Networks are ubiquitous in nature and engineering, with applications in areas such as modeling power grids, population dynamics, and social networks. Applying network science to complex dynamical systems provides us with powerful tools to study the interactions between agents. Moreover, these tools can also be used to study the nonlinear dynamics of the network itself and he combination of dynamics over networks and dynamics of networks paves the way for new models that accurately capture the rich behavior in sociological processes. In this thesis, we first study synchronization in networks of coupled oscillators. Second we model and analyze the dynamics of influence networks of human teams.

Regarding coupled oscillators, we study the frequency synchronization problem for networks of Kuramoto oscillators with arbitrary topology and heterogeneous edge weights. We propose a novel equivalent transcription for the equilibrium synchronization equation. Using this transcription, we develop a power series expansion to compute the synchronized solution of the Kuramoto model as well as a sufficient condition for the strong convergence of this series expansion. Truncating the power series provides (i) an efficient approximation scheme for computing the synchronized solution, and (ii) a simple-to-check, statistically-correct hierarchy of increasingly accurate synchronization tests. This hierarchy of tests provides a theoretical foundation for and generalizes the best-known approximate synchronization test in the literature.

Regarding the dynamics of social networks of human teams, we focus on modeling and analyzing how performance and expertise impact the level of influence team members have on each other. First, we propose a novel quantitative model describing the decentralized process by which individuals in a team learn who has what abilities, while concurrently assigning tasks to each of the team members. Our theoretical analysis characterizes a team's ability, or inability, to learn each other's skill and thus converge to a work allocation maximizing the team performance. Second, we propose a cognitive dynamical model to describe the process by interpersonal influences are adjusted in small teams over a sequence of intellective tasks with fixed workload. We provide analytical results on the model's asymptotic behavior for the case with identically performing individuals and verify the accuracy of the proposed model on experimental data.

Autonomous mobile systems are becoming more common place, and have the opportunity to revolutionize many modern application areas. They include, but are not limited to, tasks such as search and rescue operations, ad-hoc mobile wireless networks and warehouse management; each application having its own complexities and challenging problems that need addressing. In this thesis, we explore and characterize two application areas in particular.

First, we explore the problem of autonomous stochastic surveillance. In particular, we study random walks on a finite graph that are described by a Markov chain. We present strategies that minimize the first hitting time of the Markov chain, and look at both the single agent and multi-agent cases. In the single agent case, we provide a formulation and convex optimization scheme for the hitting time on graphs with travel distances. In addition, we provide detailed simulation results showing the effectiveness of our strategy versus other well-known Markov chain design strategies. In the multi-agent case, we provide the first characterization of the hitting time for multiple random walkers, which we denote the "group hitting time". We also provide a closed form solution for calculating the hitting time between specified nodes for both the single and multiple random walker cases. Our results allow for the multiple random walks to be different and, moreover, for the random walks to operate on different subgraphs. Finally, we use sequential quadratic programming to find the transition matrices that generate minimal "group hitting time".

Second, we consider the problem of optimal coverage with a group of mobile agents. For a planar environment with an associated density function, this problem is equivalent to dividing the environment into optimal subregions such that each agent is responsible for the coverage of its own region. We study this problem for the discrete time and space case and the continuous time and space case. For the discrete time and space case, we present algorithms that provide optimal coverage control in a non-convex environment when each robot has only asynchronous and sporadic communication with a base station. We introduce the notion of coverings, a generalization of partitions, to do this. For the continuous time and space case, we present a continuous-time distributed policy which allows a team of agents to achieve a convex area-constrained partition in a convex workspace. This work is related to the classic Lloyd algorithm, and makes use of generalized Voronoi diagrams. For both cases we provide detailed simulation results and discuss practical implementation issues.

