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Localization and Universality of Eigenvectors in Directed Random Graphs.
Fernando Lucas Metz1, Izaak Neri2
1Physics Institute, Federal University of Rio Grande do Sul, 91501-970 Porto Alegre, Brazil and London Mathematical Laboratory, 18 Margravine Gardens, London W6 8RH, United Kingdom.
Physical Review Letters
|February 12, 2021
Summary
We developed a general theory for directed random graphs, revealing that right eigenvectors localize at low average degrees. This localization transition is independent of degree fluctuations, unlike in undirected graphs.
Area of Science:
- Network theory
- Statistical physics
- Graph theory
Background:
- Spectral properties of random graphs are well-studied.
- Analytic treatment of right eigenvectors in directed graphs remains challenging.
Purpose of the Study:
- To present a general theory for the statistics of right eigenvector components in directed random graphs.
- To analyze localization phenomena and their dependence on degree distribution and connectivity.
Main Methods:
- Developed a general theory for directed random graphs with prescribed degree distribution and random link weights.
- Derived exact analytic expressions for the inverse participation ratio.
- Investigated eigenvector component distributions in low and high connectivity limits.
Main Results:
- Right eigenvectors of directed random graphs with small average degree are localized.
- The critical mean degree for localization is independent of degree fluctuations if the fourth moment of the degree distribution is finite.
- In the high connectivity limit, eigenvector component distributions depend solely on the degree distribution.
Conclusions:
- The study provides an exact analytic treatment for right eigenvectors in directed random graphs.
- Localization properties in directed graphs differ significantly from undirected graphs, particularly concerning degree fluctuations.
- The findings offer insights into the universal and distribution-dependent behaviors of eigenvectors in complex networks.
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