How basin stability complements the linear-stability paradigm
TL;DR: In this paper, the volume of a state's basin of attraction is used to quantify stability against substantial perturbations in real-world networks and points towards a plausible explanation for regularity in realworld networks.
Abstract: Linear-stability measures for assessing the state of a dynamical system are inherently local, and thus insufficient to quantify stability against substantial perturbations. The volume of a state’s basin of attraction offers a powerful alternative—and points towards a plausible explanation for regularity in real-world networks.
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TL;DR: In this article, B. Sonnenschein, E.R. dos Santos, P.J. Schultz, C.A. Ha, M.K. Choi and C.P.
683 citations
TL;DR: In this paper, the authors discuss the most representative examples of hidden attractors, discuss their theoretical properties and experimental observations, and also describe numerical methods which allow identification of the hidden attractor.
569 citations
TL;DR: It is found in numerical simulations of artificially generated power grids that tree-like connection schemes--so-called dead ends and dead trees--strongly diminish stability, which may indicate a topological design principle for future power grids: avoid dead ends.
Abstract: The cheapest and thus widespread way to add new generators to a high-voltage power grid is by a simple tree-like connection scheme. However, it is not entirely clear how such locally cost-minimizing connection schemes affect overall system performance, in particular the stability against blackouts. Here we investigate how local patterns in the network topology influence a power grid's ability to withstand blackout-prone large perturbations. Employing basin stability, a nonlinear concept, we find in numerical simulations of artificially generated power grids that tree-like connection schemes--so-called dead ends and dead trees--strongly diminish stability. A case study of the Northern European power system confirms this result and demonstrates that the inverse is also true: repairing dead ends by addition of a few transmission lines substantially enhances stability. This may indicate a topological design principle for future power grids: avoid dead ends.
368 citations
TL;DR: It turns out that the extent to which one can exercise control via manipulation of a subset of nodes depends sensitively on the number of nodes perturbed.
Abstract: Recent studies have made important advances in identifying sensor or driver nodes, through which we can observe or control a complex system. But the observational uncertainty induced by measurement noise and the energy required for control continue to be significant challenges in practical applications. Here we show that the variability of control energy and observational uncertainty for different directions of the state space depend strongly on the number of driver nodes. In particular, we find that if all nodes are directly driven, control is energetically feasible, as the maximum energy increases sublinearly with the system size. If, however, we aim to control a system through a single node, control in some directions is energetically prohibitive, increasing exponentially with the system size. For the cases in between, the maximum energy decays exponentially when the number of driver nodes increases. We validate our findings in several model and real networks, arriving at a series of fundamental laws to describe the control energy that together deepen our understanding of complex systems. The complex interactions inherent in real-world networks grant us precise system control via manipulation of a subset of nodes. It turns out that the extent to which we can exercise this control depends sensitively on the number of nodes perturbed.
283 citations
TL;DR: An overview of recent progress of synchronization of complex dynamical networks and its applications is presented and robustness of synchronization, controllability and observability of complex networks and synchronization of multiplex networks are focused on.
260 citations
Cites background or methods from "How basin stability complements the..."
...Actually, the method used for basin stability in Menck et al. (2013), Menck et al. (2014) is the time-domain method in Chiang et al. (1995) or the numerical integration method in Ribbens-Pavella and Evans (1985)....
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...Consequently, the time-domain method can serve as a good complement for the direct method, as manifested in Menck et al. (2013), Menck et al. (2014)....
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...1(b) a network’s mean basin stability hSBi is determined primarily by the location of its stability interval Is (Menck et al., 2013)....
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...In order to answer this fundamental question, Menck et al. (2013) employs the concept of region of attraction to measure synchronizability versus robustness....
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...The results in Menck et al. (2013), Cornelius et al. (2013), Motter et al. (2013), Rohden et al. (2012), Witthaut and Timme (2012) confirm that synchronization of power grid networks is a challenge for ongoing research on smart grids, which could establish a bridge between physics, control theory…...
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TL;DR: Simple models of networks that can be tuned through this middle ground: regular networks ‘rewired’ to introduce increasing amounts of disorder are explored, finding that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs.
Abstract: Networks of coupled dynamical systems have been used to model biological oscillators, Josephson junction arrays, excitable media, neural networks, spatial games, genetic control networks and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks 'rewired' to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them 'small-world' networks, by analogy with the small-world phenomenon (popularly known as six degrees of separation. The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.
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TL;DR: Developments in this field are reviewed, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.
Abstract: Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.
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