{"ID":2874022,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.05793","arxiv_id":"2509.05793","title":"Spectral Methods in Complex Systems","abstract":"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.","short_abstract":"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 ident...","url_abs":"https://arxiv.org/abs/2509.05793","url_pdf":"https://arxiv.org/pdf/2509.05793v2","authors":"[\"Francesco Caravelli\"]","published":"2025-09-06T18:03:31Z","proceeding":"cond-mat.stat-mech","tasks":"[\"cond-mat.stat-mech\",\"cs.LG\",\"math-ph\"]","methods":"[]","has_code":false}