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... Show moreThis 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. 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. Show less
A Chung–Lu random graph is an inhomogeneous Erdős–Rényi random graph in which vertices are assigned average degrees, and pairs of vertices are connected by an edge with a probability that is... Show moreA Chung–Lu random graph is an inhomogeneous Erdős–Rényi random graph in which vertices are assigned average degrees, and pairs of vertices are connected by an edge with a probability that is proportional to the product of their average degrees, independently for different edges. We derive a central limit theorem for the principal eigenvalue and the components of the principal eigenvector of the adjacency matrix of a Chung–Lu random graph. Our derivation requires certain assumptions on the average degrees that guarantee connectivity, sparsity and bounded inhomogeneity of the graph. Show less
