Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2601.00464 |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866914368459898880 |
|---|---|
| author | Gokavarapu, Chandrasekhar Chinnam, Komala Lakshmi |
| author_facet | Gokavarapu, Chandrasekhar Chinnam, Komala Lakshmi |
| contents | Spectral graph signal processing is traditionally built on self-adjoint Laplacians, where orthogonal eigenbases yield an energy-preserving Fourier transform and a variational frequency ordering via a real Dirichlet form. Directed networks break self-adjointness: the combinatorial directed Laplacian $L=D_{\mathrm{out}}-A$ is generally non-normal, so eigenvectors are non-orthogonal and classical Parseval identities and Rayleigh-quotient orderings do not apply. This paper develops a Laplacian-centric harmonic analysis for directed graphs that remains exact at the algebraic level while explicitly quantifying the geometric distortion induced by non-normality. We (i) define a Biorthogonal Graph Fourier Transform (BGFT) for $L$ using dual left/right eigenbases and show that vertex energy equals a Gram-metric quadratic form in BGFT coordinates, (ii) introduce a directed variational semi-norm $TV_{\mathcal{G}}(x)=\|Lx\|_2^2$ and prove sharp two-sided BGFT-domain bounds controlled by singular values of the eigenvector matrix, and (iii) derive sampling and reconstruction guarantees with explicit stability constants that separate sampling-set informativeness from eigenvector geometry. Finally, we provide reproducible simulations comparing a normal directed cycle to perturbed non-normal digraphs and show that filtering and reconstruction robustness track $κ(V)$ and the Henrici departure-from-normality $Δ(L)$, validating the theoretical predictions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_00464 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Harmonic Analysis on Directed Networks via a Biorthogonal Laplacian Calculus for Non-Normal Digraphs Gokavarapu, Chandrasekhar Chinnam, Komala Lakshmi Computational Engineering, Finance, and Science 05C50, 15A18, 94A12, 65F15 Spectral graph signal processing is traditionally built on self-adjoint Laplacians, where orthogonal eigenbases yield an energy-preserving Fourier transform and a variational frequency ordering via a real Dirichlet form. Directed networks break self-adjointness: the combinatorial directed Laplacian $L=D_{\mathrm{out}}-A$ is generally non-normal, so eigenvectors are non-orthogonal and classical Parseval identities and Rayleigh-quotient orderings do not apply. This paper develops a Laplacian-centric harmonic analysis for directed graphs that remains exact at the algebraic level while explicitly quantifying the geometric distortion induced by non-normality. We (i) define a Biorthogonal Graph Fourier Transform (BGFT) for $L$ using dual left/right eigenbases and show that vertex energy equals a Gram-metric quadratic form in BGFT coordinates, (ii) introduce a directed variational semi-norm $TV_{\mathcal{G}}(x)=\|Lx\|_2^2$ and prove sharp two-sided BGFT-domain bounds controlled by singular values of the eigenvector matrix, and (iii) derive sampling and reconstruction guarantees with explicit stability constants that separate sampling-set informativeness from eigenvector geometry. Finally, we provide reproducible simulations comparing a normal directed cycle to perturbed non-normal digraphs and show that filtering and reconstruction robustness track $κ(V)$ and the Henrici departure-from-normality $Δ(L)$, validating the theoretical predictions. |
| title | Harmonic Analysis on Directed Networks via a Biorthogonal Laplacian Calculus for Non-Normal Digraphs |
| topic | Computational Engineering, Finance, and Science 05C50, 15A18, 94A12, 65F15 |
| url | https://arxiv.org/abs/2601.00464 |