Spectral Methods in Complex Systems

• URL: http://arxiv.org/abs/2509.05793v2
• Date: Tue, 09 Sep 2025 10:39:54 GMT
• Title: Spectral Methods in Complex Systems
• Abstract summary: These notes offer a unified introduction to spectral methods for the study of complex systems.<n>The emphasis is on tools, identities, and perspectives that can be readily applied across disciplines.<n>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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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