A graph dynamical system (GDS) is a theoretical construct that can be used to simulate and analyze the dynamics of a wide spectrum of real world processes which can be modeled as networked systems. One of our goals is to compute the phase space of a system, and for this, even 30-vertex graphs present a computational challenge. This is because the number of state transitions needed to compute the phase space is exponential in the number of graph vertices. These problems thus produce memory and execution speed challenges. To address this, we devise various MapReduce programming paradigms that can be used to characterize system state transitions, compute phase spaces, functional equivalence classes, dynamic equivalence classes and cycle equivalence classes of dynamical systems. We also evaluate these paradigms and analyze their suitability for modeling different GDSs. / Master of Science

Dynamics of social processes in populations, such as the spread of emotions, influence, language, mass movements, and warfare (often referred to individually and collectively as contagions), are increasingly studied because of their social, political, and economic impacts. Discrete dynamical systems (discrete in time and discrete in agent states) are often used to quantify contagion propagation in populations that are cast as graphs, where vertices represent agents and edges represent agent interactions. We refer to such formulations as graph dynamical systems. For social applications, threshold models are used extensively for agent state transition rules (i.e., for vertex functions). In its simplest form, each agent can be in one of two states (state 0 (1) means that an agent does not (does) possess a contagion), and an agent contracts a contagion if at least a threshold number of its distance-1 neighbors already possess it. The transition to state 0 is not permitted. In this study, we extend threshold models in three ways. First, we allow transitions to states 0 and 1, and we study the long-term dynamics of these bithreshold systems, wherein there are two distinct thresholds for each vertex; one governing each of the transitions to states 0 and 1. Second, we extend the model from a binary vertex state set to an arbitrary number r of states, and allow transitions between every pair of states. Third, we analyze a recent hierarchical model from the literature where inputs to vertex functions take into account subgraphs induced on the distance-1 neighbors of a vertex. We state, prove, and analyze conditions characterizing long-term dynamics of all of these models. / Master of Science

The underlying mathematical model of many simulation models is graph dynamical systems (GDS). This dynamical system, its implementation, and analyses on each will be the focus of this paper. When using a simulation model to answer a research question, it is important to describe this underlying mathematical model in which we are operating for verification and validation. In this paper we discuss analyses commonly used in simulation models. These include sensitivity analyses and uncertainty quantification, which provide motivation for stability and structure-to-function research in GDS. We review various results in these areas, which contribute toward validation and computationally tractable analyses of our simulation model. We then present two new areas of research - stability of transient structure with respect to update order permutations, and an application of GDS in which a time-varying generalized cellular automata is implemented as a simulation model. / Master of Science

Dynamics of social processes in populations, such as the spread of emotions, influence, opinions, and mass movements (often referred to individually and collectively as contagions), are increasingly studied because of their economic, social, and political impacts. Moreover, multiple contagions may interact and hence studying their simultaneous evolution is important. Within the context of social media, large datasets involving many tens of millions of people are leading to new insights into human behavior, and these datasets continue to grow in size. Through social media, contagions can readily cross national boundaries, as evidenced by the 2011 Arab Spring. These and other observations guide our work. Our goal is to study contagion processes at scale with an approach that permits intricate descriptions of interactions among members of a population. Our contributions are a modeling environment to perform these computations and a set of approaches to predict contagion spread size and to block the spread of contagions. Since we represent populations as networks, we also provide insights into network structure effects, and present and analyze a new model of contagion dynamics that represents a person\'s behavior in repeatedly joining and withdrawing from collective action. We study variants of problems for different classes of social contagions, including those known as simple and complex contagions. / Ph. D.

