Spectral analysis of inhomogeneous network models

Abstract

This thesis investigates inhomogeneous complex network models using spectral analysis, with a focus on characterising limiting spectral distributions (LSD) of random graphs with vertex inhomogeneity. Chapter 1 provides a preface and literature review. Chapter 2 examines a generalisation of the classical Erdős–Rényi Random Graph, where edge probabilities depend on a function of the vertex weights and a sparsity parameter. The LSD is derived using the method of moments and the Stieltjes transform. Chapters 3 and 4 extend the analysis to random graphs on a discrete torus with power-law vertex weights and an underlying geometry similar to long-range percolation. Here, connection probabilities depend directly on a function of the vertex weights and are scaled inversely to the torus distance. The LSDs of the adjacency matrix (Chapter 3) and the Laplacian (Chapter 4) are characterised. Chapter 5 concludes with open problems and directions for future research.