The Perron–Frobenius and Koopman operators provide natural dual settings to investigate the dynamics of complex systems. In this thesis we focus on certain pertinent concepts and strategies for obtaining dynamical models and approximating the transfer operators from data.

First, we explain the setting and the assumptions that underlie the so-called Dynamic Mode Decomposition (DMD); this methodology relates to Koopman-operator approximation through full-state observables. The goal is to highlight caveats as well as to suggest metrics that indicate that the use of DMD on specific dataset is warranted. In many applications it is often the case that only a limited number of data samples is available for modeling an otherwise exceedingly high dimensional process. The dimensionality of the process, which may represent visual or distributional fields, in conjunction with the limited observation record requires careful analysis. It is precisely this regime of ``small data,'' i.e., ``few samples,'' that has been a challenge in traditional signal analysis since its inception. DMD is a recent development that aims to identify suitable linear dynamics that can explain the data. We show how the concept of the gap metric can be used as a tool to quantify how subspaces spanned by data impact modeling assumptions. Also, the gap metric provides guidance in selecting appropriate dimensionality for models for such processes.

Next, we formulate and solve a regression problem with time-stamped distributional data.Distributions are considered as points in the Wasserstein space of probability measures, metrized by the $2$-Wasserstein metric, and may represent images, power spectra, point clouds of particles, and so on. The data sets may be thought to represent densities of particles whose precise trajectories are not be available (e.g., partially observed). The regression seeks a curve in the Wasserstein space that passes closest to the dataset. Our regression problem allows utilizing general curves in a Euclidean setting (linear, quadratic, sinusoidal, and so on), lifted to corresponding measure-valued curves in the Wasserstein space. It represents a relaxation of geodesic regression in Wasserstein space. The apparently nonlinear primal problem can be recast as a multi-marginal optimal transport, leading to a formulation as a linear program. Entropic regularization and a generalized Sinkhorn algorithm can be effectively employed to solve this multi-marginal problem.

The proposed framework can be used to estimate correlation between given distributional snapshots. Potential applications of the theory are envisioned to aggregate data inference, estimating meta-population dynamics, power spectra tracking, and more generally, system identification.

Finally, we introduce a regression-type formulation for approximating the Perron-Frobenius operator by relying on distributional snapshots of data. %These snapshots may represent densities of particles. The Wasserstein metric is leveraged to define a suitable functional optimization in the space of distributions, weighing in distances between successive distributional snapshots. The formulation allows seeking suitable dynamics so as to interpolate the distributional flow in function space. A first-order necessary condition for optimality is derived and utilized to construct a gradient flow approximating algorithm. It should be noted that we assume no information on statistical dependence between successive pairs of distributions. The method extends to search for nonlinear dynamics assuming a suitable parametrization of the nonlinear state transition map in terms of selected basis functions.

ABSTRACT OF THE DISSERTATIONAnalysis of Social and Flow Networks By Zahra Askarzadeh Doctor of Philosophy in Mechanical and Aerospace Engineering University of California, Irvine, 2021 Professor Tryphon T. Georgiou, Chair In this thesis, we study problems in analysis of social and flow networks. Specifically, we study models of social interactions between individuals who discuss and form opinions about a sequence of issues. We also study quantifying macroscopic circulation in a given planar graph. A social network is a medium for the exchange of information, ideas, and influence among its members. In recent years, availability of large amounts of data from online social networks have drawn the attention of many researchers to study opinion formation and the evolutionary behaviors in social networks. In this thesis, we revisit several types of opinion dynamics models and review relevant results from the literature. We then present a set of new results related to both modeling and analysis of social networks. Starting with analyzing DeGroot-Friedkin model, we establish existence and uniqueness of its fixed point using local inverse function theorem and Hadamard’s global inverse function theorem. Motivated by DeGroot-Friedkin model, we then propose a group of nonlinear Markov chain models of social interaction for the purpose of assessing opinion evolu- tion in social networks. We seek and develop conditions that determine when such system display oscillations, manifest chaos, or lead to a stable equilibrium that represents consensus. We also provide extensions of proposed models to count for different subgroups of interacting individuals. Flow networks are typically used to model problems involving the transportation of mass between nodes, through routes that have limited capacity. Examples of such problems that motivate our research are modeling traffic on a network of roads and blood current in heart. Based on these examples, we introduce a new concept of maximal global circulation and explore 3-partitions that expose this type of macroscopic feature of networks. Herein, graph circulation is motivated by probabilistic flows (Markov chains) on graphs. Our goal is to quantify the large-scale imbalance of network flows and delineate key parts that mediate such global features. While we introduce and propose these notions in a general setting, here, we only work out the case of planar graphs. We explain that a scalar potential can be identified to encapsulate the concept of circulation, quite similarly as in the case of the curl of planar vector fields. Beyond planar graphs, in the general case, the problem to determine global circulation remains at present a combinatorial problem.

