Bounded-confidence models (BCMs) are a type of model of opinion dynamics with continuous-valued opinions. The two most popular BCMs are the Deffuant--Weisbuch (DW) model [DNA00] and the Hegselmann--Krause (HK) model [HK02]. In a BCM, interacting pairs of agents only influence each other if their opinions differ by less than some “confidence bound” between them.One major challenge in the study of opinion dynamics is making models more realistic and applicable to real-world data and situations. In this dissertation, we develop and study three generalizations of BCMs. They each incorporate some mechanism to make them more realistic, while maintaining tractability.

We first develop and study a generalization of the DW model that uses node weights to model heterogeneous agent-activity levels. The node weights in this BCM allow us to consider individuals in a social network that share their ideas and opinions more frequently than others. Using numerical simulations, we systematically investigate the effects of node weights, which we assign uniformly at random to the nodes. We demonstrate that introducing heterogeneous node weights results in longer convergence times and more opinion fragmentation than in an associated baseline DW model.

We then investigate BCMs in which each pair of agents has a distinct confidence bound that changes when the pair interacts. The confidence bounds in these BCMs encode the mutual willingness of agents to consider each other's opinions. We demonstrate numerically that our adaptive BCMs tend to promote consensus and yield longer convergence times than the associated baseline BCMs. We also show that these adaptive BCMs can have neighboring agents that converge to the same opinion but are not receptive to each other. This qualitative behavior does not occur in the associated baseline BCMs.

Finally, we study BCMs that have multi-dimensional opinions that consist of multiple interdependent topics. When a pair of agents interact on a topic, whether or not they compromise their opinions depends on the differences in their opinions on all topics. Using numerical simulations, we demonstrate for these BCMs that the choice of initial opinion distribution has a large effect on the amount of opinion fragmentation.

The fields of topological data analysis (TDA) and geometric data analysis (GDA) use algebraic topology and differential geometry to capture topological and geometric structural properties of data that are not captured by other methods in data science and machine learning. The primary tool of TDA---and one of the focuses of this dissertation---is persistent homology, which measures the connected components, holes, and higher-dimensional voids of a data set and tracks how those voids emerge and disappear at different scales. The objective of GDA is to extract new insights by considering geometric invariants of a manifold, such as curvature, rather than topological invariants. Previous studies have demonstrated the power of geometry and topology for analyzing data in complex systems, neuroscience, biology, and many other fields.

In my thesis, I study both the theory and applications of topological and geometric data analysis. In the first part of the dissertation, I establish and analyze a new construction, called a "persistence diagram (PD) bundle," for doing multiparameter TDA, and I develop an algorithm to compute a certain class of PD bundles. PD bundles generalize several important constructions in TDA: vineyards, the persistent homology transform, and fibered barcodes. In the second part of the dissertation, I apply TDA to several geospatial and geospatiotemporal data sets. In the last part of the dissertation, I introduce a new method for curvature estimation in point-cloud data.

The field of topological data analysis (TDA) uses tools from algebraic topology to capture quantitative structural properties in a data set. Perhaps the most popular tool in TDA is persistent homology (PH), which leverages homology theory to provide insights into the structure of data by quantifying ``holes'' across different scales. PH has been applied to many domains, including neuroscience, materials science, image processing, and social systems. In this thesis, we study PH in both theory and application.

On the theory side, we build upon the foundations of PH by studying persistence modules (which are algebraic objects that are fundamental to PH) and their interval decompositions.The existence of interval decompositions of a persistence module underlies much of PH. Interval decompositions are guaranteed to exist for persistence modules with coefficients in a field. However, interval decompositions may not exist in more general settings, such as when the coefficients are not in a field. We prove necessary and sufficient conditions for persistence modules with coefficients in a principal ideal domains to have an interval decomposition. We also formulate an algorithm to compute an interval decomposition when one exists.

