Epidemic processes in complex networks

In recent years the research community has accumulated overwhelming evidence for the emergence of complex and heterogeneous connectivity patterns in a wide range of biological and sociotechnical systems. The complex properties of real-world networks have a profound impact on the behavior of equilibrium and nonequilibrium phenomena occurring in various systems, and the study of epidemic spreading is central to our understanding of the unfolding of dynamical processes in complex networks. The theoretical analysis of epidemic spreading in heterogeneous networks requires the development of novel analytical frameworks, and it has produced results of conceptual and practical relevance. A coherent and comprehensive review of the vast research activity concerning epidemic processes is presented, detailing the successful theoretical approaches as well as making their limits and assumptions clear. Physicists, mathematicians, epidemiologists, computer, and social scientists share a common interest in studying epidemic spreading and rely on similar models for the description of the diffusion of pathogens, knowledge, and innovation. For this reason, while focusing on the main results and the paradigmatic models in infectious disease modeling, the major results concerning generalized social contagion processes are also presented. Finally, the research activity at the forefront in the study of epidemic spreading in coevolving, coupled, and time-varying networks is reported.Comment: 62 pages, 15 figures, final versio

Metastability in Stochastic Replicator Dynamics

We consider a novel model of stochastic replicator dynamics for potential games that converts to a Langevin equation on a sphere after a change of variables. This is distinct from the models studied earlier. In particular, it is ill-posed due to non-uniqueness of solutions, but is amenable to a natural selection principle that picks a unique solution. The model allows us to make specific statements regarding metastable states such as small noise asymptotics for mean exit times from their domain of attraction, and quasi-stationary measures. We illustrate the general results by specializing them to replicator dynamics on graphs and demonstrate that the numerical experiments support theoretical predictions.Comment: 39 pages, 7 figure

Asymptotic Expansions for Stationary Distributions of Perturbed Semi-Markov Processes

New algorithms for computing of asymptotic expansions for stationary distributions of nonlinearly perturbed semi-Markov processes are presented. The algorithms are based on special techniques of sequential phase space reduction, which can be applied to processes with asymptotically coupled and uncoupled finite phase spaces.Comment: 83 page

Complex Systems: Nonlinearity and Structural Complexity in spatially extended and discrete systems

Resumen Esta Tesis doctoral aborda el estudio de sistemas de muchos elementos (sistemas discretos) interactuantes. La fenomenología presente en estos sistemas esta dada por la presencia de dos ingredientes fundamentales: (i) Complejidad dinámica: Las ecuaciones del movimiento que rigen la evolución de los constituyentes son no lineales de manera que raramente podremos encontrar soluciones analíticas. En el espacio de fases de estos sistemas pueden coexistir diferentes tipos de trayectorias dinámicas (multiestabilidad) y su topología puede variar enormemente dependiendo de dos parámetros usados en las ecuaciones. La conjunción de dinámica no lineal y sistemas de muchos grados de libertad (como los que aquí se estudian) da lugar a propiedades emergentes como la existencia de soluciones localizadas en el espacio, sincronización, caos espacio-temporal, formación de patrones, etc... (ii) Complejidad estructural: Se refiere a la existencia de un alto grado de aleatoriedad en el patrón de las interacciones entre los componentes. En la mayoría de los sistemas estudiados esta aleatoriedad se presenta de forma que la descripción de la influencia del entorno sobre un único elemento del sistema no puede describirse mediante una aproximación de campo medio. El estudio de estos dos ingredientes en sistemas extendidos se realizará de forma separada (Partes I y II de esta Tesis) y conjunta (Parte III). Si bien en los dos primeros casos la fenomenología introducida por cada fuente de complejidad viene siendo objeto de amplios estudios independientes a lo largo de los últimos años, la conjunción de ambas da lugar a un campo abierto y enormemente prometedor, donde la interdisciplinariedad concerniente a los campos de aplicación implica un amplio esfuerzo de diversas comunidades científicas. En particular, este es el caso del estudio de la dinámica en sistemas biológicos cuyo análisis es difícil de abordar con técnicas exclusivas de la Bioquímica, la Física Estadística o la Física Matemática. En definitiva, el objetivo marcado en esta Tesis es estudiar por separado dos fuentes de complejidad inherentes a muchos sistemas de interés para, finalmente, estar en disposición de atacar con nuevas perspectivas problemas relevantes para la Física de procesos celulares, la Neurociencia, Dinámica Evolutiva, etc..

Economic Networks: Theory and Computation

This textbook is an introduction to economic networks, intended for students and researchers in the fields of economics and applied mathematics. The textbook emphasizes quantitative modeling, with the main underlying tools being graph theory, linear algebra, fixed point theory and programming. The text is suitable for a one-semester course, taught either to advanced undergraduate students who are comfortable with linear algebra or to beginning graduate students.Comment: Textbook homepage is https://quantecon.github.io/book-networks/intro.htm

Essays on the economics of networks

Networks (collections of nodes or vertices and graphs capturing their linkages) are a common object of study across a range of fields includ- ing economics, statistics and computer science. Network analysis is often based around capturing the overall structure of the network by some reduced set of parameters. Canonically, this has focused on the notion of centrality. There are many measures of centrality, mostly based around statistical analysis of the linkages between nodes on the network. However, another common approach has been through the use of eigenfunction analysis of the centrality matrix. My the- sis focuses on eigencentrality as a property, paying particular focus to equilibrium behaviour when the network structure is fixed. This occurs when nodes are either passive, such as for web-searches or queueing models or when they represent active optimizing agents in network games. The major contribution of my thesis is in the applica- tion of relatively recent innovations in matrix derivatives to centrality measurements and equilibria within games that are function of those measurements. I present a series of new results on the stability of eigencentrality measures and provide some examples of applications to a number of real world examples

