Dynamic network processes have surrounded people for millennia. Information spread through social networks, alliance formation in financial and organizational networks, heat diffusion through material networks, and distributed synchronization in robotic networks are just a few examples. Network processes are studies along three dimensions: analysis of network processes through the data produced by them; designing complex plausible, yet, tractable mathematical models for network processes; and designing control mechanisms that would guide network processes towards desirable evolution patterns. This thesis advances the frontier of knowledge about network processes along each of these three dimensions, emphasizing applications to social networks.

The first part of the thesis is dedicated to the design of a method for model-driven analysis of a polar opinion formation process in social networks. The core of the method is a distance measure quantifying the likelihood of a social network's transitioning between different states with respect to a chosen opinion dynamics model characterizing expected evolution of the network's state. I describe how to design such a distance measure relying upon the classical transportation problem, compute it in linear time, and use it in applications.

In the second part of the thesis, I focus on designing a model for polar opinion formation in social networks, and define a class of non-linear models that capture the dependence of the users' opinion formation behavior upon the opinions themselves. The obtained models are connected to socio-psychological theories, and their behavior is theoretically analyzed employing tools from non-smooth analysis and a generalization of LaSalle Invariance Principle.

The third part of the thesis targets the problem of defense against social control. While the existing socio-psychological theories as well as influence maximization techniques expose the opinion formation process in social networks to external attacks, I propose an algorithm that nullifies the effect of such attacks by strategically recommending a small number of new edges to the network's users. The optimization problem underlying the algorithm is NP-hard, and I provide a pseudo-linear time heuristic---drawing upon the theory of Markov chains---that solves the problem approximately and performs well in experiments.

This thesis is dedicated to learning structured representations of data, to be utilized by a diverse set of downstream tasks emerging in numerous application domains such as user modeling, document categorization, and graph representation learning. Despite their notable contextual differences, the common objective behind these studies is our desire to incorporate the rich structure of data into our learned representations (both discrete and continuous) and computations, in ways that are unique to each problem. We provide comprehensive evidence showing that the principled utilization of data structure, regardless of the problem domain, is a key step towards reaching our learning objectives.

The second half of the thesis specifically targets problems centered around a complex family of graphs called flow graphs. Besides nodes and edges, flow graphs capture edge flows of a quantity of interest (e.g., water, power, people) being transported through the graph. As these flows often possess higher-order graph-level dynamics driven by sources/destinations, hotspots, and domain-specific physics, the underlying structure of flow graphs poses unique challenges from a learning viewpoint. In particular, we study two challenging problems: (i) network flow estimation based on partial flow observations, and (ii) data-driven generation of realistic flow graphs as an alternative to domain-specific simulations. During each of these studies, we showcase the complex structural patterns emerging in many real-world flow graphs and propose novel methodologies that can account for such richly structured information.

Graphs are a powerful tool for the study of dynamic processes, where a set of interconnected entities change their states according to the time-varying behavior of an underlying complex system. For instance, in a social network, an individual's opinions are influenced by their contacts; while, in a traffic network, traffic conditions are spatially localized due to the fact that vehicles are often constrained to move along roads. Understanding the interplay between structure and dynamics in networked systems enables new models, algorithms, and data structures for managing and learning from large amounts of data arising from these processes.

This dissertation is focused on recent work on the analysis of dynamic graph processes. More specifically, we will show how sampling and spectral graph theory can be applied to effectively represent data from such processes. We also present efficient algorithms for learning Interleaved Markov Models (IHMMs). These are powerful latent variable models that enable the clustering of discrete sequences without training data and lead to interesting challenges in terms of inference. Furthermore, we also discuss how graphs can be modified in order to control a process of interest via network design and introduce three ongoing projects on the subject of this dissertation. The statement of the thesis is that mining and modeling processes on graphs leads often to problems that are not not only hard computationally but also in terms of inference. They can be solved using spectral, probabilistic, and combinatorial optimization algorithms, and must take into account the graph structure and also large amounts of data traces from these processes.

Large-scale temporal graphs are everywhere in our daily life. From online social networks, mobile networks, brain networks to computer systems, entities in these large complex systems communicate with each other, and their interactions evolve over time. Unlike traditional graphs, temporal graphs are dynamic: both topologies and attributes on nodes/edges may change over time. On the one hand, the dynamics have inspired new applications that rely on mining and managing temporal graphs. On the other hand, the dynamics also raise new technical challenges. First, it is difficult to discover or retrieve knowledge from complex temporal graph data. Second, because of the extra time dimension, we also face new scalability problems. To address these new challenges, we need to develop new methods that model temporal information in graphs so that we can deliver useful knowledge, new queries with temporal and structural constraints where users can obtain the desired knowledge, and new algorithms that are cost-effective for both mining and management tasks.

In this dissertation, we discuss our recent works on mining and managing large-scale temporal graphs.

First, we investigate two mining problems, including node ranking and link prediction problems. In these works, temporal graphs are applied to model the data generated from computer systems and online social networks. We formulate data mining tasks that extract knowledge from temporal graphs. The discovered knowledge can help domain experts identify critical alerts in system monitoring applications and recover the complete traces for information propagation in online social networks. To address computation efficiency problems, we leverage the unique properties in temporal graphs to simplify mining processes. The resulting mining algorithms scale well with large-scale temporal graphs with millions of nodes and billions of edges. By experimental studies over real-life and synthetic data, we confirm the effectiveness and efficiency of our algorithms.