Building from existing work on Virtual Oscillator Control (VOC), this reports details design, analysis, and simulation of a standalone inverter-based microgrid composed of variable renewable energy sources and battery storage. VOC for inverters relies on local voltage and frequency measurements to allow for proportional loading and voltage/frequency synchronization, presenting a real-time decentralized control strategy that mimics droop control characteristics. The design specifications, derivations, models, and simulations demonstrate that VOC can operate bidirectionally (producing and consuming power) for resistive and capacitive loads. The inverters are shown to operate in a variety of more complex settings than have previously been examined: in parallel, under fluctuating loads, and under noisy loads. To compensate for deviations in steady-state frequency and voltage, low-bandwidth communication in the form distributed averaging proportional integral (DAPI) control is adapted, derived, and simulated for a VOC-based microgrid. Lastly, battery models representing distributed Energy Storage Systems (ESS) are incorporated to demonstrate grid robustness in times of excess load relative to generation available.

The focus of this work is path planning algorithms for autonomous agents. Specifically, we study problems in three areas where path planning to direct the motion of autonomous agents is critical for their performance. The first problem is a vehicle routing problem in which mobile demands appear in an environment and the task of the autonomous agent is to stop the demands from escaping the environment boundary. We first propose two fundamental performance bounds for the proposed problem. We then propose routing algorithms for this problem with performance guarantees. We examine the gap between these guarantees and the fundamental performance bounds. The second problem is a surveillance problem in a networked environment. The tasks of the autonomous surveillance agent in this problem are to (1) detect unknown intruder locations and (2) detect anomalies based on noisy measurements. We propose Markov chain based routing algorithms for the surveillance agent to achieve these goals. We parameterize these routing algorithms using a property of Markov chains called the mean first passage time. We also frame optimization problems to obtain optimal algorithms for the two surveillance tasks. The third problem studied in this work is a boundary guarding problem in which the task of a set of patrolling agents constrained to move on a ring is to achieve synchronization using only local communication. We propose a coordination algorithm to solve this problem and identify initial agent configurations under which synchronization is guaranteed.

This thesis contains two parts. In the first part, we discuss the robotic surveillance problems, with a focus on the Markov chain-based stochastic approaches. In chapter 2, we study the problem of maximizing the return time entropy of a Markov chain, subject to graph and stationary distribution constraints. We first obtain complete characterizations for the return time distribution and show its convergence. We then establish upper and lower bounds for the return time entropy. We also provide gradients of the truncated entropy function for computational purposes. Finally, we present the numerical comparisons between the proposed and existing Markov strategies.

In chapter 3, we analyze the meeting time between a pursuer and an evader. First, by analyzing multiple random walks on a common graph as a single random walk on the Kronecker product graph, we provide a closed-form expression for the expected meeting time. This novel expression leads to necessary and sufficient graph-theoretic conditions for the meeting time to be finite. Second, we study the minimization problem for the expected capture time for the pursuer. We finally report theoretical and numerical results on basic case studies.

In chapter 4, we study the problem where the mobile robot tries to capture a strategic intruder who knows the current location and the surveillance strategy of the mobile agent. We model the scenario by a Stackelberg game and study optimal and suboptimal surveillance strategies in star, complete and line graphs. We first derive a universal upper bound on the capture probability, which is shown to be tight in the complete graph and provides suboptimality guarantees. For the star and line graphs, we characterize dominant strategies and rigorously prove the optimal surveillance plan.

In the second part, chapter 5, we study the graph-theoretic conditions for stability of positive monotone systems. We first establish necessary and sufficient conditions for the stability of linear positive systems described by Metzler matrices. Specifically, we derive two sets of stability conditions based on two forms of input-to-state stability gains, namely max-interconnection gains and sum-interconnection gains. We then extend our results to nonlinear monotone systems.

The scientific world has witnessed a significant paradigm shift in recent years: that the networks should not be studied in isolation from the processes taking place over them and the large amounts of data derived and generated by them. Science has now embraced a systems approach that captures the effect of the interconnections between individual units and the behavior of a network system. This dissertation provides modeling and analysis of dynamical phenomena over interconnected network systems.

