### Citations

3069 | On random graphs
- Erdös, Rényi
- 1959
(Show Context)
Citation Context ...at the choice & = 0.2 is sufficient, and is what we use in the examples below. Control of random networks We now demonstrate our approach by considering two types of random networks: Erdős-Rényi (ER) =-=(49)-=- networks and scale-free (SF) networks. Each ER network is constructed using a fixed link probability p, and each SF network is built using the configuration model (50) with a degree sequence drawn fr... |

2600 | The structure and function of complex networks, - Newman - 2003 |

674 | Exploring complex networks,” - Strogatz - 2001 |

664 |
Chemical Oscillations, Waves, and Turbulence.
- Kuramoto
- 1984
(Show Context)
Citation Context ...ent of Mathematics, Trinity College, Hartford, CT 06106, USA. Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain.coupled dissipative oscillators =-=(39)-=-.*Corresponding author. E-mail: persebastian.skardal@trincoll.edu Skardal and Arenas Sci. Adv. 2015;1:e1500339 21 August 2015odel for the entrainment of many(35). A microgrid consists of a relatively ... |

507 | A critical point for random graphs with a given degree sequence.
- Molloy, Reed
- 1995
(Show Context)
Citation Context ...ndom networks: Erdős-Rényi (ER) (49) networks and scale-free (SF) networks. Each ER network is constructed using a fixed link probability p, and each SF network is built using the configuration model =-=(50)-=- with a degree sequence drawn from the distribution P(k) º k−g with g = 3 and enforced minimum degree k0. To tune the mean degree 〈k〉 of each F), we present the degree of ph parameter r(t) for both so... |

434 | Complex networks: Structure and dynamics. - Boccaletti, Latora, et al. - 2006 |

276 |
Controlling chaos,
- Ott, Grebogi, et al.
- 1990
(Show Context)
Citation Context ...llators arrhythmias (37) and treatments for pathological brain dynamics (38).been made in the control of nonlinear systems, for instance, the control of chaotic systems using unstable periodic orbits =-=(23)-=-, control via pinning (24–26), control and rescue of networks using compensatory perturbations (27, 28), and control via structural adaptation (29). Implicit in all such network control problems are t... |

245 |
Catastrophic shifts in ecosystems. Nature 413
- Scheffer, Carpenter, et al.
- 2001
(Show Context)
Citation Context ... the dynamical processes vital to our lives (1–3). The failure of such large-scale systems to operate in the desired way can thus lead to catastrophic events such as power outages (4, 5), extinctions =-=(6, 7)-=-, and economic collapses (8, 9). Thus, the development and design of efficient and effective control mechanisms for such systems is not only a question of theoretical interest to mathematicians but al... |

177 |
Decentralized control of complex systems.
- Siljak
- 1991
(Show Context)
Citation Context ... such systems is not only a question of theoretical interest to mathematicians but also has a wide range of important applications in physics, chemistry, biology, engineering, and the social sciences =-=(10, 11)-=-. The roots of modern linear and nonlinear control reach back several decades, but recently, research in this direction has seen a revival in physics and engineering communities. For instance, the con... |

173 |
Circuit implementation of synchronized chaos with applications to communications,”
- Cuomo, Oppenheim
- 1993
(Show Context)
Citation Context ...s the average over 100 network realizations of size N = 1000, each averaged over 100 natural frequency realizations. Skardal and Arenas Sci. Adv. 2015;1:e1500339 21 August 2015communications security =-=(57)-=-. Given this broad range of applications, we hypothesize that our findings here may potentially shed some light on the control of synchronization in other contexts, such as cardiac physiology and neur... |

157 |
Catastrophic cascade of failures in interdependent networks.
- Buldyrev, Parshani, et al.
- 2010
(Show Context)
Citation Context ...t our world and host the dynamical processes vital to our lives (1–3). The failure of such large-scale systems to operate in the desired way can thus lead to catastrophic events such as power outages =-=(4, 5)-=-, extinctions (6, 7), and economic collapses (8, 9). Thus, the development and design of efficient and effective control mechanisms for such systems is not only a question of theoretical interest to m... |

124 |
Normal and pathological oscillatory communication in the brain.
- Schnitzler, Gross
- 2005
(Show Context)
Citation Context ...gies has shown that certain power grid networks called “microgrids” evolve and can be treated as networks of Kuramoto phase oscillators arrhythmias (37) and treatments for pathological brain dynamics =-=(38)-=-.been made in the control of nonlinear systems, for instance, the control of chaotic systems using unstable periodic orbits (23), control via pinning (24–26), control and rescue of networks using comp... |

