Dissertation

Spectral signatures of breaking of ensemble equivalence

This thesis explores Breaking of Ensemble Equivalence (BEE) in random graph models by examining the spectral properties of adjacency matrices. The goal is to identify spectral characteristics that differentiate random graph ensembles, enhancing the understanding of complex network structures and behaviors.

Author

P. Dionigi

Date

19 June 2024

Links

Thesis in Leiden Repository

The research encompasses both theoretical analysis and practical applications, including a chapter on simulations and sampling methods.Chapter 1 introduces basic random graph theory and emphasizes the importance of maximum entropy models in real-world network modeling. It defines BEE and its characterization within statistical mechanics, highlighting the natural differences between canonical and microcanonical ensembles. It then introduces spectral theory of random graphs and why it is utilized to investigate BEE.Chapter 2 proposes a conjecture linking BEE to a gap between the expectations of the largest eigenvalue in the canonical and microcanonical ensembles, proving it for homogeneous graphs.Chapter 3 examines Chung-Lu random graphs, establishing central limit theorems for the largest eigenvalue and its eigenvector.Chapter 4 verifies the conjecture for inhomogeneous graphs, computing the expected largest eigenvalue of the configuration model.Chapter 5 offers numerical evidence through simulations, after a brief introduction to graph sampling. The thesis concludes with a summary of findings and suggestions for future research.

• central limit theorem
• large deviation theory
• random graphs
• random matrix theory
• statistical mechanics