Mathematical programming based approaches for classes of complex network problems : economical and sociological applications

The thesis deals with the theoretical and practical study of mathematical programming methodologies to the analysis complex networks and their application in economic and social problems. More specifically, it applies models and methods for solving linear and integer programming problems to network models exploiting the matrix structure of such models, resulting in efficient computational procedures and small processing time. As a consequence, it allows the study of larger and more complex networks models that arise in many economical and sociological applications. The main efforts have been addressed to the development of a rigorous mathematical programming based framework, which is able to capture many classes of complex network problems. Such a framework involves a general and flexible modeling approach, based on linear and integer programmin, as well as a collection of efficient probabilistic procedures to deal with these models. The computer implementation has been carried out by high level programming languages, such as Java, MatLab, R and AMPL. The final chapter of the thesis introduced an extension of the analyzed model to the case of microeconomic interaction, providing a fruitful mathematical linkage between its optimization-like properties and its multi-agents properties. The theoretical and practical use of optimization methods represents the trait-de-union of the different chapters. The overall structure of the thesis manuscript contains three parts: Part I: The fine-grained structure of complex networks: theories, models and methods; Chapter 1 and Chapter 2. Part II: Mathematical Programming based approaches for random models of network formation; Chapter 3, Chapter 4 and Chapter 5. Part III: Strategic models of network formation. Chapter 6. Results of this research have generated four working papers in quality scientific journals: one has been accepted and three are under review. Some results have been also presented in four international conferences.La tesis aborda el estudio teórico y práctico de las metodologías de programación matemática para el análisis de redes complejas y su aplicación a problemas económicos y sociales. Más específicamente, se aplica modelos y métodos para resolver problemas de programación lineal y de programación lineal entera explotando las estructuras matriciales de tales modelos, lo que resulta en procedimientos computacionales eficientes y bajo coste de procesamiento. Como consecuencia de ello, las metodologías propuestas permiten el estudio de modelos complejos de gran dimensión, para redes complejas que surgen en muchas aplicaciones económicas y sociológicas. Los principales esfuerzos se han dirigido al desarrollo de un marco teórico basado en la programación matemática, que es capaz de capturar muchas clases de problemas de redes complejas. Dicho marco teórico envuelve un sistema general y flexible de modelado y una colección de procedimientos probabilísticos para solucionar eficientemente dichos modelos, basados en la programación linear y entera. Las implementaciones informáticas se han llevado a cabo mediante lenguajes de programación de alto nivel, como Java, Matlab, R y AMPL. El último capítulo de la tesis introduce una extensión de los modelos analizados, para el caso de la interacción microeconómica, con el objetivo de establecer un nexo metodológico entre sus propiedades de optimización y sus propiedades multi-agentes. El uso teórico y práctico de los métodos de optimización representa el elemento de conjunción de los distintos capítulos. Parte I: The fine-grained structure of complex networks: theories, models and methods; - Capitulo 1 y Capitulo 2. Parte II: Mathematical Programming based approaches for random models of network formation; - Capitulo 3, Capitulo 4 y Capitulo 5. Parte III: Strategic models of network formation. - Capitulo 6. Los resultados de esta investigación han generado cuatro papers en revistas científicas indexadas: uno ha sido aceptado, tres están en revisión. Algunos resultados han sido también presentados en cuatro conferencias internacionale

An Initial Framework Assessing the Safety of Complex Systems

Trabajo presentado en la Conference on Complex Systems, celebrada online del 7 al 11 de diciembre de 2020.Atmospheric blocking events, that is large-scale nearly stationary atmospheric pressure patterns, are often associated with extreme weather in the mid-latitudes, such as heat waves and cold spells which have significant consequences on ecosystems, human health and economy. The high impact of blocking events has motivated numerous studies. However, there is not yet a comprehensive theory explaining their onset, maintenance and decay and their numerical prediction remains a challenge. In recent years, a number of studies have successfully employed complex network descriptions of fluid transport to characterize dynamical patterns in geophysical flows. The aim of the current work is to investigate the potential of so called Lagrangian flow networks for the detection and perhaps forecasting of atmospheric blocking events. The network is constructed by associating nodes to regions of the atmosphere and establishing links based on the flux of material between these nodes during a given time interval. One can then use effective tools and metrics developed in the context of graph theory to explore the atmospheric flow properties. In particular, Ser-Giacomi et al. [1] showed how optimal paths in a Lagrangian flow network highlight distinctive circulation patterns associated with atmospheric blocking events. We extend these results by studying the behavior of selected network measures (such as degree, entropy and harmonic closeness centrality)at the onset of and during blocking situations, demonstrating their ability to trace the spatio-temporal characteristics of these events.This research was conducted as part of the CAFE (Climate Advanced Forecasting of sub-seasonal Extremes) Innovative Training Network which has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 813844

Complexity in Economic and Social Systems

There is no term that better describes the essential features of human society than complexity. On various levels, from the decision-making processes of individuals, through to the interactions between individuals leading to the spontaneous formation of groups and social hierarchies, up to the collective, herding processes that reshape whole societies, all these features share the property of irreducibility, i.e., they require a holistic, multi-level approach formed by researchers from different disciplines. This Special Issue aims to collect research studies that, by exploiting the latest advances in physics, economics, complex networks, and data science, make a step towards understanding these economic and social systems. The majority of submissions are devoted to financial market analysis and modeling, including the stock and cryptocurrency markets in the COVID-19 pandemic, systemic risk quantification and control, wealth condensation, the innovation-related performance of companies, and more. Looking more at societies, there are papers that deal with regional development, land speculation, and the-fake news-fighting strategies, the issues which are of central interest in contemporary society. On top of this, one of the contributions proposes a new, improved complexity measure