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Flow and elastic networks on the N-torus: Geometry, analysis, and computation

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arXiv 2019年

作者： Jafarpour, Saber Huang, Elizabeth Y. Smith, Kevin D. Bullo, Francesco Center for Control Dynamical Systems and Computation Departments of Mechanical Engineering and of Electrical and Computer Engineering University of California Santa Barbara

Networks with phase-valued nodal variables are central in modeling several important societal and physical systems, including power grids, biological systems, and coupled oscillator networks. One of the distinctive features of phase-valued networks is the existence of multiple operating conditions corresponding to critical points of an energy function or feasible flows of a balance equation. For a network with phase-valued states, it is not yet fully understood how many operating conditions exist, how to characterize them, and how to compute them efficiently. A deeper understanding of feasible operating conditions, including their dependence upon network structures, may lead to more reliable and efficient network systems. This paper introduces flow and elastic network problems on the n-torus and provides a rigorous and comprehensive framework for their study. Based on a monotonicity assumption, this framework localizes the solutions, bounds their number, and leads to an algorithm to compute them. Our analysis is based on a novel winding partition of the n-torus into winding cells, induced by Kirchhoff's angle law for undirected graphs. The winding partition has several useful properties, including notably that, each winding cell contains at most one solution. The proposed algorithm is based on a novel contraction mapping and is guaranteed to compute all solutions. Finally, we apply our results to numerically study the active power flow equations in several test cases and estimate power capacity and congestion of a power network. Copyright © 2019, The Authors. All rights reserved.

关键词： Winding

Social power evolution in influence networks with stubborn individuals

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arXiv 2019年

作者： Tian, Ye Jia, Peng Mirtabatabaei, Anahita Wang, Long Friedkin, Noah E. Bullo, Francesco Center for Complex Systems School of Mechanoelectronic Engineering Xidian University Xi’an710071 China Mechanical Engineering Department Center of Control Dynamical Systems and Computation UC Santa BarbaraCA93106 United States Mechanical Engineering Department Center of Control Dynamical Systems and Computation UC Santa BarbaraCA93106 United States Center for Systems and Control College of Engineering Peking University Beijing100871 China Department of Sociology Center for Control Dynamical Systems and Computation UC Santa BarbaraCA93106 United States

—This paper studies the evolution of social power in influence networks with stubborn individuals. Based on the Friedkin-Johnsen opinion dynamics and the reflected appraisal mechanism, two models are proposed over issue sequences and over a single issue, respectively. These models generalize the original DeGroot-Friedkin (DF) model by including stubbornness. To the best of our knowledge, this paper is the first attempt to investigate the social power evolution of stubborn individuals basing on the reflected appraisal mechanism. Properties of equilibria and convergence are provided. We show that the models have same equilibrium social power and convergence property, where the equilibrium social power depends only upon interpersonal influence and individuals’ stubbornness. Roughly speaking, more stubborn individual has more equilibrium social power. Moreover, unlike the DF model without stubbornness, we prove that for the models with stubbornness, autocracy can never be achieved, while democracy can be achieved under any network topology. Copyright © 2019, The Authors. All rights reserved.

关键词： dynamical systems

System size identification from sinusoidal probing in diffusive complex networks

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arXiv 2020年

作者： Tyloo, Melvyn Delabays, Robin School of Engineering University of Applied Sciences of Western Switzerland HES-SO SionCH-1951 Switzerland Automatic Control Laboratory ETH Zurich ZurichCH-8092 Switzerland Center for Control Dynamical Systems and Computation UC Santa Barbara Santa BarbaraCA93106-5070 United States

One of the most fundamental characteristic of a complex system is its size (or volume), which, in many modelling, is represented by the number of its individual components. Complex systems under investigation nowadays are typically large and/or time-varying, rendering their identification challenging. We propose here an accurate and efficient method to determine the size of (i.e., number of agents in) a complex, diffusively coupled dynamical system, that leverages the response of the system to an injected probing signal. For our method to be applicable, we rely on some assumptions on system's characteristics, namely, on the spectrum of the coupling graph and on the basin stability of its steady state(s). Even though such assumptions imply that our method cannot be applied to any instance of diffusively coupled group of dynamical agents, we argue that it covers relevant and interesting examples. Furthermore, the simplicity of the approach and its low computational complexity renders it very interesting for the systems to which it applies. © 2020, CC BY.

