Structural engineering of evolving complex dynamical networks

Networks are ubiquitous in nature and many natural and man-made systems can be modelled as networked systems. Complex networks, systems comprising a number of nodes that are connected through edges, have been frequently used to model large-scale systems from various disciplines such as biology, ecology, and engineering. Dynamical systems interacting through a network may exhibit collective behaviours such as synchronisation, consensus, opinion formation, flocking and unusual phase transitions. Evolution of such collective behaviours is highly dependent on the structure of the interaction network. Optimisation of network topology to improve collective behaviours and network robustness can be achieved by intelligently modifying the network structure. Here, it is referred to as "Engineering of the Network". Although coupled dynamical systems can develop spontaneous synchronous patterns if their coupling strength lies in an appropriate range, in some applications one needs to control a fraction of nodes, known as driver nodes, in order to facilitate the synchrony. This thesis addresses the problem of identifying the set of best drivers, leading to the best pinning control performance. The eigen-ratio of the augmented Laplacian matrix, that is the largest eigenvalue divided by the second smallest one, is chosen as the controllability metric. The approach introduced in this thesis is to obtain the set of optimal drivers based on sensitivity analysis of the eigen-ratio, which requires only a single computation of the eigenvector associated with the largest eigenvalue, and thus is applicable for large-scale networks. This leads to a new "controllability centrality" metric for each subset of nodes. Simulation results reveal the effectiveness of the proposed metric in predicting the most important driver(s) correctly.     Interactions in complex networks might also facilitate the propagation of undesired effects, such as node/edge failure, which may crucially affect the performance of collective behaviours. In order to study the effect of node failure on network synchronisation, an analytical metric is proposed that measures the effect of a node removal on any desired eigenvalue of the Laplacian matrix. Using this metric, which is based on the local multiplicity of each eigenvalue at each node, one can approximate the impact of any node removal on the spectrum of a graph. The metric is computationally efficient as it only needs a single eigen-decomposition of the Laplacian matrix. It also provides a reliable approximation for the "Laplacian energy" of a network. Simulation results verify the accuracy of this metric in networks with different topologies. This thesis also considers formation control as an application of network synchronisation and studies the "rigidity maintenance" problem, which is one of the major challenges in this field. This problem is to preserve the rigidity of the sensing graph in a formation during motion, taking into consideration constraints such as line-of-sight requirements, sensing ranges and power limitations. By introducing a "Lattice of Configurations" for each node, a distributed rigidity maintenance algorithm is proposed to preserve the rigidity of the sensing network when failure in a sensing link would result in loss of rigidity. The proposed algorithm recovers rigidity by activating, almost always, the minimum number of new sensing links and considers real-time constraints of practical formations. A sufficient condition for this problem is proved and tested via numerical simulations. Based on the above results, a number of other areas and applications of network dynamics are studied and expounded upon in this thesis

Networked Dynamical Systems: Privacy, Control, and Cognition

Many natural and man-made systems, ranging from thenervous system to power and transportation grids to societies, exhibitdynamic behaviors that evolve over a sparse and complex network. This networked aspect raises significant challenges and opportunities for the identification, analysis, and control of such dynamic behaviors. While some of these challenges emanate from the networked aspect \emph{per se} (such as the sparsity of connections between system components and the interplay between nodal \emph{communication} and network dynamics), various challenges arise from the specific application areas (such as privacy concerns in cyber-physical systems or the need for \emph{scalable} algorithm designs due to the large size of various biological and engineered networks). On the other hand, networked systems provide significant opportunities and allow for performance and robustness levels that are far beyond reach for centralized systems, with examples ranging from the Internet (of Things) to the smart grid and the brain. This dissertation aims to address several of these challenges and harness these opportunities. The dissertation is divided into three parts. In the first part, we study privacy concerns whose resolution is vital for the utility of networked cyber-physical systems. We study the problems of average consensus and convex optimization as two principal distributed computations occurring over networks and design algorithm with rigorous privacy guarantees that provide a \emph{best achievable} tradeoff between network utility and privacy. In the second part, we analyze networks with resource constraints. More specifically, we study three problems of stabilization under communication (bandwidth and latency) limitations in sensing and actuation, optimal time-varying control scheduling problem under limited number of actuators and control energy, and the structure identification problem of under-sensed networks (i.e., networks with latent nodes). Finally in the last part, we focus on the intersection of networked dynamical systems and neuroscience and draw connections between brain network dynamics and two extensively studied but yet not fully understood neuro-cognitive phenomena: goal-driven selective attention and neural oscillations. Using a novel axiomatic approach, we establish these connections in the form of necessary and/or sufficient conditions on the network structure that match the network output trajectories with experimentally observed brain activity

Contagion dynamics in multilevel and structured populations.

