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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.13513 |
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| _version_ | 1866908713042837504 |
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| author | Gokavarapu, Chandrasekhar |
| author_facet | Gokavarapu, Chandrasekhar |
| contents | Classical spectral graph theory relies on the symmetry of the adjacency and Laplacian operators, which guarantees orthogonal eigenbases and energy-preserving Fourier transforms. However, real-world networks are intrinsically directed and asymmetric, resulting in non-normal operators where standard orthogonality assumptions fail. In this paper, we develop a rigorous harmonic analysis framework for directed graphs centered on the \emph{Combinatorial Directed Laplacian} ($L = D_{out} - A$). We construct a \emph{Biorthogonal Graph Fourier Transform} (BGFT) using dual left and right eigenbases, and introduce a directed variational semi-norm based on the operator norm $\|Lx\|_2$ rather than the quadratic form. We derive exact Parseval-type bounds that quantify the energy distortion induced by the non-normality of the graph, explicitly linking signal reconstruction stability to the condition number of the eigenvector matrix, $κ(V)$. Finally, we present experimental validation comparing normal directed cycles against non-normal perturbed topologies, demonstrating that while the BGFT provides exact reconstruction in ideal regimes, the geometric departure from normality acts as the fundamental limit on filter stability in directed networks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_13513 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Harmonic Analysis on Directed Networks: A Biorthogonal Laplacian Framework for Non-Normal Graphs Gokavarapu, Chandrasekhar Rings and Algebras Primary: 05C50, Secondary: 15A18, 94A12, 65F15 Classical spectral graph theory relies on the symmetry of the adjacency and Laplacian operators, which guarantees orthogonal eigenbases and energy-preserving Fourier transforms. However, real-world networks are intrinsically directed and asymmetric, resulting in non-normal operators where standard orthogonality assumptions fail. In this paper, we develop a rigorous harmonic analysis framework for directed graphs centered on the \emph{Combinatorial Directed Laplacian} ($L = D_{out} - A$). We construct a \emph{Biorthogonal Graph Fourier Transform} (BGFT) using dual left and right eigenbases, and introduce a directed variational semi-norm based on the operator norm $\|Lx\|_2$ rather than the quadratic form. We derive exact Parseval-type bounds that quantify the energy distortion induced by the non-normality of the graph, explicitly linking signal reconstruction stability to the condition number of the eigenvector matrix, $κ(V)$. Finally, we present experimental validation comparing normal directed cycles against non-normal perturbed topologies, demonstrating that while the BGFT provides exact reconstruction in ideal regimes, the geometric departure from normality acts as the fundamental limit on filter stability in directed networks. |
| title | Harmonic Analysis on Directed Networks: A Biorthogonal Laplacian Framework for Non-Normal Graphs |
| topic | Rings and Algebras Primary: 05C50, Secondary: 15A18, 94A12, 65F15 |
| url | https://arxiv.org/abs/2512.13513 |