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Eigenvalue Outliers of Non-Hermitian Random Matrices with a Local Tree Structure
Izaak Neri1,2, Fernando Lucas Metz3
1Max Planck Institute for the Physics of Complex Systems, Nöthnitzerstraße 38, 01187 Dresden, Germany.
Physical Review Letters
|December 8, 2016
Summary
We developed a theory for eigenvalue outliers in sparse non-Hermitian random matrices with tree structures. This theory precisely predicts spectral properties crucial for understanding complex network dynamics and stability.
Area of Science:
- Graph Theory
- Random Matrix Theory
- Network Dynamics
Background:
- Sparse non-Hermitian random matrices are crucial for modeling complex processes on graphs.
- Eigenvalue outliers significantly influence the stationary state and stability of these dynamical processes.
Purpose of the Study:
- To develop a general and exact theory for eigenvalue outliers in random matrices with local tree structures.
- To derive analytical expressions for spectral observables in oriented random graphs.
Main Methods:
- Analysis of adjacency and Laplacian matrices of oriented random graphs.
- Derivation of exact analytical expressions for spectral properties.
Main Results:
- Analytical expressions for eigenvalue outliers, eigenvector element moments, spectral density support, and spectral gap.
- Demonstration of universal expressions for these spectral observables across various oriented random matrices.
Conclusions:
- The developed theory provides exact predictions for spectral outliers in tree-structured random matrices.
- These findings offer a universal framework for analyzing the dynamics and stability of complex networks.
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