Existen numerosos procesos de contagio sobre redes, como la propagación de epidemias, los rumores, la información u otros fenómenos no lineales propios de los sistemas complejos humanos. Desde la perspectiva de la modelización matemática, los procesos de contagio en poblaciones estructuradas se están volviendo cada vez más sofisticados en lo que respecta al tipo de interacciones no triviales involucradas en ellos. Los modelos han evolucionado desde los simples métodos compartimentales a modelos estructurados en los que se tienen en cuenta las heterogeneidades de la población. Además, para visualizar estas jerarquías y heterogeneidades de los sistemas complejos humanos, también consideramos la representación multicapa de las poblaciones. En esta tesis, intentamos explorar la punta del iceberg en lo que respecta a procesos de contagio sobre poblaciones basándonos en varios modelos matemáticos. Nuestro objetivo es entender la complejidad de las dinámicas de contagio en poblaciones estructuradas y multinivel.En el primer capítulo, nos centramos en presentar el desarrollo de algunas de las teorías principales que se utilizan para estudiar los sistemas complejos. El descubrimiento de las interacciones no lineales hizo que le método del reduccionismo fuese cuestionado, dado que el comportamiento general no puede describirse como una simple superposición de pequeñas escalas. La ciencia de redes busca caracterizar los sistemas complejos de diversos campos. Al mismo tiempo, la teoría de grafos proporciona las herramientas matemáticas necesarias para describir redes realistas. Discutiremos algunas de las cantidades fundamentales y las métricas más relevantes para la caracterización de la estructura de la red, así como varios ejemplos de modelos de red. Además, repasaremos brevemente los principios básicos de las redes multicapa que rompen la limitación de un solo tipo de conexión existente en las redes monocapa, estableciendo la base para explorar y generalizar estos conceptos.A continuación, estudiaremos procesos dinámicos comenzando por una breve introducción a los modelos matemáticos que se usarán durante el resto de la tesis. En el caso de la ecuación maestra, resaltaremos el rol de los procesos de Markov así como la aproximación de campo medio, sin centrarnos en sus soluciones completas. Los métodos de modelización y las reglas de actualización que se utilizan en las simulaciones numéricas también se presentan en detalle. En esta tesis, nos centraremos en el problema de la propagación de epidemias sobre redes, un tema que despierta gran interés en el campo de los procesos de propagación y contagio. Después de revisar las propiedades y los resultados teóricos de algunos de los modelos epidemiológicos típicos, con varias simplificaciones desde el punto de vista matemático, exploraremos varias medidas importantes en el campo de la epidemiología, i.e., el número reproductivo básico y la inmunidad de grupo. Después, implementaremos un modelo clásico de epidemias sobre redes multicapa para explorar el papel que juega la direccionalidad utilizando funciones generatrices. Terminaremos el capítulo 2 modelizando un tipo especial de procesos de contagio social, en particular, utilizaremos un modelo compartimental para estudiar la propagación de la corrupción. Prestaremos atención a las condiciones críticas para que surja este tipo de comportamiento desarrollando la aproximación de campo medio y comparando sus predicciones con simulaciones. Es más, extenderemos el modelo de corrupción a un sistema de dos capas en el que los flujos de contagio pueden ser diferentes en cada capa para investigar el papel que juega el solapamiento de enlaces y las correlaciones de grado entre capas en la evolución de las actividades honestas y corruptas.Resulta evidente que la complejidad de los sistemas humanos del mundo real afecta la precisión con la que se pueden predecir las epidemias y algunas propiedades específicas de los sistemas. Sin embargo, debido al desarrollo de la ciencia de datos, fuentes de datos masivas y muy informativas pueden utilizarse para enriquecer la topología de la red de forma que se acerque a los sistemas reales. En al tercera parte de esta tesis, comenzaremos describiendo los retos y las oportunidades que han surgido durante el desarrollo de la ciencia de datos. A continuación, intentaremos conseguir una imagen más realista de la estructura interna de las redes de contacto utilizando datos reales. Además, ilustraremos la importancia de utilizar una perspectiva conducida por los datos en lo que respecta a la modelización de redes a la hora de estudiar la propagación de epidemias en redes de contacto. En este caso, la variabilidad de patrones de interacción que surge de la heterogeneidad de la población, sus comportamientos sociales, etc. puede ser capturada correctamente.Bajo este mismo desarrollo teórico, consideraremos la edad de los individuos y sus patrones de interacción social para generar redes multicapa con estructura de edad para estudiar el problema de la inmunidad de grupo del SARS-CoV-2 y evaluar el impacto que tres estrategias de vacunación pueden tener a la hora de eliminar la transmisión de la panedmia. Después, para explorar la dinámica de las enfermedades que se propagan en entornos hospitalarios (HAI, por sus siglas en inglés) cuando los pacientes están recibiendo tratamiento en ellos, utilizaremos una colección de datos espacio-temporales recogida en tres hospitales de Canadá para generar las redes de interacción entre los trabajadores hospitalarios (HCWs). Nos centraremos en determinar cuantitativemente el riesgo de que las HAIs se propaguen por las diferentes unidades de un hospital y los varios grupos de HCWs, respectivamente. Calcularemos el riesgo de las unidades espaciales usando el tiempo de llegada de la enfermedad y el número de infecciones producidas en cada unidad. En el caso de los HCWs, la probabilidad de infectarse y el número de reproducción efectivo son usados como indicador del riesgo de HCWs.Concluiremos la tesis presentando nuestras conclusiones y discutiendo algunos de los restos que quedan por explorar en el futuro.<br /

