Analysis of contagion maps on a class of networks that are spatially embedded in a torus

A spreading process on a network is influenced by the network's underlying spatial structure, and it is insightful to study the extent to which a spreading process follows such structure. We consider a threshold contagion on a network whose nodes are embedded in a manifold and where the network has both `geometric edges', which respect the geometry of the underlying manifold, and `non-geometric edges' that are not constrained by that geometry. Building on ideas from Taylor et al. \cite{Taylor2015}, we examine when a contagion propagates as a wave along a network whose nodes are embedded in a torus and when it jumps via long non-geometric edges to remote areas of the network. We build a `contagion map' for a contagion spreading on such a `noisy geometric network' to produce a point cloud; and we study the dimensionality, geometry, and topology of this point cloud to examine qualitative properties of this spreading process. We identify a region in parameter space in which the contagion propagates predominantly via wavefront propagation. We consider different probability distributions for constructing non-geometric edges --- reflecting different decay rates with respect to the distance between nodes in the underlying manifold --- and examine the effect of such choices on the qualitative properties of the spreading dynamics. Our work generalizes the analysis in Taylor et al. and consolidates contagion maps both as a tool for investigating spreading behavior on spatial networks and as a technique for manifold learning

Topological tools for understanding complex systems

The behavior of complex systems is often influenced by their structure. In mathematics, the field of algebraic topology has been especially useful for characterizing mathematical structures. Topological data analysis (TDA) is a growing field in which methods from algebraic topology are applied to studying the structure of data. TDA has been used in a variety of applications, including biological data, granular materials, and demography. Social interactions are heavily informed by space and have complex structure due to patterns in the way humans arrange themselves geographically. Consequently, social applications can benefit from the application of TDA.In this dissertation, I develop topological methods for studying spatial networks and apply them to a wide variety of data sets. In particular, I study methods for building topological spaces (specifically, simplicial complexes) based on data. I present two novel simplicial-complex constructions, the adjacency complex and the level-set complex, for spatial data. I apply both constructions to random networks, cities, voting, and scientific images, gaining insights into the structure of these systems. I also propose a novel simplicial complex construction for studying patterns of neighborhood formation based on combining demographic and spatial data. I present case studies in neighborhood segregation for two U.S. cities. In addition to my topological research, I discuss two projects in the study of social systems using methods from network analysis. I present an extension to multilayer networks of the Hegselmann--Krause model for opinion dynamics and discuss preliminary findings on its convergence properties. I also present a framework for estimating homelessness underreporting in California Local Education agencies (LEAs)

Diffusion and Supercritical Spreading Processes on Complex Networks

Die große Menge an Datensätzen, die in den letzten Jahren verfügbar wurden, hat es ermöglicht, sowohl menschlich-getriebene als auch biologische komplexe Systeme in einem beispiellosen Ausmaß empirisch zu untersuchen. Parallel dazu ist die Vorhersage und Kontrolle epidemischer Ausbrüche für Fragen der öffentlichen Gesundheit sehr wichtig geworden. In dieser Arbeit untersuchen wir einige wichtige Aspekte von Diffusionsphänomenen und Ausbreitungsprozeßen auf Netzwerken. Wir untersuchen drei verschiedene Probleme im Zusammenhang mit Ausbreitungsprozeßen im überkritischen Regime. Zunächst untersuchen wir die Reaktionsdiffusion auf Ensembles zufälliger Netzwerke, die durch die beobachteten Levy-Flugeigenschaften der menschlichen Mobilität charakterisiert sind. Das zweite Problem ist die Schätzung der Ankunftszeiten globaler Pandemien. Zu diesem Zweck leiten wir geeignete verborgene Geometrien netzgetriebener Streuprozeße, unter Nutzung der Random-Walk-Theorie, her und identifizieren diese. Durch die Definition von effective distances wird das Problem komplexer raumzeitlicher Muster auf einfache, homogene Wellenausbreitungsmuster reduziert. Drittens führen wir durch die Einbettung von Knoten in den verborgenen Raum, der durch effective distances im Netzwerk definiert ist, eine neuartige Netzwerkzentralität ein, die ViralRank genannt wird und quantifiziert, wie nahe ein Knoten, im Durchschnitt, den anderen Knoten im Netzwerk ist. Diese drei Studien bilden einen einheitlichen Rahmen zur Charakterisierung von Diffusions- und Ausbreitungsprozeßen, die sich auf komplexen Netzwerken allgemein abzeichnen, und bieten neue Ansätze für herausfordernde theoretische Probleme, die für die Bewertung künftiger Modelle verwendet werden können.The large amount of datasets that became available in recent years has made it possible to empirically study humanly-driven, as well as biological complex systems to an unprecedented extent. In parallel, the prediction and control of epidemic outbreaks have become very important for public health issues. In this thesis, we investigate some important aspects of diffusion phenomena and spreading processes unfolding on networks. We study three different problems related to spreading processes in the supercritical regime. First, we study reaction-diffusion on ensembles of random networks characterized by the observed Levy-flight properties of human mobility. The second problem is the estimation of the arrival times of global pandemics. To this end, we derive and identify suitable hidden geometries of network-driven spreading processes, leveraging on random-walk theory. Through the definition of network effective distances, the problem of complex spatiotemporal patterns is reduced to simple, homogeneous wave propagation patterns. Third, by embedding nodes in the hidden space defined by network effective distances, we introduce a novel network centrality, called ViralRank, which quantifies how close a node is, on average, to the other nodes. These three studies constitute a unified framework to characterize diffusion and spreading processes unfolding on complex networks in very general settings, and provide new approaches to challenging theoretical problems that can be used to benchmark future models

