Asymmetry in Spectral Graph Theory: Harmonic Analysis on Directed Networks via Biorthogonal Bases (Adjacency-Operator Formulation)

Classical spectral graph theory and graph signal processing rely on a symmetry principle: undirected graphs induce symmetric (self-adjoint) adjacency/Laplacian operators, yielding orthogonal eigenbases and energy-preserving Fourier expansions. Real-world networks are typically directed and hence asymmetric, producing non-self-adjoint and frequently non-normal operators for which orthogonality fails and spectral coordinates can be ill-conditioned. In this paper we develop an original harmonic-analysis framework for directed networks centered on the \emph{adjacency} operator. We propose a \emph{Biorthogonal Graph Fourier Transform} (BGFT) built from left/right eigenvectors, formulate directed ``frequency’’ and filtering in the non-Hermitian setting, and quantify how asymmetry and non-normality affect stability via condition numbers and a departure-from-normality functional. We prove exact synthesis/analysis identities under diagonalizability, establish sampling-and-reconstruction guarantees for BGFT-bandlimited signals, and derive perturbation/stability bounds that explain why naive orthogonal-GFT assumptions break down on non-normal directed graphs. A simulation protocol compares undirected versus directed cycles (asymmetry without non-normality) and a perturbed directed cycle (genuine non-normality), demonstrating that BGFT yields coherent reconstruction and filtering across asymmetric regimes.

💡 Research Summary

The paper addresses a fundamental limitation of classical graph signal processing (GSP): the reliance on symmetric (undirected) graphs that yield self‑adjoint adjacency or Laplacian operators with orthogonal eigenbases. Real‑world networks—traffic, citation, social, biological—are inherently directed, producing adjacency matrices that are asymmetric and often non‑normal (i.e., A A* ≠ A* A). In such cases orthogonal Fourier bases disappear, eigenvectors become ill‑conditioned, and standard graph Fourier transforms (GFT) lose energy preservation, stability, and reliable filtering.

To overcome these obstacles, the authors retain the adjacency matrix as the primitive shift operator and construct a Biorthogonal Graph Fourier Transform (BGFT) based on left and right eigenvectors of A. For a diagonalizable A, right eigenvectors vₖ satisfy A vₖ = λₖ vₖ, left eigenvectors uₖ satisfy uₖ* A = λₖ uₖ*. By scaling them to satisfy the bi‑orthonormal condition uₖ* vⱼ = δₖⱼ, the analysis operator is defined as  x̂ = U* x and the synthesis operator as x = V x̂, where V =