关键词： Complex networks

Distributionally robust formulation and model selection for the graphical lasso

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arXiv 2019年

作者： Cisneros-Velarde, Pedro Oh, Sang-Yun Petersen, Alexander Center for Control Dynamical Systems and Computation Department of Statistics and Applied Probability University of California Santa Barbara United States

Building on a recent framework for distributionally robust optimization in machine learning, we develop a similar framework for estimation of the inverse covariance matrix for multivariate data. We provide a novel notion of a Wasserstein ambiguity set specifically tailored to this estimation problem, from which we obtain a representation for a tractable class of regularized estimators. Special cases include penalized likelihood estimators for Gaussian data, specifically the graphical lasso estimator. As a consequence of this formulation, a natural relationship arises between the radius of the Wasserstein ambiguity set and the regularization parameter in the estimation problem. Using this relationship, one can directly control the level of robustness of the estimation procedure by specifying a desired level of confidence with which the ambiguity set contains a distribution with the true population covariance. Furthermore, a unique feature of our formulation is that the radius can be expressed in closed-form as a function of the ordinary sample covariance matrix. Taking advantage of this finding, we develop a simple algorithm to determine a regularization parameter for graphical lasso, using only the bootstrapped sample covariance matrices, meaning that computationally expensive repeated evaluation of the graphical lasso algorithm is not necessary. Alternatively, the distributionally robust formulation can also quantify the robustness of the corresponding estimator if one uses an off-the-shelf method such as cross-validation. Finally, we numerically study the obtained regularization criterion and analyze the robustness of other automated tuning procedures used in *** Codes 62F35, 62F40, 90C90 Copyright © 2019, The Authors. All rights reserved.

关键词： Parameterization

Structural balance via gradient flows over signed graphs

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arXiv 2019年

作者： Cisneros-Velarde, Pedro Friedkin, Noah E. Proskurnikov, Anton V. Bullo, Francesco Center for Control Dynamical Systems and Computation University of California Santa Barbara United States Politecnico di Torino Turin Italy

Structural balance is a classic property of signed graphs satisfying Heider’s seminal axioms. Mathematical sociologists have studied balance theory since its inception in the 1940s. Recent research has focused on the development of dynamic models explaining the emergence of structural balance. In this paper, we introduce a novel class of parsimonious dynamic models for structural balance based on an interpersonal influence process. Our proposed models are gradient flows of an energy function, called the dissonance function, which captures the cognitive dissonance arising from the violations of Heider’s axioms. Thus, we build a new connection with the literature on energy landscape minimization. This gradient-flow characterization allows us to study the transient and asymptotic behaviors of our model. We provide mathematical and numerical results describing the critical points of the dissonance function. Copyright © 2019, The Authors. All rights reserved.

关键词： Dynamic models

Graph-theoretic small-gain theorems for metzler matrices and monotone systems

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arXiv 2019年

作者： Duan, Xiaoming Jafarpour, Saber Bullo, Francesco Mechanical Engineering Department Center of Control Dynamical Systems and Computation UC Santa BarbaraCA93106-5070 United States

This paper studies the graph-theoretic conditions for stability of positive monotone systems. Using concepts from the input-to-state stability and network small-gain theory, we first establish necessary and sufficient conditions for the stability of linear positive systems described by Metzler matrices. Specifically, we define and compute two forms of input-to-state stability gains for Metzler systems, namely max-interconnection gains and sum-interconnection gains. Then, based on the max-interconnection gains, we show that the cyclic small-gain theorem becomes necessary and sufficient for the stability of Metzler systems;based on the sum-interconnection gains, we obtain novel graph-theoretic conditions for the stability of Metzler systems. All these conditions highlight the role of cycles in the interconnection graph and unveil how the structural properties of the graph affect stability. Finally, we extend our results to the nonlinear monotone system and obtain similar sufficient conditions for global asymptotic stability. Copyright © 2019, The Authors. All rights reserved.

