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.