113 |
Sync: The Emerging Science of Spontaneous Order. Hyperion,
- Strogatz
- 2003
(Show Context)
Citation Context ...hat is, irreducible. Over the last few decades, the Kuramoto model has proven to be very useful for modeling real-world systems (36, 40), uncovering the mechanisms behind emergent collective behavior =-=(41, 42)-=-, exploring additional effects such as time delays (43) and community structure (44), and finding optimal network structure (45). Depending on the coupling strength K, as well as the frequency vector ... |

111 |
Trophic cascades revealed in diverse ecosystems.
- Pace, Cole, et al.
- 1999
(Show Context)
Citation Context ... the dynamical processes vital to our lives (1–3). The failure of such large-scale systems to operate in the desired way can thus lead to catastrophic events such as power outages (4, 5), extinctions =-=(6, 7)-=-, and economic collapses (8, 9). Thus, the development and design of efficient and effective control mechanisms for such systems is not only a question of theoretical interest to mathematicians but al... |

99 |
Controllability of complex networks.
- Liu, Slotine, et al.
- 2011
(Show Context)
Citation Context ...nce, the concept of “structural controllability,” which is based on the paradigm of linear homogeneous dynamical systems, was first introduced by Lin (12) and more recently investigated by Liu et al. =-=(13)-=- and Yuan et al. (14). These advances have enabled further progress related to structural controllability such as centrality (15), energy (16), effect of correlations (17), emergence of bimodality (18... |

97 |
Pathological synchronization in Parkinson's disease: networks, models and treatments,
- Hammond, Bergman, et al.
- 2007
(Show Context)
Citation Context ... behavior such as cardiac fibrillation (58) and the promotion of normal brain oscillations (59) while repressing disorders such as Parkinson’s disease, which are associated with abnormal oscillations =-=(60)-=-. MATERIALS AND METHODS Steady-state solution To derive the steady-state solution q* = K−1L†w, we begin with Eq. 2, which represents the linearized dynamics of Eq. 1. Recall that this linearization re... |

94 |
Modulation of long-range neural synchrony reflects temporal limitations of visual attention in humans.
- ross, Schmitz, et al.
- 2004
(Show Context)
Citation Context ... the control method we present above cantreatments that require minimal shock to knock out fatal asynchronous behavior such as cardiac fibrillation (58) and the promotion of normal brain oscillations =-=(59)-=- while repressing disorders such as Parkinson’s disease, which are associated with abnormal oscillations (60). MATERIALS AND METHODS Steady-state solution To derive the steady-state solution q* = K−1L... |

89 | Synchronization in complex networks.
- Arenas, Dıaz-Guilera, et al.
- 2008
(Show Context)
Citation Context ... “frequency drooping,” the dynamics of microgrids become equivalent to Kuramoto oscillator networks—a class of system for which a large body of literature detailing various dynamical phenomena exists =-=(36)-=-. Here, we develop a control mechanism for such coupled oscillator networks, thus providing a solution with potentially direct application to the control of certain power grids. Our goal is to induce ... |

78 | Structural controllability,”
- Lin
- 1974
(Show Context)
Citation Context ... in physics and engineering communities. For instance, the concept of “structural controllability,” which is based on the paradigm of linear homogeneous dynamical systems, was first introduced by Lin =-=(12)-=- and more recently investigated by Liu et al. (13) and Yuan et al. (14). These advances have enabled further progress related to structural controllability such as centrality (15), energy (16), effect... |

72 |
Applied Nonlinear Control, Prentice-Hall International, Taylor and Francis,
- Slotine, Li
- 1991
(Show Context)
Citation Context ... such systems is not only a question of theoretical interest to mathematicians but also has a wide range of important applications in physics, chemistry, biology, engineering, and the social sciences =-=(10, 11)-=-. The roots of modern linear and nonlinear control reach back several decades, but recently, research in this direction has seen a revival in physics and engineering communities. For instance, the con... |

72 | Synchronization and transient stability in power networks and non-uniform Kuramoto oscillators,”
- Dorfler, Bullo
- 2010
(Show Context)
Citation Context ...research. Here, we have focused on the control of synchronization (that is, consensus) in coupled oscillator networks. Our primary inspiration has been advances in the research of power grid networks =-=(52, 53)-=-. In particular, recent studies have shown that certain power grids known as microgrids can be treated as Kuramoto oscillator networks (35, 39). Here, we have presented a control method that can easil... |