Second, we focus on temporal graph management problems. In these study, temporal graphs are used to model datacenter networks, mobile networks, and subscription relationships between stream queries and data sources. We formulate graph queries to retrieve knowledge that supports applications in cloud service placement, information routing in mobile networks, and query assignment in stream processing system. We investigate three types of queries, including subgraph matching, temporal reachability, and graph partitioning. By utilizing the relatively stable components in these temporal graphs, we develop flexible data management techniques to enable fast query processing and handle graph dynamics. We evaluate the soundness of the proposed techniques by both real and synthetic data.

Through these study, we have learned valuable lessons. For temporal graph mining, temporal dimension may not necessarily increase computation complexity; instead, it may reduce computation complexity if temporal information can be wisely utilized. For temporal graph management, temporal graphs may include relatively stable components in real applications, which can help us develop flexible data management techniques that enable fast query processing and handle dynamic changes in temporal graphs.

Network structure exists in various types of data in the real world, such as online and offline social networks, traffic networks, computer networks, brain networks, and countless other cases where there are relationships between different entities in the data. What are the roles of network structures in these data? First, the network captures inherent characteristics of the data themselves. This is clear from the definition of the network, which represents the relationship between entities: e.g., the social links among people in a social network describe how they interact with each other; a road network summarizes how the roads are laid out geographically; a brain network obtained from fMRI images represents pairs of brain regions that are active at the same time; a computer network constrains the paths via which internet packages and thus information or viruses can spread. Second, the network structures affect the evolution of the data over time. For example, new friendship links in an online social network are frequently created between friends of friends. Similarly, the current road network structure is without a doubt taken into consideration when roads are added or temporarily closed. As we grow, our brains also grow, including the additions of useful links or the clean up of unnecessary links between brain regions. Third, the network structures act as guidance for many different processes happening in the data. For instance, the links between users on social network dictate how gossips can spread; the roads influence how traffic flows in a city; the links between brain regions affects the way we think and how effectively we do things; the connections between computers route the transfer of any information on the internet.

In this thesis, I studied the network effect in various networked behaviors, including analyzing such effect, finding its patterns, and predicting future networked behaviors. First, I gained insights into the data by analyzing the accompanied network structures as well as its evolution. Second, I proposed algorithms for mining different network patterns that help summarize the effect of the network structures on different networked behaviors. Finally, I proposed models to predict the evolution of networked behaviors over time. Toward these tasks, I explored a wide variety of network data, including protein-protein interaction networks, online social networks, collaboration networks, chemical compounds, and traffic networks. Overall, I tackled these network data in different aspects and developed a number of methods for effectively mining and forecasting networked behaviors in data.

We have witnessed massive advances in predictive modeling within the past decade, with machine learning models achieving superhuman performance in a variety of tasks. However, the notion that such models are a 'black-box' with little to no explanatory power has limited their impact on fields where erroneous data-driven decisions can have severe consequences, such as to do with our health. Health data in particular has the potential for transformational impact to our lives with the improved efficiency machine learning models have brought to other domains. There is a need for new computational approaches that can derive insight from health data while addressing these concerns.

In this dissertation, we describe novel computational methods on health data that do not just achieve high performance on singular performance metrics but chiefly to characterise the data. The methods combine insights from statistical and machine learning, network science, and Bayesian uncertainty quantification to improve our understanding of human health data through the lens of two main sources of data: the human brain as imaged by diffusion MRI, and human physiology as measured by wearable sensors. (1) We show how to find brain regions that are more cohesive within a population of interest, discovering that nearly 4% of white matter is associated with genetic similarity. (2) We quantify how informative an individual's attributes are for generative regions of white matter, and (3) also the dual problem of measuring how predictive a region of white matter is of an attribute. (4) Lastly, we demonstrate how to measure a cardiovascular transfer function from digital activity traces by learning from 'natural experiments' during daily living conditions, and show that these transfer functions are predictive of variables associated with cardiovascular health.

The field of machine learning has seen great strides in the past two decades, leading to groundbreaking advancements in various domains. Yet many challenges remain in AI research. Scalability remains a persistent hurdle, with current models struggling to efficiently manage increasing amounts of data and complexity. The challenge of scalability is further highlighted by the power of the human brain, which, with its remarkable efficiency, underscores the limitations of current machine learning models. Finally, both industry and academia have faced considerable difficulties in advancing robotics, particularly in enabling neural networks to interact effectively with the physical world.

In this work, we explore how the lens of dynamical systems can address all three of these challenges. We tackle memory constraints by framing a quantization algorithm as a fixed-point equation, enabling efficient differentiation. We investigate an energy-based, biological model of human memory and reinterpret the widely used self-attention mechanism through this model. Lastly, we leverage contraction theory to train a neural network that can follow a trajectory with stability and robustness. In the first two problems, we utilize dynamical systems to differentiate neural networks, while in the latter, we employ neural networks to learn a dynamical system. Through our understanding of dynamical systems, we build on theoretical advancements and practical applications in AI, offering new insights into both memory optimization and robust robotic manipulation.