With the growing demand for computation and the increasing prevalence of resource-constrained agents, the importance of leveraging a network of agents to solve complex problems has become often more pronounced. A multi-agent system consists of interconnected agents with computing capabilities, working collaboratively towards a shared objective. Distributed computation using a multi-agent system provides benefits with regards to privacy, reduction of computational load and resources. In this dissertation, we study two problems that benefit from being solved with the help of a multi-agent system namely (i) distributed convex optimization and (ii) distributed fact-checking.

In part I, we consider a set of agents collaboratively solving a distributed convex optimization problem, asynchronously, under stringent communication constraints. In such situations, when an agent is activated and is allowed to communicate with only one of its neighbors, we would like to pick the one holding the most informative local estimate. We propose new algorithms where the agents with maximal dissent average their estimates, leading to an information mixing mechanism that often displays faster convergence to an optimal solution compared to randomized gossip.

In Part II, we explore a distributed fact-checking system to detect fake news using inexpert agents. Each agent labels news as true or false based on its reliability, modeled as a Binary Symmetric Channel (BSC) with some error probability. We develop an algorithm that estimates statement validity by thresholding a linear combination of agents' labels and deriving optimal weights and thresholds to minimize error probability. Moreover, we present an adaptive algorithm to learn the agents' unreliability parameters and prove the convergence of the adaptive estimator. We also propose a broader class of adaptive estimators for the agents' unreliability parameters, providing the necessary conditions for convergence. We show that estimators for ensembles of two and three agents adhere to a consistent update rule, while hard-decoded estimates fail to converge for any number of agents.

This dissertation contributes to the theoretical aspects of distributed optimization and fact-checking in multi-agent systems, offering novel algorithms and insights for efficient and reliable distributed decision-making.

Mathematical models of networked dynamical systems are ubiquitous - they are used to study power grids, networks of webpages, robotic and sensor networks, and social networks, to name a few. Importantly, most real-world networks are time-varying and are affected by stochastic phenomena such as adversarial attacks and communication link failures. Time-varying networks, therefore, have been under study for a few decades. However, our current understanding of the dynamical processes evolving over such networks is limited. This observation motivates the two-pronged objective of this dissertation: (i) to use theoretical and empirical methods to analyze certain networked dynamical systems that cannot be studied using standard tools and techniques, and (ii) to develop suitable mathematical techniques for the systematic study of such systems.

As our main contribution resulting from (i), we use the properties of random time-varying networks to provide a rigorous theoretical foundation for the age-structured Susceptible-Infected-Recovered model, a model of epidemic spreading. We then use system identification to show that the age-structured SIR dynamics accurately model the spread of COVID-19 at city and state levels in two different parts of the world – Tokyo and California.

As for our contributions resulting from (ii), we extend two assertions of the Perron-Frobenius theorem to time-varying networks described by strongly aperiodic stochastic chains, thereby widening the applicability of the fundamental result that is foundational to probability theory and to the studies of complex networks, population dynamics, internet search engines, etc. Our results enable us to extend several known results on distributed learning and averaging. Moreover, they promise to advance our understanding of dynamical processes over real-world networks.

As an application of these results, we study non-Bayesian social learning on random time-varying networks that violate the standard connectivity criterion of uniform strong connectivity. In doing so, we also make a methodological contribution: we show how the theory of absolute probability sequences and martingale theory can be combined to analyze nonlinear dynamics that approximate linear dynamics asymptotically in time.

Finally, we study the convergence properties of social Hegselmann-Krause dynamics (which is a variant of the classical Hegselmann-Krause model of opinion dynamics and incorporates state-dependence into distributed averaging). As our main contribution here, we provide nearly tight necessary and sufficient conditions for a given connectivity graph to exhibit unbounded epsilon-convergence times for such dynamics.