CLADAG 2021 BOOK OF ABSTRACTS AND SHORT PAPERS

The book collects the short papers presented at the 13th Scientific Meeting of the Classification and Data Analysis Group (CLADAG) of the Italian Statistical Society (SIS). The meeting has been organized by the Department of Statistics, Computer Science and Applications of the University of Florence, under the auspices of the Italian Statistical Society and the International Federation of Classification Societies (IFCS). CLADAG is a member of the IFCS, a federation of national, regional, and linguistically-based classification societies. It is a non-profit, non-political scientific organization, whose aims are to further classification research

Spatio-temporal analysis of vegetation dynamics of selected successional stages of dry acidic grasslands : experimental studies and model simulations

Austenfeld M. Spatio-temporal analysis of vegetation dynamics of selected successional stages of dry acidic grasslands : experimental studies and model simulations. Bielefeld (Germany): Bielefeld University; 2009.Eine allgemeine Entwicklungsumgebung wurde für die Analyse und Simulation von räumlich-zeitlichen Phänomenen in ökologischen Systemen entwickelt. Die gesamte Plattform basiert auf einer "Rich-Client-Platform" (RCP), die neue Konzepte der Modularisierung und allgemeinen Programmarchitektur mitbringt. Damit bietet sie die Grundlage für eine nachhaltige Weiterentwicklung und ist somit eine solide Basis für eine integrierte Entwicklungsumgebung für ökologische Modelle. Die Integration verschiedener statistischer Werkzeuge, Methoden der Bildverarbeitung und spezielle Visualisierungen qualifizieren diese Umgebung besonders für die Analyse der oben genannten räumlich-zeitlichen Prozesse. Aufgrund ihrer vergleichsweise geringen Komplexität wurden Sandlebensräume wiederholt für Studien von Vegetationsmustern und ihrer zugrunde liegenden biotischen Interaktionen genutzt. Für einen integrativen Überblick und weitere integrative Ansätze mit Hilfe von Simulationsmodellen wurde die oben genannte Plattform genutzt, um eine individuenbasierte Modellstruktur für die Analyse von Langzeiteffekten aufgrund von Umweltveränderungen auf die Stabilität von Sandlebensräumen zu entwickeln, die typischerweise von zwei Pionierarten, Corynephorus canescens und Polytrichum piliferum, dominiert werden. Das Modell wurde mit experimentellen Daten verifiziert, und die vom Modell erzeugten räumlich-zeitlichen Muster zeigten eine hohe Übereinstimmung mit natürlich gemessenen Mustern. Das Modell wurde dann genutzt, um Langzeiteffekte von Veränderungen der Temperatur, Nährstoffversorgung und Störungsraten in diesem System zu untersuchen. Die Ergebnisse zeigten eine generell hohe Stabilität des Systems unter veränderten Temperatur- und Nährstoffbedingungen, wobei temporal wiederkehrende, kleinräumige Störungen als Grundlage notwendig waren. Schließlich wurde noch eine Untersuchung über die Auswirkungen von Herbivorie und Konkurrenz auf Corynephorus canescens durchgeführt. In einem kontrollierten Freilandexperiment wurden die Auswirkungen von entfernter Biomasse von Blättern sowie die An- oder Abwesenheit eines intraspezifischen und interspezifischen Konkurrenten (Hieracium pilosella) auf die überirdische und unterirdische Allokation von Biomasse in der folgenden Regenerationsphase analysiert. Die Ergebnisse zeigten, dass Corynephorus canescens die Fähigkeit besitzt, leichte bis mittlere Blattverluste (die typische natürliche Herbivorie von Kaninchen und Paarhufern simulieren sollten) zu kompensieren, ohne dabei an Konkurrenzstärke zu verlieren. Unterirdisch konnten keine Auswirkungen der simulierten Herbivorie bzw. Konkurrenz festgestellt werden. Aufgrund dieser zu vernachlässigenden Effekte wurde Herbivorie nicht in dem Modell berücksichtigt.A generic modeling environment for the analysis and simulation of spatio-temporal phenomena in ecosystems was developed. This framework was built upon a Rich Client Platform (RCP) which uses new concepts of extensibility and software architecture for sustainable development and provides a solid basis for an Integrated Development Environment (IDE) for ecological models. The integration of various statistical tools, imaging routines and several specialized drawing panels makes this environment particularly suitable for the analysis of the above mentioned spatio-temporal ecological processes. Because of their comparatively low complexity, dry acidic grassland ecosystems have been repeatedly used for studying vegetation pattern formation and the underlying biotic interactions. In order to obtain an integrative view of the existing knowledge as well as to provide a possibility for further integrative analysis with the help of model simulations, the above described platform was used to develop an individual based Model structure for the investigation of long term effects of environmental changes on the stability of early successional stages of such dry acidic grasslands which are typically dominated by the two pioneer species Corynephorus canescens and Polytrichum piliferum. The model was validated with experimental data and the spatio-temporal patterns created by the model were in good accordance with the measured natural patterns. The model was then used to analyze the effect of changes in temperature, nutrient supply and disturbance rate on the long term behavior of this ecosystem. The results showed an overall high stability of this system under different temperature and nutrient scenarios as long as an intermediate disturbance frequency is assured. Finally, an experimental study on the effect of herbivory and competition on the Corynephorus canescens was conducted. In a controlled field experiment, the effects of the removal of various amounts of aboveground biomass on the above and belowground biomass allocation during the following regeneration phase was analyzed in the presence or absence of an intraspecific and interspecific competitor (Hieracium pilosella). The results show a rather high ability of C. canescens to compensate low to medium amounts of foliage loss (reflecting the typical natural herbivory induced by grasshoppers and rabbits) without significant changes in its competitive ability. Belowground, no biomass effects of foliage removal and/or competition could be detected. Because of these negligible effects, herbivory was not implemented in the above described model

