Statistical Mechanics for Network Structure and Evolution

In this thesis, we address problems in complex networks using the methods of statistical mechanics and information theory. We particularly focus on the thermodynamic characterisation of networks and entropic analysis on statistics and dynamics of network evolution. After a brief introduction of background and motivation behind the thesis in Chapter 1, we provide a review of relevant literature in Chapter 2, and elaborate the main methods from Chapter 3 to Chapter 6.

In Chapter 3, we explore the normalised Laplacian matrix as the Hamiltonian operator of the network which governs the particle occupations corresponding to Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics. The relevant partition functions derive the thermodynamic quantities in revealing network structural characterisations. Chapter 4 further decomposes the global network entropy in three statistics on edge-connection components. This decompensation reflects the detailed distribution of entropy across the edges of a network, and is particularly useful if the analysis of non-homogeneous networks with a strong community and hub structure is being attempted.

Furthermore, Chapter 5 and Chapter 6 provide the theoretical approaches to analyse the dynamic network evolution and the application of the real-world networks. In Chapter 5, we investigate both undirected and directed network evolution using the Euler-Lagrange equation. This variational principle is based on the von Neumann entropy for time-varying network structure. The presented model not only provides an accurate simulation of the degree statistics in network evolution, but also captures the topological variations taking place when the structure of a network changes violently. Chapter 6 studies the fMRI regional brain interaction networks using directed graphs. We further develop a novel method for characterising networks using Bose-Einstein entropy and the Jensen-Shannon divergence. It offers a high discrimination among patients with suspected Alzheimer's disease. Finally, Chapter 7 concludes the thesis and discusses the limitations of our methodologies, which also supplies the potential research in the future.