# Spectral Methods in Complex Systems

arXiv: https://arxiv.org/abs/2509.05793 · HTML: https://mlsanity.com/paper/2509.05793/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Sep 2025.

## Built on by (most cited first, 1 of 1)

- [Learning Nonlinear Heterogeneity in Physical Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2601.15340/) (2026)

## Abstract (arXiv)

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.