The structure and dynamics of multilayer networks

In the past years, network theory has successfully characterized the interaction among the constituents of a variety of complex systems, ranging from biological to technological, and social systems. However, up until recently, attention was almost exclusively given to networks in which all components were treated on equivalent footing, while neglecting all the extra information about the temporal- or context-related properties of the interactions under study. Only in the last years, taking advantage of the enhanced resolution in real data sets, network scientists have directed their interest to the multiplex character of real-world systems, and explicitly considered the time-varying and multilayer nature of networks. We offer here a comprehensive review on both structural and dynamical organization of graphs made of diverse relationships (layers) between its constituents, and cover several relevant issues, from a full redefinition of the basic structural measures, to understanding how the multilayer nature of the network affects processes and dynamics.Comment: In Press, Accepted Manuscript, Physics Reports 201

Topology Reconstruction of Dynamical Networks via Constrained Lyapunov Equations

The network structure (or topology) of a dynamical network is often unavailable or uncertain. Hence, we consider the problem of network reconstruction. Network reconstruction aims at inferring the topology of a dynamical network using measurements obtained from the network. In this technical note we define the notion of solvability of the network reconstruction problem. Subsequently, we provide necessary and sufficient conditions under which the network reconstruction problem is solvable. Finally, using constrained Lyapunov equations, we establish novel network reconstruction algorithms, applicable to general dynamical networks. We also provide specialized algorithms for specific network dynamics, such as the well-known consensus and adjacency dynamics.Comment: 8 page

Statistical physics of vaccination

Historically, infectious diseases caused considerable damage to human societies, and they continue to do so today. To help reduce their impact, mathematical models of disease transmission have been studied to help understand disease dynamics and inform prevention strategies. Vaccination–one of the most important preventive measures of modern times–is of great interest both theoretically and empirically. And in contrast to traditional approaches, recent research increasingly explores the pivotal implications of individual behavior and heterogeneous contact patterns in populations. Our report reviews the developmental arc of theoretical epidemiology with emphasis on vaccination, as it led from classical models assuming homogeneously mixing (mean-field) populations and ignoring human behavior, to recent models that account for behavioral feedback and/or population spatial/social structure. Many of the methods used originated in statistical physics, such as lattice and network models, and their associated analytical frameworks. Similarly, the feedback loop between vaccinating behavior and disease propagation forms a coupled nonlinear system with analogs in physics. We also review the new paradigm of digital epidemiology, wherein sources of digital data such as online social media are mined for high-resolution information on epidemiologically relevant individual behavior. Armed with the tools and concepts of statistical physics, and further assisted by new sources of digital data, models that capture nonlinear interactions between behavior and disease dynamics offer a novel way of modeling real-world phenomena, and can help improve health outcomes. We conclude the review by discussing open problems in the field and promising directions for future research

