Spectral Methods in Complex Systems
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912577609531392 |
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| author | Caravelli, Francesco |
| author_facet | Caravelli, Francesco |
| contents | These notes offer a unified introduction to spectral methods for the study of complex systems. They are intended as an operative manual rather than a theorem-proof textbook: the emphasis is on tools, identities, and perspectives that can be readily applied across disciplines. Beginning with a compendium of matrix identities and inversion techniques, the text develops the connections between spectra, dynamics, and structure in finite-dimensional systems. Applications range from dynamical stability and random walks on networks to input-output economics, PageRank, epidemic spreading, memristive circuits, synchronization phenomena, and financial stability. Throughout, the guiding principle is that eigenvalues, eigenvectors, and resolvent operators provide a common language linking problems in physics, mathematics, computer science, and beyond. The presentation is informal, accessible to advanced undergraduates, yet broad enough to serve as a reference for researchers interested in spectral approaches to complex systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_05793 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral Methods in Complex Systems Caravelli, Francesco Statistical Mechanics Machine Learning Mathematical Physics These notes offer a unified introduction to spectral methods for the study of complex systems. They are intended as an operative manual rather than a theorem-proof textbook: the emphasis is on tools, identities, and perspectives that can be readily applied across disciplines. Beginning with a compendium of matrix identities and inversion techniques, the text develops the connections between spectra, dynamics, and structure in finite-dimensional systems. Applications range from dynamical stability and random walks on networks to input-output economics, PageRank, epidemic spreading, memristive circuits, synchronization phenomena, and financial stability. Throughout, the guiding principle is that eigenvalues, eigenvectors, and resolvent operators provide a common language linking problems in physics, mathematics, computer science, and beyond. The presentation is informal, accessible to advanced undergraduates, yet broad enough to serve as a reference for researchers interested in spectral approaches to complex systems. |
| title | Spectral Methods in Complex Systems |
| topic | Statistical Mechanics Machine Learning Mathematical Physics |
| url | https://arxiv.org/abs/2509.05793 |