Nonlinear opinion models and other networked systems

Networks play a critical role in many physical, biological, and social systems. In this thesis, we investigate tools to model and analyze networked systems. We first examine some of the ways in which we can model social dynamics that take place on networks. We then study two recently developed data-analysis methods that employ a network framework and explore new ways in which they can be used to find meaningful signals in large data sets. In the first half of the thesis, we study opinion dynamics on networks. We begin by examining a class of opinion models, known as coevolving voter models (CVM), that couple the mechanisms of opinion formation and changing social connections. We then propose a version of CVMs that incorporates nonlinearity. In our models, we assume that individuals strive to achieve harmony and avoid disagreement, both by changing their social connections to reflect their opinions and by changing their opinions to reflect their social connections. By taking a minimalist approach to modeling social dynamics, we hope to gain a deeper understanding of how these two mechanisms can give rise to social phenomena such as the ``majority illusion''. Comparing several versions of CVMs, we find that seemingly small changes in update rules can lead to strikingly different behaviors. A particularly interesting feature of our nonlinear CVMs is that, under certain conditions, the opinion state that is held initially by a minority of the nodes can effectively spread to almost every node in a network if the minority nodes view themselves as the majority. We then discuss an ongoing project that involves another class of opinion models called bounded-confidence models. Specifically, we examine extensions of bounded-confidence models on hypergraphs and discuss some preliminary findings. In the second half of the thesis, we study problems in data analysis. We begin by considering topological structures as a tool to study integrated circuit (IC) devices. In particular, we examine a problem in the design and manufacturing of IC devices using topological data analysis (TDA), which is based on network structures called simplicial complexes. Failures in IC devices generally occur near the tolerance limits of photolithography systems, such as at the minimum separation distance between adjacent electronic components. However, for complex arrangements of electronic components, simply ensuring minimal separation is insufficient to guarantee that one can manufacture an IC design accurately and reliably. We apply tools from TDA to compare data from IC designs. Without inputting domain knowledge, we are able to infer several results about the IC design-manufacturing process. Finally, we discuss an ongoing project in the analysis of network data. Specifically, we explore applications of a recently developed algorithm called network dictionary learning (NDL) and discuss problems of network reconstruction and denoising using NDL on both synthetic and real-world networks