In chapter 1, we review a class of deterministic nonlinear models for the propagation of infectious diseases over contact networks with strongly-connected topologies. We consider network models for susceptible-infected (SI), susceptible-infected-susceptible (SIS), and

susceptible-infected-recovered (SIR) settings. In each setting, we provide a comprehensive nonlinear analysis of equilibria, stability properties, convergence, monotonicity, positivity, and threshold conditions.

The recent convergence of research in social sciences, dynamic modeling, and network science has encouraged reexamining the collective team behavior from a quantitative perspective. Research shows that teams cannot be understood fully by studying their members in isolation. To study the coordination and control features of a group task, the multiple subgroups' performances must be fitted together. On such decomposed tasks, group performance is more than a simple union of subgroup performances. This work aims to understand how patterns of interactions among teams impact performance.

In chapter 2, we investigate the implications of dierent forms of multi-group connecviiitivity. Four multi-group connectivity modalities are considered: co-memberships, edge bundles, bridges, and liaison hierarchies. We propose generative models to generate these four modalities. Our models are variants of planted partition or stochastic block models conditioned under certain topological constraints. We report findings of a comparative analysis in which we evaluate these structures, controlling for their edge densities and sizes, on mean rates of information propagation, convergence times to consensus, and steady state deviations from the consensus value in the presence of noise as network size increases.

In chapter 3, we present a strategic network formation model predicting the emergence of multigroup structures. Individuals decide to form or remove links based on the benets and costs those connections carry; we focus on bilateral consent for link formation. We are interested in structures that arise to resolve coordination issues and, specifically, structures in which groups are linked through bridging, redundant, and co-membership interconnections. We characterize the conditions under which certain structures are stable and study their efficiency as well as the convergence of formation dynamics.

Networks are ubiquitous in natural and engineered systems, from critical infrastructures (like power grids and water distribution systems) to contact networks in epidemiological models. Managing these systems requires a broad array of tools to monitor the configuration and state of the network, identify optimal operating points, and design controls. This thesis examines a collection of topics broadly related to this theme.

In Part 1, we consider problems related to control and optimization in network systems. Chapter 1 studies the problem of safety-critical control in networks of grid-forming inverters. Coupling a physically-meaningful Lyapunov-like function with an optimization approach to identifying forward-invariant sets, we propose a method to certify that a post-fault trajectory achieves frequency synchronization while respecting safety constraints. In Chapter 2, we consider the network resource allocation problem of optimally distributing resources to mitigate the spread of an epidemic. We propose a convex optimization framework for minimizing the basic reproduction number for general compartmental epidemiological models. Chapter 3 addresses optimal control in infinitesimally contracting systems. We provide new convergence criteria for a common indirect optimal control algorithm, and we establish the uniqueness of the optimal control in the limits of large contraction rates and short time horizons.

In Part 2, we use tools from statistical inference and machine learning to solve estimation problems in network systems. Chapter 4 examines the problem of inferring routing topologies from endpoint data in communication networks. Extending a technique called network tomography to use higher-order statistics, and using Mobius inversion to disentangle the interactions between different network paths that are reflected in these statistics, we are able to estimate routing matrices without the cooperation of intermediate routers. Finally, in Chapter 5, we use machine learning to predict edge flows. We propose an implicit neural network that incorporates two fundamental physical principles to estimate flows on unlabeled edges, and we provide a contraction mapping to evaluate the model and backpropagate loss gradients.

In this thesis, I will first present mathematical models that explain the evolution of interpersonal relationships in a social network, represented by a signed graph, converging to structures that have a long history in sociology - namely, structural and clustering balance. Then, I will present a simple model for the evolution of opinions over signed graphs, including the aforementioned special structures. Then, I will present an important phenomenon that occurs on the susceptible-infected-susceptible (SIS) model of epidemics: the emergence of a new epidemic domain of bistability when higher-order interaction among individuals are considered on the contact network. Then, I will present an algorithm for the computation of Wasserstein barycenters, and show a connection with the theory of opinion dynamics. Finally, the last part of this thesis is devoted to the study and application of contraction theory, an important tool that certifies incremental stability. We study its expansion to dynamical systems on Hilbert spaces, as well as its application to various optimization problems and settings.