The transition to a sustainable and resilient power grid is essential for addressing the environmental and socio-economic challenges posed by climate change. While the integration of renewable energy resources offers significant benefits, it also introduces complexities in grid management due to their non-dispatchable nature and the need for advanced energy storage solutions. This dissertation focuses on optimizing grid operations, managing renewable integration, and ensuring reliability under various conditions, presenting novel methodologies and insights into our future energy infrastructure. Grid decarbonization is a primary goal, driven by the need to reduce greenhouse gas emissions, mitigate climate change, and promote public health. It also fosters economic benefits such as green job creation and energy independence. However, the shift to a cleaner energy grid presents challenges, including the increased penetration of non-dispatchable resources like wind and solar, the necessity for advanced energy storage solutions, and the requirement for economic analyses to evaluate new technologies and policies. Additionally, upgrading the existing grid infrastructure to accommodate renewable energy sources is crucial to ensure reliability and resilience. Enhancing grid resilience is vital for mitigating the impacts of natural disasters, improving power system reliability and security, and ensuring continuity of supply to customers and critical loads. Achieving resilience involves modeling and incorporating risk into grid operations, preparing for natural disasters, and addressing the exponential complexity of optimization problems. This research introduces wildfire risk-aware planning and the strategic relocation of electric vehicles (EVs) for resilient grid operation as key components of a robust resilience framework. To further improve grid operations, this dissertation proposes an agile decision-support system that accelerates power grid optimization. Enhanced grid resilience is achieved by making the system more robust against disruptions, and improved decision-making is facilitated through faster and more accurate analyses. Real-time optimization becomes feasible, allowing for immediate adjustments to changing conditions, reducing costs, increasing grid flexibility, and improving the accuracy of predictions and planning. This leads to more effective utilization of renewable energy sources. This dissertation presents comprehensive strategies to enhance the resilience and sustainability of a decarbonized power grid. By leveraging advanced optimization techniques and incorporating dynamic models of natural gas and renewable energy interactions, it addresses the critical need for real-time optimization, improved decision-making, and effective renewable energy utilization, ultimately contributing to a more resilient and sustainable energy future.

In this thesis, we study Distributed Averaging Dynamics and its main application, i.e. Distributed Optimization. More specifically, the results of this thesis can be divided into two main parts: 1) Ergodicity of distributed averaging dynamics, and 2) Distributed optimization over dependent random networks.

First, we study both discrete-time and continuous-time time-varying distributed averaging dynamics. We show a necessary and a sufficient condition for ergodicity of such dynamics. We extend a well-known result in ergodicity of time-homogeneous (time-invariant) averaging dynamics and we show that ergodicity of a dynamics necessitates that its (directed) infinite flow graph has a spanning rooted tree. Then, we show that if groups of agents are connected using a rooted tree and the averaging dynamics restricted to each group is $\Pst$ and ergodic, then the dynamics over the whole networks is ergodic. In particular, this provides a general condition for convergence of consensus dynamics where \textit{groups} of agents capable of reaching consensus follow each other on a time-varying network.

Then, we study the averaging-based distributed optimization solvers over random networks for both convex and strongly convex functions. We show a general result on the convergence of such schemes for a broad class of dependent weight-matrix sequences. In addition to implying many of the previously known results on this domain, our work shows the robustness of distributed optimization results to link-failure. Also, it provides a new tool for synthesizing distributed optimization algorithms. To prove our main theorems, we establish new results on the rate of convergence analysis of averaging dynamics and non-averaging dynamics over (dependent) random networks. These secondary results, along with the required martingale-type results to establish them, might be of interest to broader research endeavors in distributed computation over random networks.

Emerging sustainable solutions in smart grids such as Distributed Energy Resources(DERs), Vertical Farms (VFs), and Electric Vehicles (EV s) enhances the resilience of the Active Distribution Networks (ADNs). We propose an operational framework to integrate emerging technologies into ADN under solar and demand uncertainty to reduce the operation cost, carbon emissions, and power quality issues of the smart communities. We primarily focus on enhancing the reliability and efficiency of power systems through advanced optimization techniques. A key contribution of this work is the development of a second-order cone programming (SOCP) relaxation approach that addresses cycle constraints in the optimal xxi power flow (OPF) problem. We demonstrate how this method improves the tightness of solutions and reduces computation burdens compared to traditional convex relaxation methods. Additionally, we introduced a strategic bidding framework in electricity markets based on the convexified AC market clearing problem (MCP) for market participants to optimize their bids under various market conditions. Our results demonstrate that the proposed framework renders 52.3% more profit for the market participants. The increase in utilization of solar generation units and EV s in past decades, especially in California, lead to power quality and uncertainty issues in the grid. We proposed the model of Smart Inverters (SI s) to leverage the potential of DERs to offer grid services through reactive power compensation. The results show that solar dispatchability increases by 12% and voltage violation is mitigated when the proposed volt/VAR model is utilized for IEEE 33-bus system. Furthermore, we propose the model of various systems in VFs as an emerging sustainable demand responsive asset in ADNs to mitigate the carbon emissions of the smart communities and increase the resiliency of ADNs against uncertainties in demand and solar generation. To quantify the uncertainties more accurate based on the real data and decrease the conservatism of the robust optimization (RO) method, we propose the data-driven mean robust optimization (MRO) approach for the vertical farming expansion planning problem in ADNs. It is shown that the proposed MRO method can mitigate the carbon emissions of the smart city by up to 10% and decrease the total operation and planning cost of the system compared to utilizing the more conservative RO method.