional methods.

• •Develops a harmonic analysis framework for directed graphs using a Biorthogonal Graph Fourier Transform (BGFT).
• •Addresses the non-self-adjoint nature of the Markov transition operator in directed networks.
• •Provides sampling and reconstruction theorems for bandlimited signals.
• •Quantifies noise amplification through eigenvector conditioning.
• •Demonstrates that non-normality and eigenvector ill-conditioning drive reconstruction sensitivity.