Measuring Scales: Integration and Modularity in Complex Systems

Abstract

In this thesis, we investigate novel methods for studying complex systems at multiple scales. First, we develop an information-theoretic measure of multi-scale integration in multivariate systems. This quantifies the strength of interactions in subsystems of different sizes, where size is defined relative to some underlying distance metric. We apply this measure to MRI recordings of the human brain and show that it provides meaningful characterization of different brain areas. Second, we consider modularity, a pattern of organization in which systems are composed of weakly-coupled subsystems. We propose two different methods to identify modular decompositions of multivariate dynamical systems. The first is based on statistical learning of dynamics; it uses model selection to choose decompositions that optimize a trade-off between capturing dependencies (favoring large-scale modules) and model simplicity (favoring small-scale modules), with the trade-off controlled by a parameter representing the amount of available data. The second method is based on perturbations to system dynamics; it finds decompositions having subsystems that maximally constrain the spread of perturbations over time, with the time scale acting a resolution parameter. We apply the method to several non-linear dynamical systems and also show that it is a generalization of community-detection techniques based on random walks on graphs. These approaches to studying multi-scale integration and modularity are novel: while measures of multi-scale integration have been investigated theoretically, they have not been extended to consider real-world, spatially-embedded systems; on the other hand, while much work has been done on identifying modular decompositions in real-world graphs, the problem has received little attention in the domain of multivariate dynamical systems. Thus, our work provides new tools for analyzing multivariate dynamics and time-series.