When capturing a real-world, networked system using a simulation model, features are usually omitted or represented by probability distributions. Verification and validation (V and V) of such models is an inherent and fundamental challenge. Central to V and V, but also to model analysis and prediction, are uncertainty quantification (UQ), sensitivity analysis (SA) and design of experiments (DOE). In addition, network-based computer simulation models, as compared with models based on ordinary and partial differential equations (ODE and PDE), typically involve a significantly larger volume of more complex data. Efficient use of such models is challenging since it requires a broad set of skills ranging from domain expertise to in-depth knowledge including modeling, programming, algorithmics, high- performance computing, statistical analysis, and optimization. On top of this, the need to support reproducible experiments necessitates complete data tracking and management. Finally, the lack of standardization of simulation model configuration formats presents an extra challenge when developing technology intended to work across models. While there are tools and frameworks that address parts of the challenges above, to the best of our knowledge, none of them accomplishes all this in a model-independent and scientifically reproducible manner. In this dissertation, we present a computational framework called GENEUS that addresses these challenges. Specifically, it incorporates (i) a standardized model configuration format, (ii) a data flow management system with digital library functions helping to ensure scientific reproducibility, and (iii) a model-independent, expandable plugin-type library for efficiently conducting UQ/SA/DOE for network-based simulation models. This framework has been applied to systems ranging from fundamental graph dynamical systems (GDSs) to large-scale socio-technical simulation models with a broad range of analyses such as UQ and parameter studies for various scenarios. Graph dynamical systems provide a theoretical framework for network-based simulation models and have been studied theoretically in this dissertation. This includes a broad range of stability and sensitivity analyses offering insights into how GDSs respond to perturbations of their key components. This stability-focused, structure-to-function theory was a motivator for the design and implementation of GENEUS. GENEUS, rooted in the framework of GDS, provides modelers, experimentalists, and research groups access to a variety of UQ/SA/DOE methods with robust and tested implementations without requiring them to necessarily have the detailed expertise in statistics, data management and computing. Even for research teams having all the skills, GENEUS can significantly increase research productivity. / Ph. D.

Moore and Shannon's reliability polynomial can be used as a global statistic to explore the behavior of diffusive processes on a graph dynamical system representing a finite sized interacting system. It depends on both the network topology and the dynamics of the process and gives the probability that the system has a particular desired property. Due to the complexity involved in evaluating the exact network reliability, the problem has been classified as a NP-hard problem. The estimation of the reliability polynomials for large graphs is feasible using Monte Carlo simulations. However, the number of samples required for an accurate estimate increases with system size. Instead, an adaptive method using Bernstein polynomials as kernel density estimators proves useful. Network reliability has a wide range of applications ranging from epidemiology to statistical physics, depending on the description of the functionality. For example, it serves as a measure to study the sensitivity of the outbreak of an infectious disease on a network to the structure of the network. It can also be used to identify important dynamics-induced contagion clusters in international food trade networks. Further, it is analogous to the partition function of the Ising model which provides insights to the interpolation between the low and high temperature limits. / Ph. D. / The research presented here explores the effects of the structural properties of an interacting system on the outcomes of a diffusive process using Moore-Shannon network reliability. The network reliability is a finite degree polynomial which provides the probability of observing a certain configuration for a diffusive process on networks. Examples of such processes analyzed here are outbreak of an epidemic in a population, spread of an invasive species through international trade of commodities and spread of a perturbation in a physical system with discrete magnetic spins. Network reliability is a novel tool which can be used to compare the efficiency of network models with the observed data, to find important components of the system as well as to estimate the functions of thermodynamic state variables.

Network science emerged as an interdisciplinary field over the last 20 years, and played a central role to address fundamental problems in other fields, e.g., epidemiology, public health, and transportation, and is now part of most university curriculums. Network dynamics is a major area within network science where researchers study different forms of processes in networked populations, such as the spread of emotions, influence, opinions, flu, ebola, and mass movements. These processes often referred to individually and collectively as contagions. Contagions are increasingly studied because of their economic, social, and political impacts. Yet, resources for studying network dynamics are largely dispersed and stand-alone. Furthermore, many researchers interested in the study of networks are not computer scientists. As a result, they do not have easy access to computing and data resources. Even with the presence of software or tools, it is challenging to install, build, and maintain software. These challenges create a barrier for researchers and domain scientists. The goal of this work is the design and implementation of a research framework for modeling contagions on large networked populations. The framework consists of various systems and services that provide support for researchers and domain scientists at different stages of their research workflow. / Ph. D.