关键词： Graph theory

Network inference using sinusoidal probing

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arXiv 2020年

作者： Delabays, Robin Tyloo, Melvyn Automatic Control Laboratory Swiss Federal Institute of Technology ZurichCH-8092 Switzerland Center for Control Dynamical Systems and Computation UCSB Santa BarbaraCA93106-5070 United States School of Engineering University of Applied Sciences of Western Switzerland HES-SO SionCH-1951 Switzerland LausanneCH-1015 Switzerland

The aim of this manuscript is to present a non-invasive method to recover the network structure of a dynamical system. We propose to use a controlled probing input and to measure the response of the network, in the spirit of what is done to determine oscillation modes in large electrical networks. For a large class of dynamical systems, we show that this approach is analytically tractable and we confirm our findings by numerical simulations of networks of Kuramoto oscillators. Our approach also allows us to determine the number of agents in the network by probing and measuring a single one of them. © 2020, CC BY.

关键词： dynamical systems

Policy evaluation and seeking for multi-agent reinforcement learning via best response

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arXiv 2020年

作者： Yan, Rui Duan, Xiaoming Shi, Zongying Zhong, Yisheng Marden, Jason R. Bullo, Francesco Department of Automation Tsinghua University Beijing100084 China Mechanical Engineering Department Center of Control Dynamical Systems and Computation UC Santa BarbaraCA93106-5070 United States Department of Electrical and Computer Engineering UC Santa Barbara CA93106-5070 United States

This paper introduces two metrics (cycle-based and memory-based metrics), grounded on a dynamical game-theoretic solution concept called sink equilibrium, for the evaluation, ranking, and computation of policies in multi-agent learning. We adopt strict best response dynamics (SBRD) to model selfish behaviors at a meta-level for multi-agent reinforcement learning. Our approach can deal with dynamical cyclical behaviors (unlike approaches based on Nash equilibria and Elo ratings), and is more compatible with single-agent reinforcement learning than α-rank which relies on weakly better responses. We first consider settings where the difference between largest and second largest underlying metric has a known lower bound. With this knowledge we propose a class of perturbed SBRD with the following property: only policies with maximum metric are observed with nonzero probability for a broad class of stochastic games with finite memory. We then consider settings where the lower bound for the difference is unknown. For this setting, we propose a class of perturbed SBRD such that the metrics of the policies observed with nonzero probability differ from the optimal by any given tolerance. The proposed perturbed SBRD addresses the opponent-induced non-stationarity by fixing the strategies of others for the learning agent, and uses empirical game-theoretic analysis to estimate payoffs for each strategy profile obtained due to the perturbation. Copyright © 2020, The Authors. All rights reserved.

关键词： Game theory

Persistence of excitation for koopman operator represented dynamical systems

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arXiv 2019年

作者： Boddupalli, Nibodh Hasnain, Aqib Yeung, Enoch Department of Mechanical Engineering and the Center for ControlDynamical Systems and Computation University of California Santa Barbara Santa BarbaraCA93106 United States

The increase in complexity of engineered systems and naturally occurring systems, coupled with increasingly data-driven representations of these systems motivates the need for design of experiment frameworks for nonlinear system identification. In the case of a general nonlinear autonomous system (without inputs or control), designing an experiment for identification relies on the selection of persistently exciting flows and their corresponding initial conditions. This work investigates persistence of excitation conditions for discrete-time nonlinear systems that can be represented using Koopman operators. The Koopman operator presents a linear lifted representation of the evolution of a nonlinear system over a space of observable functions. We use this framework to derive spectral conditions for persistence of excitation, relating classical notions of persistence of excitation to the spectral properties of initial conditions, seen through the lens of Koopman observable functions. We illustrate these concepts on a repressilator circuit, a classical biological system from the field of synthetic biology, to motivate the proper selection of initial conditions. We show that when the rank properties of the flow's spectra are degenerate, the trained Koopman operator does not produce robust predictions on test data. Copyright © 2019, The Authors. All rights reserved.

关键词： Design of experiments