Quasi-stationary distributions as centrality measures of reducible graphs

Random walk can be used as a centrality measure of a directed graph. However, if the graph is reducible the random walk will be absorbed in some subset of nodes and will never visit the rest of the graph. In Google PageRank the problem was solved by introduction of uniform random jumps with some probability. Up to the present, there is no clear criterion for the choice this parameter. We propose to use parameter-free centrality measure which is based on the notion of quasi-stationary distribution. Specifically we suggest four quasi-stationary based centrality measures, analyze them and conclude that they produce approximately the same ranking. The new centrality measures can be applied in spam detection to detect ``link farms'' and in image search to find photo albums

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

Coevolutionary systems and PageRank

Coevolutionary systems have been used successfully in various problem domains involving situations of strategic decision-making. Central to these systems is a mechanism whereby finite populations of agents compete for reproduction and adapt in response to their interaction outcomes. In competitive settings, agents choose which solutions to implement and outcomes from their behavioral interactions express preferences between the solutions. Recently, we have introduced a framework that provides both qualitative and quantitative characterizations of competitive coevolutionary systems. Its two main features are: (1) A directed graph (digraph) representation that fully captures the underlying structure arising from pairwise preferences over solutions. (2) Coevolutionary processes are modeled as random walks on the digraph. However, one needs to obtain prior, qualitative knowledge of the underlying structures of these coevolutionary digraphs to perform quantitative characterizations on coevolutionary systems and interpret the results. Here, we study a deep connection between coevolutionary systems and PageRank to address this issue. We develop a principled approach to measure and rank the performance (importance) of solutions (vertices) in a given coevolutionary digraph. In PageRank formalism, B transfers part of its authority to A if A dominates B (there is an arc from B to A in the digraph). In this manner, PageRank authority indicates the importance of a vertex. PageRank authorities with suitable normalization have a natural interpretation of long-term visitation probabilities over the digraph by the coevolutionary random walk. We derive closed-form expressions to calculate PageRank authorities for any coevolutionary digraph. We can precisely quantify changes to the authorities due to modifications in restart probability for any coevolutionary system. Our empirical studies demonstrate how PageRank authorities characterize coevolutionary digraphs with different underlying structures

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

Modeling adaptive dynamics in microbial populations with applications to the evolution of cellular resource allocation trade-offs

Adaptive evolution is the process by which natural selection, acting on variation within a population, promotes the survival of individuals that are more successful at reproducing and contributing to future generations. Evolutionary processes in microbes occur at the intersection of population genetics, natural selection, and underlying mechanistic constraints, to give rise to the repertoire of adaptation observed in nature. Understanding microbial adaptive evolution is of critical importance for human health for example, through the emergence of pathogenicity and antibiotic resistance. Moreover, the stability and function of natural and artificial ecosystems is contingent on the evolving interactions between microbes, and between microbes and the environment. We present a modelling framework, based on the theory of adaptive dynamics, to investigate how cellular resource allocation trade-offs affect the adaptation process. We used resource-consumer theory, which explicitly models the interactions between cells and their environment, together with matrix models of structured populations, to implement phenotype-determined cellular strategies of resource allocation between mutually exclusive processes. We then analyse the outcome of competitions between different phenotypes across environmental and competitive conditions. We applied our methods to the evolution of strategies (phenotypes) for resource allocation between two competing cellular process in microbial populations growing in chemostat-like environments. We calculated the adaptively stable strategies for several models and showed how state-structured population models can be mapped to simpler chemostat models on invariant manifolds. We then extended our analysis to the case where a limiting nutrient may be utilized using two alternative metabolic pathways. We described how the total population fitness of a metabolic strategy can be constructed from the individual decisions of its constituent members. We developed numerical methods to simulate and analyse general models of adaptive dynamics using principles from graph theory and discrete Markov processes. The methods were used to explore the evolution of nutrient use strategies for microbial populations growing on two and three substitutable nutrients. We highlight the importance of the ancestral phenotype in channelling the adaptation process, which, together with the choice of the mutational kernel, influences the adaptively stable strategies and modes of co-existence. In a related finding, we show how some phenotypes are adaptively stable only in the presence of a competitor lineage that modifies the environment in a manner that permits another phenotype to invade. Our methods also reveal instances where historical contingency and chance have an important effect on determining the stable nutrient use strategies. Finally, we demonstrate the existence of adaptively stable periodic solutions whereby, under some conditions, phenotype successions are cyclical. Our work builds on the foundation of adaptive dynamics theory to provide a general framework for analysing models of microbial adaptation. We focused on understanding the implications of underlying constraints and cellular resource allocation trade-offs in the context of adaptive evolution

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

Dynamics of life - self-organisation, co-adaptation & the evolution of resilience in complex ecosystems

Ecosystems are highly ordered, non-random systems, yet neither are they designed from a specific construction plan determining their structure and composition, nor are they directly predictable from the traits and behaviour of the species they are comprised of. The order arises from self-organising forces stemming from the interactions between individuals - yet the concrete principles behind self-organisation, the role of community integrity versus single species traits and the feedbacks between species and the systems they form are largely unknown. In this thesis, complex-systems modelling is used as a tool to bridge the gap between microscopic behaviour and emergent macroscopic systems and concepts from information theory are applied to quantify the arising, development and evolution of order within ecological communities. Generally, the information content within an ecosystem, the information content the ecosystem has about itself and its future states as well as the general coadaptation and organisation tend to increase with advanced developmental state. This goes along with a shift from bottom-up control from micro- to the macrolevel in earlier developmental stages to top-down control the system tends to impose on its species and individuals in late-successional stages. Coadaptation, next to individual fitness, also plays a role for species abundance and survival in communities undergoing disturbances, emphasizing the role of community integrity for its persistence. Overall, the results illustrate that species interactions are the guiding force behind ecosystem organisation, that coadaptation shapes functionality, stability and resilience and that ecosystem development is not either guided by fundamental laws \emph{or} global optimisation criteria, but by both - individual fitness and the system's state determine ecosystem development in a dialogue of bottom-up and top-down control.Open Acces

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..

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