Operator-Based Detecting, Learning, and Stabilizing Unstable Periodic Orbits of Chaotic Attractors

This paper examines the use of operator-theoretic approaches to the analysis of chaotic systems through the lens of their unstable periodic orbits (UPOs). Our approach involves three data-driven steps for detecting, identifying, and stabilizing UPOs. We demonstrate the use of kernel integral operators within delay coordinates as an innovative method for UPO detection. For identifying the dynamic behavior associated with each individual UPO, we utilize the Koopman operator to present the dynamics as linear equations in the space of Koopman eigenfunctions. This allows for characterizing the chaotic attractor by investigating its principal dynamical modes across varying UPOs. We extend this methodology into an interpretable machine learning framework aimed at stabilizing strange attractors on their UPOs. To illustrate the efficacy of our approach, we apply it to the Lorenz attractor as a case study.Comment: arXiv admin note: text overlap with arXiv:2304.0783

Complex network analysis using modulus of families of walks

Doctor of PhilosophyDepartment of Electrical and Computer EngineeringPietro Poggi-CorradiniCaterina M. ScoglioThe modulus of a family of walks quantifies the richness of the family by favoring having many short walks over a few longer ones. In this dissertation, we investigate various families of walks to study new measures for quantifying network properties using modulus. The proposed new measures are compared to other known quantities. Our proposed method is based on walks on a network, and therefore will work in great generality. For instance, the networks we consider can be directed, multi-edged, weighted, and even contain disconnected parts. We study the popular centrality measure known in some circles as information centrality, also known as effective conductance centrality. After reinterpreting this measure in terms of modulus of families of walks, we introduce a modification called shell modulus centrality, that relies on the egocentric structure of the graph. Ego networks are networks formed around egos with a specific order of neighborhoods. We then propose efficient analytical and approximate methods for computing these measures on both directed and undirected networks. Finally, we describe a simple method inspired by shell modulus centrality, called general degree, which improves simple degree centrality and could prove to be a useful tool for practitioners in the applied sciences. General degree is useful for detecting the best set of nodes for immunization. We also study the structure of loops in networks using the notion of modulus of loop families. We introduce a new measure of network clustering by quantifying the richness of families of (simple) loops. Modulus tries to minimize the expected overlap among loops by spreading the expected link-usage optimally. We propose weighting networks using these expected link-usages to improve classical community detection algorithms. We show that the proposed method enhances the performance of certain algorithms, such as spectral partitioning and modularity maximization heuristics, on standard benchmarks. Computing loop modulus benefits from efficient algorithms for finding shortest loops, thus we propose a deterministic combinatorial algorithm that finds a shortest cycle in graphs. The proposed algorithm reduces the worst case time complexity of the existing combinatorial algorithms while visiting at most the cycle basis. For most empirical networks with sublinear average degree our algorithm is subcubic

Network clustering and community detection using modulus of families of loops

Citation: Shakeri, H., Poggi-Corradini, P., Albin, N., & Scoglio, C. (2017). Network clustering and community detection using modulus of families of loops. Physical Review E, 95(1), 7. doi:10.1103/PhysRevE.95.012316We study the structure of loops in networks using the notion of modulus of loop families. We introduce an alternate measure of network clustering by quantifying the richness of families of (simple) loops. Modulus tries to minimize the expected overlap among loops by spreading the expected link usage optimally. We propose weighting networks using these expected link usages to improve classical community detection algorithms. We show that the proposed method enhances the performance of certain algorithms, such as spectral partitioning and modularity maximization heuristics, on standard benchmarks

Contra-Analysis: Prioritizing Meaningful Effect Size in Scientific Research

At every phase of scientific research, scientists must decide how to allocate limited resources to pursue the research inquiries with the greatest potential. This prioritization dictates which controlled interventions are studied, awarded funding, published, reproduced with repeated experiments, investigated in related contexts, and translated for societal use. There are many factors that influence this decision-making, but interventions with larger effect size are often favored because they exert the greatest influence on the system studied. To inform these decisions, scientists must compare effect size across studies with dissimilar experiment designs to identify the interventions with the largest effect. These studies are often only loosely related in nature, using experiments with a combination of different populations, conditions, timepoints, measurement techniques, and experiment models that measure the same phenomenon with a continuous variable. We name this assessment contra-analysis and propose to use credible intervals of the relative difference in means to compare effect size across studies in a meritocracy between competing interventions. We propose a data visualization, the contra plot, that allows scientists to score and rank effect size between studies that measure the same phenomenon, aid in determining an appropriate threshold for meaningful effect, and perform hypothesis tests to determine which interventions have meaningful effect size. We illustrate the use of contra plots with real biomedical research data. Contra-analysis promotes a practical interpretation of effect size and facilitates the prioritization of scientific research.Comment: 4 figures, 8000 word

Maximizing algebraic connectivity in interconnected networks

Citation: Shakeri, H., Albin, N., Sahneh, F. D., Poggi-Corradini, P., & Scoglio, C. (2016). Maximizing algebraic connectivity in interconnected networks. Physical Review E, 93(3), 6. doi:10.1103/PhysRevE.93.030301Algebraic connectivity, the second eigenvalue of the Laplacian matrix, is a measure of node and link connectivity on networks. When studying interconnected networks it is useful to consider a multiplex model, where the component networks operate together with interlayer links among them. In order to have a well-connected multilayer structure, it is necessary to optimally design these interlayer links considering realistic constraints. In this work, we solve the problem of finding an optimal weight distribution for one-to-one interlayer links under budget constraint. We show that for the special multiplex configurations with identical layers, the uniform weight distribution is always optimal. On the other hand, when the two layers are arbitrary, increasing the budget reveals the existence of two different regimes. Up to a certain threshold budget, the second eigenvalue of the supra-Laplacian is simple, the optimal weight distribution is uniform, and the Fiedler vector is constant on each layer. Increasing the budget past the threshold, the optimal weight distribution can be nonuniform. The interesting consequence of this result is that there is no need to solve the optimization problem when the available budget is less than the threshold, which can be easily found analytically