# Hamiltonian System Approach to Distributed Spectral Decomposition in Networks

## TL;DR

This paper introduces a novel Hamiltonian system approach for distributed spectral decomposition in large networks, modeling graph spectra as physical systems and applying symplectic integrators for higher resolution eigenvalue detection.

## Contribution

It develops efficient distributed algorithms based on Hamiltonian and Lagrangian dynamics to improve spectral resolution and stability in eigenvalue computations for large symmetric graph matrices.

## Key findings

- Effective detection of closely spaced eigenvalues.
- Improved stability in numerical spectral computation.
- Validated on real-world large networks.

## Abstract

Because of the significant increase in size and complexity of the networks, the distributed computation of eigenvalues and eigenvectors of graph matrices has become very challenging and yet it remains as important as before. In this paper we develop efficient distributed algorithms to detect, with higher resolution, closely situated eigenvalues and corresponding eigenvectors of symmetric graph matrices. We model the system of graph spectral computation as physical systems with Lagrangian and Hamiltonian dynamics. The spectrum of Laplacian matrix, in particular, is framed as a classical spring-mass system with Lagrangian dynamics. The spectrum of any general symmetric graph matrix turns out to have a simple connection with quantum systems and it can be thus formulated as a solution to a Schr\"odinger-type differential equation. Taking into account the higher resolution requirement in the spectrum computation and the related stability issues in the numerical solution of the underlying differential equation, we propose the application of symplectic integrators to the calculation of eigenspectrum. The effectiveness of the proposed techniques is demonstrated with numerical simulations on real-world networks of different sizes and complexities.