Synchronization in Complex Networks Under Uncertainty
- Autores: Lluís Arola-Fernández
- Directores de la Tesis: Alejandro Arenas Moreno (dir. tes.)
- Lectura: En la Universitat Rovira i Virgili ( España ) en 2022
- Idioma: español
- Tribunal Calificador de la Tesis: C. Pérez Vicente (presid.), Marta Sales Pardo (secret.), Ernesto Estrada (voc.)
- Programa de doctorado: Programa de Doctorado en Ingeniería Informática y Matemáticas de la Seguridad por la Universidad Rovira i Virgili
- Materias:
- Enlaces
- Tesis en acceso abierto en: TDX
- Resumen
Non-linearity, complex interaction patterns and interdependencies, emergent properties like collective behaviors, but also chaos and unpredictability, are some of the features that many complex systems share, ranging from the brain to the power-grid and society. An open question that persists throughout different fields is to understand the rich interplay between structure and dynamics. In this thesis, we focus on some theoretical aspects of a paradigmatic complex problem, the synchronization phenomena in networks of coupled oscillators. Collective rhythms emerging from many interacting oscillators appear across all scales of nature, from the steady heartbeat and the recurrent patterns in neuronal activity to the decentralized synchrony in power-grids. The mathematical models describing these processes are remarkably solid and have significantly advanced lately, especially in the mean-field problem, where oscillators are all mutually connected. However, most real networks have complex interactions that difficult the analytical treatment and challenge our ability to predict and control these systems. A general framework is still missing and most existing results rely on numerical and spectral black-boxes that hinder interpretation and mechanistic insights. Also, the information obtained from measurements is usually incomplete and inacurate. Motivated by these limitations, in this thesis we propose a theoretical study of network-coupled oscillators under uncertainty, dealing with partial information in the complex network of interactions. We will borrow some ideas from classical physics and its mathematics and methods, including standard error propagation techniques, Lagrange optimization and entropy maximization, linear algebra, truncated expansions, stability analysis of dynamical systems, etc…. We will combine these classical tools with numerical simulations and more recent techniques suitable to model coupled oscillators, including minimal models of phase oscillators like the Kuramoto model, network generative models like preferential attachment, small-world, stochastic block models and several more, tools from random matrix theory and spectral graph theory, exact and approximate dimensional reductions, mean-field, perturbative and algebraic approaches to non-linear dynamical systems and also datasets of empirical networks to validate our theoretical results.
The aim of the work is to find quantitative predictions, but there is an intencionated bias towards the search for mechanistic explanations that deepen our understanding of the global problem, i.e. the interplay between the structure and the dynamics in networks of coupled oscillators. The most relevant contributions and findings include: i) an error propagation analysis to unveil how a complex network can non-linearly amplify noise in the weights towards the macroscopic onset of synchronization, ii) a composite Laplacian framework applied to the study of optimal synchronization in networks that can balance pair-wise and higher-order interactions, iii) an extended mean-field approach to approximate functionally invariant dynamics by weight-tuning networks with different degree distributions, iv) an exact geometric unfolding of the synchronized state to unveil the mechanistic interplay between structure and dynamics and to tackle decentralized systems, v) a model of a synchronization bomb, where abrupt synchronization transitions emerge by adding or removing single links following an optimal local rule and vi) an explicit connection between the recent algebraic solution of a linear complex-valued oscillator system and the heuristic origin of the Kuramoto model, which leads to a direct estimation of the onset of synchronization in complex networks. The document of the thesis is organized as follows:
Chapter 1 introduces the main ideas behind complex systems and a review of previous research on the theory of synchronization and complex networks. We also discuss the largely unexplored roles of network uncertainty in the modeling aspects, motivating our work in detail. We also summarize the main findings that the reader will find here.
Chapter 2 reviews several well-known mathematical techniques suitable to deal with network-coupled phase-oscillators. We go through the classical solution of the Kuramoto model, the well-known approximations of the onset of synchronization and the tools to deal with optimization problems. We also introduce the Ott-Antonsen and the Collective Coordinate model reductions.
Chapter 3 focuses on the question: how do structural constraints in the complex network affect the range of possible synchronization behaviors? We consider fixed structures and allow fine-tuning or fluctuations in the link weights to explain how network properties as degree heterogeneity and higher-order interactions affect the dynamical range. Some of these heuristic approaches pave the way for further works, including our next results.
Chapter 4 presents the results on the geometric expansion of the synchronized state and the associated proofs. We discuss its implications in our problems and related ones. We perform a convergence analysis and derive a local approximation of synchrony to explain several features that were only understood numerically. In this context, we also predict the Braess’ Paradox (the effect of links removals that improve synchrony) in directed networks exploiting only decentralized information.
Chapter 5 introduces a model of a synchronization bomb, where explosive transitions are induced by perturbations of single links. The framework is a competitive percolation process driven by a local rule, derived using the machinery introduced in chapter 4. We find that phenomena hold in models of chaotic oscillators and cardiac pacemakers, and we provide an analytical characterization in the Kuramoto case. We also discuss the benefits of noise in this decentralized process and the link between explosive synchronization, percolation and optimization in complex networks.
Chapter 6 shows an explicit connection between a recent algebraic approach to the Kuramoto model, and the heuristic derivation of Kuramoto starting from a coupled system of complex oscillators in 1975. These results point towards the potential of using a matrix, complex-valued formalism in several open problems of network synchronization, as the prediction of the critical threshold. As a first step in this line, a novel derivation of the mean-field synchronization onset is presented using a linear algebra framework and a rank-reduction of the complex network.
Chapter 7 concludes this thesis. Overall, this dissertation provides analytical solutions to several key open questions dealing with the interplay structure-dynamics in oscillator networks. We develop robust methods to manage uncertainty and partial information, which lead to theoretical discoveries in different aspects of synchronization dynamics. We reveal the crucial role of prevalent network features like heterogeneity, moludar and bipartite patterns or higher-order interactions in the amplification of