In an increasingly interconnected world, the efficient and economical transportation of individuals and commodities has emerged as a cornerstone of modern society. Optimizing transportation plans has a huge potential for journey planning, congestion reduction, supply chain management, and data exchanges. These strategies hold immense relevance not onlyin the realm of engineering and transportation, but also in other fields, such as physics, computer science, economics, and several subject areas in mathematics. The present thesis aims to elucidate the optimization of transportation strategies, with a particular focus on two classical problems, finding short paths in large networks and solving optimal transport problems with flow-rate constraints.

The so-called shortest path problem seeks an optimal path of transporting one unit of mass between pairs of vertices on graphs. We present a novel formulation of the problem as an l1-regularized regression, often referred to as lasso (Least Absolute Shrinkage and Selection Operator). Based on this formulation, we draw a connection specifically between trees that grow as active edge-sets in the least angle regression (LARS) algorithm of the lasso problem, and respective shortest-path trees that emerge using the bi-directional Dijkstra algorithm. Then, to overcome the dimensionality challenge in large graphs, we explore the alternating direction method of multipliers (ADMM) in the lasso formulation. The resulting derivativeproximal algorithm speeds up the search for the short paths, trading off optimality (i.e., finding shortest paths) that may not be absolutely essential in a variety applications.

The basic transport problem is motivated by the need to transport resources/mass between end-point distributions (supply and demand). We consider the classical Monge-Kantorovich optimal transport problem with a quadratic cost functional to penalize distance of transport, with an added constraint that transported mass is required to pass through constriction points while abiding by specified allowable flow-rate; constriction points may be conceptualized as toll stations with limited throughput. Our contributions in this topic are as follows: (1) we provide a precise Monge formulation for the optimal transport problem with flux constraint at constriction sites along the path that is amenable to generalization in higher dimensions. We work out in detail the case of transport in one dimension by proving existence and uniqueness of solutions. Under suitable regularity assumptions we give an explicit construction of the transport plan; (2) we provide a Kantorovich-type reformulation of the problem by introducing a marginal probability density for the time that mass-elements cross toll stations –a probability density that is to be determined so as to meet given flow-rate constraints. Interestingly, the Kantorovich-type formalism leads to multi-marginal optimal transport problem that is readily solvable by using linear programming. Moreover, existence and uniqueness of solutions are also established in this setting. Then, (3) we propose an entropic penalty term to regularize and reduce the computational cost of resulting multi-marginal problems. Entropic regularization of standard optimal transport leads to an efficient algorithm, the Sinkhorn algorithm, which applies in the present case as well. Leveraging the splittable nature of the cost in our formulation, we proposed a Gluing Sinkhorn algorithm for the multi-marginal optimal transport problem, which reduces the computational cost to a level comparable to that in standard two-marginal problems.

Classical thermodynamics is aimed at quantifying the efficiency of thermodynamic engines by bounding the maximal amount of mechanical energy produced compared to the amount of heat required under quasi-static operation. While this was accomplished early on, by Carnot and Clausius, the more practical problem of quantifying limits of output power remained elusive due to the fact that quasi-static processes require infinitely slow cycling, resulting in a vanishing power. Recent insights, drawn from stochastic thermodynamics, bridge the gap between theory and practice by presenting fresh approaches that lead to general laws applicable to the non-equilibrium system. Remarkably, the problem of minimizing dissipation over a finite time window can be expressed as a stochastic control problem leading to physically meaningful expressions for the dissipation cost in thermodynamic engines.

Building on the framework of stochastic thermodynamics, we derive bounds on the maximal power that can be drawn by cycling an overdamped ensemble of particles via a time-varying potential while alternating contact with heat baths of different temperature (Tc cold, and Th hot). The previous work focused on the setting where the potential is quadratic andthe distributions are Gaussian. Our analysis relaxes this assumption and asks about fundamental bounds on the power under arbitrary distribution. Specifically, first by casting the optimization problem of the power into the Benamou-Brenier formulated Optimal mass transport problem, we show that the power output is bounded by the Fisher information of the boundary distribution, which can become unbounded with complicated and irregular control. However, it is unreasonable to expect technological solutions to such demands, and therefore, a constraint on the complexity of the potential seems meaningful. Assuming a suitable bound M on the spatial gradient of the controlling potential, we show that the maximal achievable power is bounded by M/8(Th/Tc-1). Moreover, we show that this bound can be reached to within a factor of (Th/Tc-1)/(Th/Tc+1) by operating the cyclic thermodynamic process with a quadratic potential.

While the majority of works in the literature are concerned with thermodynamic engines operating in the setting of Carnot's cyclic contact with alternating heat baths, the natural processes in living organisms do not follow this setting. Instead, it is the periodic fluctuations in chemical concentrations in conjunction with the variability of electrochemical potentials that provide the universal source of cellular energy. Thus, energy exchange is often mediated by continuous processes and energy differentials, whereas the Carnot cycle reflects the switching mechanics of an idealized engine. We herein propose to consider thermodynamic processes driven by a heat bath with periodic and continuous temperature profile, and study questions of maximal power and efficiency at maximal power. Our results state that themaximal power satisfies a bound proportional to the average fluctuations in the temperature profile. Moreover, we show the surprising result that the efficiency at maximal power does not depend on the specific temperature profile, only the maximum and minimum.