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Eigenvalue tunneling and decay of quenched random network.
V Avetisov1,2, M Hovhannisyan3, A Gorsky4,5
1N. N. Semenov Institute of Chemical Physics of the Russian Academy of Sciences, 119991 Moscow, Russia.
Physical Review. E
|January 14, 2017
Summary
Large random graphs defragment into cliques above a critical threshold. This study analyzes the spectral properties and block-diagonal structure of these emergent cliques in Erdős-Rényi networks.
Area of Science:
- Statistical physics and network science.
- Graph theory and random matrix theory.
Background:
- Erdős-Rényi (ER) random graphs are fundamental models in network science.
- Understanding the structural properties of large random networks is crucial.
Purpose of the Study:
- To investigate the defragmentation process in large-N ER random topological graphs.
- To analyze the spectral density evolution and ground state structure with increasing fugacity.
Main Methods:
- Canonical ensemble of N-vertex ER random topological graphs.
- Analysis of adjacency matrix spectral density (ρ(λ)) and eigenvalue tunneling.
- Study of network ground state properties and block-diagonal structure.
Main Results:
- Complete defragmentation into almost full subgraphs (cliques) observed above critical fugacity (μc).
- Multizonal support formation in spectral density for μ > μc.
- Ground state adjacency matrix exhibits block-diagonal form with fluctuating block sizes.
Conclusions:
- The study reveals a phase transition towards defragmentation in random graphs.
- Spectral properties and eigenvalue tunneling provide insights into clique formation.
- Findings offer a framework for understanding complex network structures and potential physical applications.
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