We then use PH to quantify and assess the accessibility of resources in a geographic area. This allows us to identify regions that have poor resource access. Our work focuses on the accessibility of polling sites across six geographic regions (five cities and Los Angeles County). We adapt traditional approaches in PH to incorporate important factors of polling-site accessibility, including travel times to and from polling sites and waiting times at those sites.

Finally, we discuss the modelling of opinion dynamics on networks. We consider bounded-confidence models (BCMs), which are models of opinion dynamics in which agents are receptive only to agents whose opinions are sufficiently similar (i.e., within a confidence bound). We formulate and analyze two BCMs with adaptive confidence bounds; our BCMs generalize the Hegselmann--Krause and Deffuant--Weisbuch models. Using mathematical analysis and numerical simulations, we demonstrate that our adaptive BCMs exhibit quantitatively and qualitatively different behavior than the associated baseline (i.e., nonadaptive) HW and DW BCMs. This includes fewer major opinion clusters, longer convergence times, and adjacent nodes that converge to the same opinion but are not receptive to each other.

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).

In experiments that date back to at least the 1930s, psychologists have observed the tendency of subjects to change their beliefs to match those of others. From this observation, researchers have hypothesized that the beliefs of populations should move towards consensus. However, despite marked similarities between the beliefs of individuals in social groups, large-scale consensus is often not observed. This disparity between psychological and sociological observations, called the ``community-cleavage problem'', is one of many problems that have been examined by researchers through the development and analysis of mathematical models of belief changes. These mathematical models, often in the form of models for opinion dynamics or social influence, attract much interest from researchers for explaining how the behaviors of individuals combine to produce sociological phenomena.

In this dissertation, we draw from psychological studies to develop a consensus model of social influence. Our model is distinct from most previous models in that (1) it includes both direct and indirect social influence and (2) it incorporates group interactions of any number of individuals. We study the convergence time to consensus of our model under various assumptions on how groups form. In doing so, we make some hypotheses about how the size and composition of groups of interacting individuals affect the speed of convergence to consensus.

In various applications---such as optimizing both marketing and political campaigns and disrupting the formation of echo chambers in online social networks---researchers have used mathematical models to develop strategies for influencing opinions. These strategies prescribe how to modify factors like individuals' opinions, susceptibilities to influence, and social connections. In this dissertation, we apply our model of opinion dynamics to conflict resolution, and we propose group-selection strategies for choosing groups of individuals to meet for the purpose of accelerating convergence to consensus. We show through analytical estimates, simulations, and examples that our strategies generate an improvement in convergence time over random choices of groups.

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.

Analyzing public opinions on political affairs never fails to attract researchers’ attention. One popular application is analyzing ideologies of the ordinary citizens. Traditionally, researchers collect public opinions by conducting surveys, interviews, or estimate via the roll call data. It took a lot of time to come to a conclusion, and it was also hard to find representative objects and keep the objects’ opinions unaffected by researchers who design the survey questions. By using social media data, some of the previously-mentioned problems are mitigated due to the unprecedented massive scale. However, we need to keep in mind that social media data might be systematically biased in some other ways. For instance, those who never use social media could not be included.

In recent years, social media, such as Twitter, plays an increasingly important role in people’s life. People express opinions, digest information, and interact on social medias. All these behaviors left massive amount of observable clues online, which we collect as our data. Centered around the problems on political ideology analysis, we start from collecting a list of politicians who have verified accounts on Twitter. Then we build our data sets from the Twitter accounts who are not further than one hop away from the politicians, directly following or are followed by the politicians. Although it becomes much easier to collect massive amount of data in an efficient and timely manner, social media data bring us unique challenges. For example: (1) we need to deal with the data size which is typically large; (2) there are seldom ground-truth knowledge on social media data, which leads to the lack of labels; (3) we need to consider how to deal with the multi-modality nature of our data.