A systems approach to analyze the robustness of infrastructure networks to complex spatial hazards

Ph. D. ThesisInfrastructure networks such as water supply systems, power networks, railway networks, and road networks provide essential services that underpin modern society’s health, wealth, security, and wellbeing. However, infrastructures are susceptible to damage and disruption caused by extreme weather events such as floods and windstorms. For instance, in 2007, extensive disruption was caused by floods affecting a number of electricity substations in the United Kingdom, resulting in an estimated damage of GBP£3.18bn (US$4bn). In 2017, Hurricane Harvey hit the Southern United States, causing an approximated US$125bn (GBP£99.35bn) in damage due to the resulting floods and high winds. The magnitude of these impacts is at risk of being compounded by the effects of Climate Change, which is projected to increase the frequency of extreme weather events. As a result, it is anticipated that an estimated US$3.7tn (GBP£2.9tn) in investment will be required, per year, to meet the expected need between 2019 and 2035. A key reason for the susceptibility of infrastructure networks to extreme weather events is the wide area that needs to be covered to provide essential services. For example, in the United Kingdom alone there are over 800,000 km of overhead electricity cables, suggesting that the footprint of infrastructure networks can be as extended as that of an entire Country. These networks possess different spatial structures and attributes, as a result of their evolution over long timeframes, and respond to damage and disruption in different and complex ways. Existing approaches to understanding the impact of hazards on infrastructure networks typically either (i) use computationally expensive models, which are unable to support the investigation of enough events and scenarios to draw general insights, or (ii) use low complexity representations of hazards, with little or no consideration of their spatial properties. Consequently, this has limited the understanding of the relationship between spatial hazards, the spatial form and connectivity of infrastructure networks, and infrastructure reliability. This thesis investigates these aspects through a systemic modelling approach, applied to a synthetic and a real case study, to evaluate the response of infrastructure networks to spatially complex hazards against a series of robustness metrics. In the first case study, non-deterministic spatial hazards are generated by a fractal method which allows to control their spatial variability, resulting in spatial configurations that very closely resemble natural phenomena such as floods or windstorms. These hazards are then superimposed on a range of synthetic network layouts, which have spatial structures consistent with real infrastructure networks reported in the literature. Failure of network components is initially determined as a function of hazard intensity, and cascading failure of further components is also investigated. The performance of different infrastructure configurations is captured by an array of metrics which cover different aspects of robustness, ranging from the proneness to partitioning to the ability to process flows in the face of disruptions. Whereas analyses to date have largely adopted low complexity representations of hazards, this thesis shows that consideration of a high complexity representation which includes hazard spatial variability can reduce the robustness of the infrastructure network by nearly 40%. A “small-world” network, in which each node is within a limited number of steps from any other node, is shown to be the most robust of all the modelled networks to the different structures of spatial hazard. The second case study uses real data to assess the robustness of a power supply network operating in the Hull region in the United Kingdom, which is split in high and low voltage lines. The spatial hazard is represented by a high-resolution wind gust model and tested under current and future climate scenarios. The analysis reveals how the high and low voltage lines interact with each other in the event of faults, which lines would benefit the most from increased robustness, and which are most exposed to cascading failures. The second case study also reveals the importance of the spatial footprint of the hazard relative to the location of the infrastructure, and how particular hazard patterns can affect low voltage lines that are more often located in exposed areas at the edge of the network. The impact of Climate Change on windstorms is highly uncertain, although it could further reduce network robustness due to more severe events. Overall the two case studies provide important insights for infrastructure designers, asset managers, the academic sector, and practitioners in general. In fact, in the first case study, this thesis defines important design principles, such as the adoption of a small-world network layout, that can integrate the traditional design drivers of demand, efficiency, and cost. In the second case study, this thesis lays out a methodology that can help identify assets requiring increased robustness and protection against cascading failures, resulting in more effective prioritized infrastructure investments and adaptation plans