Expedition in Data and Harmonic Analysis on Graphs

The graph Laplacian operator is widely studied in spectral graph theory largely due to its importance in modern data analysis. Recently, the Fourier transform and other time-frequency operators have been defined on graphs using Laplacian eigenvalues and eigenvectors. We extend these results and prove that the translation operator to the i’th node is invertible if and only if all eigenvectors are nonzero on the i’th node. Because of this dependency on the support of eigenvectors we study the characteristic set of Laplacian eigenvectors. We prove that the Fiedler vector of a planar graph cannot vanish on large neighborhoods and then explicitly construct a family of non-planar graphs that do exhibit this property. We then prove original results in modern analysis on graphs. We extend results on spectral graph wavelets to create vertex-dyanamic spectral graph wavelets whose support depends on both scale and translation parameters. We prove that Spielman’s Twice-Ramanujan graph sparsifying algorithm cannot outperform his conjectured optimal sparsification constant. Finally, we present numerical results on graph conditioning, in which edges of a graph are rescaled to best approximate the complete graph and reduce average commute time.

Data-Aware Scheduling in Datacenters

Datacenters have emerged as the dominant form of computing infrastructure over the last two decades. The tremendous increase in the requirements of data analysis has led to a proportional increase in power consumption and datacenters are now one of the fastest growing electricity consumers in the United States. Another rising concern is the loss of throughput due to network congestion. Scheduling models that do not explicitly account for data placement may lead to a transfer of large amounts of data over the network causing unacceptable delays. In this dissertation, we study different scheduling models that are inspired by the dual objectives of minimizing energy costs and network congestion in a datacenter. As datacenters are equipped to handle peak workloads, the average server utilization in most datacenters is very low. As a result, one can achieve huge energy savings by selectively shutting down machines when demand is low. In this dissertation, we introduce the network-aware machine activation problem to find a schedule that simultaneously minimizes the number of machines necessary and the congestion incurred in the network. Our model significantly generalizes well-studied combinatorial optimization problems such as hard-capacitated hypergraph covering and is thus strongly NP-hard. As a result, we focus on finding good approximation algorithms. Data-parallel computation frameworks such as MapReduce have popularized the design of applications that require a large amount of communication between different machines. Efficient scheduling of these communication demands is essential to guarantee efficient execution of the different applications. In the second part of the thesis, we study the approximability of the co-flow scheduling problem that has been recently introduced to capture these application-level demands. Finally, we also study the question, "In what order should one process jobs?'' Often, precedence constraints specify a partial order over the set of jobs and the objective is to find suitable schedules that satisfy the partial order. However, in the presence of hard deadline constraints, it may be impossible to find a schedule that satisfies all precedence constraints. In this thesis we formalize different variants of job scheduling with soft precedence constraints and conduct the first systematic study of these problems.

A Visual Analytics Approach to Comparing Cohorts of Event Sequences

Sequences of timestamped events are currently being generated across nearly every domain of data analytics, from e-commerce web logging to electronic health records used by doctors and medical researchers. Every day, this data type is reviewed by humans who apply statistical tests, hoping to learn everything they can about how these processes work, why they break, and how they can be improved upon. To further uncover how these processes work the way they do, researchers often compare two groups, or cohorts, of event sequences to find the differences and similarities between outcomes and processes. With temporal event sequence data, this task is complex because of the variety of ways single events and sequences of events can differ between the two cohorts of records: the structure of the event sequences (e.g., event order, co-occurring events, or frequencies of events), the attributes about the events and records (e.g., gender of a patient), or metrics about the timestamps themselves (e.g., duration of an event). Running statistical tests to cover all these cases and determining which results are significant becomes cumbersome. Current visual analytics tools for comparing groups of event sequences emphasize a purely statistical or purely visual approach for comparison. Visual analytics tools leverage humans' ability to easily see patterns and anomalies that they were not expecting, but is limited by uncertainty in findings. Statistical tools emphasize finding significant differences in the data, but often requires researchers have a concrete question and doesn't facilitate more general exploration of the data. Combining visual analytics tools with statistical methods leverages the benefits of both approaches for quicker and easier insight discovery. Integrating statistics into a visualization tool presents many challenges on the frontend (e.g., displaying the results of many different metrics concisely) and in the backend (e.g., scalability challenges with running various metrics on multi-dimensional data at once). I begin by exploring the problem of comparing cohorts of event sequences and understanding the questions that analysts commonly ask in this task. From there, I demonstrate that combining automated statistics with an interactive user interface amplifies the benefits of both types of tools, thereby enabling analysts to conduct quicker and easier data exploration, hypothesis generation, and insight discovery. The direct contributions of this dissertation are: (1) a taxonomy of metrics for comparing cohorts of temporal event sequences, (2) a statistical framework for exploratory data analysis with a method I refer to as high-volume hypothesis testing (HVHT), (3) a family of visualizations and guidelines for interaction techniques that are useful for understanding and parsing the results, and (4) a user study, five long-term case studies, and five short-term case studies which demonstrate the utility and impact of these methods in various domains: four in the medical domain, one in web log analysis, two in education, and one each in social networks, sports analytics, and security. My dissertation contributes an understanding of how cohorts of temporal event sequences are commonly compared and the difficulties associated with applying and parsing the results of these metrics. It also contributes a set of visualizations, algorithms, and design guidelines for balancing automated statistics with user-driven analysis to guide users to significant, distinguishing features between cohorts. This work opens avenues for future research in comparing two or more groups of temporal event sequences, opening traditional machine learning and data mining techniques to user interaction, and extending the principles found in this dissertation to data types beyond temporal event sequences.