Asymmetry in Spectral Graph Theory: Harmonic Analysis on Directed Networks via Biorthogonal Bases (Adjacency-Operator Formulation)
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910020416831488 |
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| author | Gokavarapu, Chandrasekhar |
| author_facet | Gokavarapu, Chandrasekhar |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_12226 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymmetry in Spectral Graph Theory: Harmonic Analysis on Directed Networks via Biorthogonal Bases (Adjacency-Operator Formulation) Gokavarapu, Chandrasekhar Rings and Algebras Information Theory Primary:94A12, Secondary: 05C50, 15A18, 65F15, 94A20 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. |
| title | Asymmetry in Spectral Graph Theory: Harmonic Analysis on Directed Networks via Biorthogonal Bases (Adjacency-Operator Formulation) |
| topic | Rings and Algebras Information Theory Primary:94A12, Secondary: 05C50, 15A18, 65F15, 94A20 |
| url | https://arxiv.org/abs/2512.12226 |