Visual Similarity Analysis for Dynamic Systems

Dynamical systems are ubiquitous in multiple fields, such as mathematics, physics, chemistry, neuroscience, sociology, and system biology. The dynamical processes of those systems are defined by many factors, such as the intrinsic rules of the processes, the properties of the systems, the parameter settings, and the initial conditions of the processes. Understanding how those factors shape these dynamical processes is one of the fundamental questions in the aforementioned fields. To explain the role of those factors, purely analytical approaches focus on individual aspects of some hypothesis that is being tested, without making use of visualization methods. However, visualization can discover unknown organizational principles that other standard analysis methods may miss, since standard analysis methods typically are not targeted at discovering the unknown. This thesis proposes novel visual analysis approaches to determine how topology and simulation parameter configurations influence the dynamics in dynamical systems, such as network dynamics and chaotic systems. We particularly focus on how excitable network dynamics and chaos-based pseudo-random number generators (PRNGs) are impacted by these influences. In the dynamical systems, sets of time series are achieved from simulations. Depending on the context, the time series can be distributed spatially over a network. Those time series are also influenced by global parameters while the simulations are executed. Ensembles of such time series are retrieved by carrying out the simulations with different parameter configurations. Given the data, we develop methods to (1) extract dynamical features from the time series and visually encode them, (2) extract relevant topological features from the network data when given and visually encode them, (3) relate the features to each other using coordinated views, and (4) develop ensemble visualization methods to analyze the influence of simulation parameters. The core idea of our work is the similarity-based visualization: motivated from applications throughout this thesis, dynamical features are extracted based on different similarity measures that reveal different properties of the time series and topological features are extracted based on similarities of nodes in the network. The features are then encoded by using suitable visual encodings. To relate the features to each other, relevant linking techniques between coordinated views of the encodings are adopted. To learn how the parameters influence the dynamical features, we build up a system of 2D scatterplots to visually encode the similarity of ensemble members. The system allows for analysis in a top-down manner that starts with an overview and supports iteratively zooming into regions of interest to explore the similarity features. To explain how the parameters shape the relation between the topological features and the dynamical ones, we encode the matching between those features as graphs of the function of those parameters and relate the topological features to the dynamical encounters for selected parameter values in a 2D plot based on their suitable 1D encodings. In this vein, we present a novel user-aided approach for creating 1D embedding of a multi-dimensional space as a suggestion in supporting the task of analyzing the parameter space. By applying our visual approaches, we visually confirm co-activation waves around hubs and synchronization of nodes in modules in excitable network dynamics. We discover the interference waves among hubs and detect new relevant dynamical features such as excitation ratio. We form a hypothesis about dependency of the excitation ratio over node’s degree, which is confirmed by a domain expert using mean-field theory. We visually explore the parameter space of the excitable network dynamics in the context of synthetic and real brain-connectome networks. Using our approaches in chaos-based PRNGs, we identify the dependency of their randomness on a global parameter and can visually compare the rate of convergence to the perfect randomness case among the chaos-based PRNGs with respect to the parameter. Overall, our visual approaches illustrate how interactive visualization can support understanding of dynamical systems in terms of discovering new findings and forming new hypotheses for domain experts.