Network Formation and Dynamics under Economic Constraints

Networks describe a broad range of systems across a wide variety of topics from social and economic interactions over technical infrastructures such as power grids and the internet to biological contexts such as food webs or neural networks. A number of large scale failures and events in these interconnected systems in recent years has shown that understanding the behavior of individual units of these networks is not necessarily sufficient to handle the increasing complexity of these systems. Many theoretical models have been studied to understand the fundamental mechanisms underlying the formation and function of networked systems and a general framework was developed to describe and understand networked systems. However, most of these models ignore a constraint that affects almost all realistic systems: limited resources. In this thesis I study the effects of economic constraints, such as a limited budget or cost minimization, both on the control of network formation and dynamics as well as on network formation itself. I introduce and analyze a new coupling scheme for coupled dynamical systems, showing that synchronization of chaotic units can be enhanced by restricting the interactions based on the states of the individual units, thus saving interactions costs. This new interaction scheme guarantees synchronizability of arbitrary networks of coupled chaotic oscillators, independent of the network topology even with strongly limited interactions. I then propose a new order parameter to measure the degree of phase coherence of networks of coupled phase oscillators. This new order parameter accurately describes the phase coherence in all stages of incoherent movement, partial and full phase locking up to full synchrony. Importantly, I analytically relate this order parameter directly to the stability of the phase locked state. In the second part, I consider the formation of networks under economic constraints from two different points of view. First I study the effects of explicitly limited resources on the control of random percolation, showing that optimal control can have undesired side effects. Specifically, maximal delay of percolation with a limited budget results in a discontinuous percolation transition, making the transition itself uncontrollable in the sense that a single link can have a macroscopic effect on the connectivity. Finally, I propose a model where network formation is driven by cost minimization of the individual nodes in the network. Based on a simple economically motivated supply problem, the resulting network structure is given as the solution of a large number of individual but interaction optimization problem. I show that these network states directly correspond to the final states of a local percolation algorithm and analyze the effects of local optimization on the network formation process. Overall, I reveal mechanisms and phenomena introduced by these economic constraints that are typically not considered in the standard models, showing that economic constraints can strongly alter the formation and function of networked systems. Thereby, I extend the theoretical understanding that we have of networked systems to economic considerations. I hope that this thesis enables better prediction and control networked systems in realistic settings

Quantitative Methods For Guiding Epilepsy Surgery From Intracranial Eeg

Despite advances in intracranial EEG (iEEG) technique, technology and neuroimaging, patients today are no more likely to achieve seizure freedom after epilepsy surgery than they were 20 years ago. These poor outcomes are in part due to the difficulty and subjectivity associated with interpreting iEEG recordings, and have led to widespread interest in developing quantitative methods to localize the epileptogenic zone. Approaches to computational iEEG analysis vary widely, spanning studies of both seizures and interictal periods, and encompassing a range of techniques including electrographic signal analysis and graph theory. However, many current methods often fail to generalize to new data and are sensitive to differences in pathology and electrode placement. Indeed, none have completed prospective clinical trials. In this dissertation, I develop and validate tools for guiding epilepsy surgery through the quantitative analysis of intracranial EEG. Specifically, I leverage methods from graph theory for mapping network synchronizability to predict surgical outcome from ictal recordings, and also investigate the effects of sampling bias on network models. Finally, I construct a normative intracranial EEG atlas as a framework for objectively identifying patterns of abnormal neural activity and connectivity. Overall, the methods and results of this dissertation support the implementation of quantitative iEEG analysis in epilepsy surgical evaluation

Development of structural correlations and synchronization from adaptive rewiring in networks of Kuramoto oscillators

L.P. acknowledges support from the National Science Foundation Graduate Research Fellowship Program. J.K. acknowledges support from the National Science Foundation Graduate Research Fellowship Program and NIH T32-EB020087, PD: Felix W. Wehrli. D.S.B. also acknowledges support from the John D. and Catherine T. MacArthur Foundation, the Alfred P. Sloan Foundation, and the National Science Foundation (BCS-1441502, CAREER PHY-1554488, BCS-1631550, and CNS-1626008). We also thank two anonymous reviewers whose comments greatly improved the quality of this work. The content is solely the responsibility of the authors and does not necessarily represent the official views of any of the funding agencies.Peer reviewedPublisher PD

Applied Harmonic Analysis and Data Processing

Massive data sets have their own architecture. Each data source has an inherent structure, which we should attempt to detect in order to utilize it for applications, such as denoising, clustering, anomaly detection, knowledge extraction, or classification. Harmonic analysis revolves around creating new structures for decomposition, rearrangement and reconstruction of operators and functions—in other words inventing and exploring new architectures for information and inference. Two previous very successful workshops on applied harmonic analysis and sparse approximation have taken place in 2012 and in 2015. This workshop was the an evolution and continuation of these workshops and intended to bring together world leading experts in applied harmonic analysis, data analysis, optimization, statistics, and machine learning to report on recent developments, and to foster new developments and collaborations

Evolving and adaptive strategies for consensus and synchronization of multi-agent systems

We investigate evolving and adaptive strategies, in network of dynamical agents, for solving general types of consensus and synchronization. First, we analyse the problem of max/min consensus in directed networks of integrators. Extending edge snapping method with a three-well potential, we are able to show the effectiveness of our strategy to achieve general types of consensus, different from the average. Theoretical results are validated via a number of numerical examples. Then we move to synchronization of coupled non identical oscillators. We design an evolutionary strategy for network synchronization. Our results suggest that heterogeneity is the driving force determining the evolution of state-dependent functional networks. Minimal emergent networks show enhanced synchronization properties and high levels of degree-frequency assortativity. We analyse networks of N = 100 and N = 1000 Kuramoto oscillators showing that hubs in the network tend to emerge as nodes' heterogeneity is increased. Finally, we study synchronization of multi-agent systems from a contraction theory viewpoint. Contraction theory is a useful tool to study convergence of dynamical systems and networks, recently proposed in the literature. In detail, we recall three strategies: virtual systems method, convergence to a flow-invariant subspace and hierarchical approach. While the former is simple to apply, the latter is suited for larger networks