If we view each user account as a node, their interactive behaviors could be modeled as links in between. Since they interact in multiple ways, the graph structure we form is heterogeneous. If we view each account as an individual information source, we could represent its feature by the collection of tweets it posted in the past. If we consider temporal information as well, the whole system could also be regarded as a multi-agent dynamic system. Based on the observation of the data, we have proposed the following research problems to answer:

(1) Can we rely solely on the user behavior data, represented as heterogeneous types of links in the graph structure, to reveal the users’ ideologies?

(2) Can we rely solely on the text information from tweets, to reveal the users’ political polarities?

(3) By learning from the historical data on social media, containing text, link, and temporal information, can we predict the future trend?

To answer the first research problem, we proposed a model that successfully uses the heterogeneous types of relations, called TIMME (Twitter Ideology-detection via Multitask Multi-relational Embedding). Challenges come from (1) the extreme sparsity of the labels, (2) the incompleteness of the input features, and (3) the heterogeneous types of links. TIMME is overall better than the state-of-the-art models for ideology detection on Twitter. In theory, it could be extended to other data sets, and could be apply to tasks other than ideology detection.

The work we’ve done to answer the second problem resulted in an embedding approach called PEM (i.e., Polarity-Aware Embeddings Using Multi-Task Learning). The ideological divisions in the United States have become increasingly prominent in daily communication, and a lot of research has been conducted. We propose to quantify political polarity in social-media text data using a polarity-aware method of learning word representations. By learning a word embedding with an explicit polarity dimension, one can infer the polarity of a post and therefore of the social-media account that produced it. Decomposing a traditional embedding into a polarity-neutral semantic component and a polarity-aware component is a major challenge. Very sparse labels is another key challenge. Our experimental results demonstrate that our model can successfully learn high-quality polarity-aware embeddings.

The third research question is answered by our next project, a social dynamic system model that captures the updating patterns in real-world social network data sets. We faced challenges on both the data end and the model end. From the data set, there is no existing publicly-available data sets on real-world social network observations. From the model perspective, modeling continuous time is trickier than modeling discrete time by nature. After decided that we should consider using following the Neural-ODE framework, other challenges came about, such as the data size, and the selection of an appropriate decoder task. Our experimental results show that the framework we propose is capable of learning the dynamic changes in the social network data sets. And our framework could also be used to verify how well an opinion dynamic model captures the changes of a specific real-world data set, under a given task.

Mathematical models can be very useful for improving understanding of human behavior and helping forecasting it. In this thesis, we develop and explore three models of human dynamics. One is a model of disease spread coupled to opinion dynamics, another is a model of opinion dynamics, and the third is a model of illegal logging. We build models in the form of dynamical processes on networks for the first two applications and employ optimal control theory to study the third application.

We first introduce a network model of the spread of a disease under the influence of the spread of competing opinions. We describe the network structure in this model as a two-layer multilayer network. On one layer, two opinions --- pro-physical-distancing and anti-physical-distancing --- spread concurrently and compete with each other. On the other layer, the disease evolves and individuals are less likely (respectively, more likely) to become infected if they adopt the pro-physical-distancing (respectively, anti-physical-distancing) opinion. We explore both beneficial and harmful effects of the spread of opinions on disease transmission with mean-field approximations and direct numerical simulations. We also examine how heterogeneous networks with specified interlayer and intralyer degree--degree correlations influence the dynamics.

We then develop an opinion-dynamics model that extends the majority-vote model to two-layer multilayer networks with community structure. We assume that neighbors from different social relationships have different abilities to change an individual's opinions. We find three patterns of steady-state opinion distributions and study phase transitions in the model with a mean-field approximation and direct numerical simulations.

Finally, we study a model of the behavior of uncontrolled loggers in the presence of law-enforcement agencies. We assume that loggers want to maximize a profit function that incorporates the benefit of logging, travel cost, and the the risk of capture by finding optimal travel trajectories and optimal logging duration. We formulate the problem as a static Hamilton--Jacobi equation, which we solve using a fast sweeping method. We use Brazilian rainforest data and demonstrate the importance of geographically targeted patrol strategies using numerical experiments.