42 | Pinning control of scale-free dynamical networks,” - Wang, Chen - 2002 |

41 |
Pinning a complex dynamical network to its equilibrium,”
- Li, Wang, et al.
- 2004
(Show Context)
Citation Context ...and ultimately control synchronization in power grid networks [see, in particular, (5, 32)], and more generally complement important work on the control of network-coupled nonlinear dynamical systems =-=(26, 28, 29)-=-. Although our central inspiration and target application are in the area of power grid technology, synchronization phenomenon plays a vital role in a variety of complex processes that occur in both n... |

36 |
Synchronization in complex oscillator networks and smart grids.”
- Dorfler, Chertkov, et al.
- 2013
(Show Context)
Citation Context ...wing questions: (i) What form(s) of control should one choose? (ii) How much effort is needed to attain a desired state (30)? Motivated by ongoing studies on the stability and function of power grids =-=(31, 32)-=-, we study the control of heterogeneous coupled oscillator networks (33, 34). Recent research into smart grid technolnamical parameters. Although Kuramoto osci our primary system of interest due to bo... |

34 |
Generalized inverses
- Ben-Israel, Greville
- 2003
(Show Context)
Citation Context ...lta. A straightforward analysis yields a “target” synchronized state (within the rotating reference frame q ↦ q + 〈w〉t) given by the vector q* = K−1L†w, where L† is the pseudoinverse of the Laplacian =-=(46)-=-. (We summarize a derivation of this result inMaterials andMethods.)We note that, because the system is assumed to be partially incoherent, the fixed point q = q* either does not exist or is unstable.... |

30 | Synchronization and power sharing for droopcontrolled inverters in islanded microgrids,”
- Simpson-Porco, Dörfler, et al.
- 2013
(Show Context)
Citation Context ... Spain.coupled dissipative oscillators (39).*Corresponding author. E-mail: persebastian.skardal@trincoll.edu Skardal and Arenas Sci. Adv. 2015;1:e1500339 21 August 2015odel for the entrainment of many=-=(35)-=-. A microgrid consists of a relatively small number of localized sources and loads that, while typically operating in connection to a RESULTS The Kuramoto model1 2ogies has shown that certain power gr... |

25 |
Complex systems: Ecology for bankers. Nature
- May, Levin, et al.
(Show Context)
Citation Context ...to our lives (1–3). The failure of such large-scale systems to operate in the desired way can thus lead to catastrophic events such as power outages (4, 5), extinctions (6, 7), and economic collapses =-=(8, 9)-=-. Thus, the development and design of efficient and effective control mechanisms for such systems is not only a question of theoretical interest to mathematicians but also has a wide range of importan... |

24 |
Low dimensional behavior of large systems of globally coupled oscillators. Chaos: An Interdisciplinary
- Ott, Antonsen
(Show Context)
Citation Context ...hat is, irreducible. Over the last few decades, the Kuramoto model has proven to be very useful for modeling real-world systems (36, 40), uncovering the mechanisms behind emergent collective behavior =-=(41, 42)-=-, exploring additional effects such as time delays (43) and community structure (44), and finding optimal network structure (45). Depending on the coupling strength K, as well as the frequency vector ... |

22 |
Controlling edge dynamics in complex networks
- Nepusz, Vicsek
- 2012
(Show Context)
Citation Context ...odeling and studying sy contexts, we hypothesize that our results may s on the control of synchronization processes an insight into other important applications such as tedges in switchboard dynamics =-=(22)-=-. Significant advances have also more sensitive to the average connectivity of the network and the dyllator networks serve asapplication to microgrid technolo Per Sebastian Skardal1,2* and Alex Arenas... |

19 | A sensing array of radically coupled genetic /`biopixels/’,”
- Prindle, Samayoa, et al.
- 2012
(Show Context)
Citation Context ...nds on both the system’s 3 of 6 K20% for which, on average, a fraction nD = 0.05, 0.1, and 0.2 will achieve synchronization as a function of the average degree 〈k〉. We plot tionality of cell circuits =-=(55)-=-, stability of pedestrian bridges (56), and Fig. 3. Driver and free nodes. (A and B) [L w]i versus wi/ki for driver (red crosses) and free (blue circles) nodes in the ER (A) and SF (B) networks used i... |

18 |
Controllability transition and nonlocality in network control
- Sun, Motter
(Show Context)
Citation Context ...dvances have enabled further progress related to structural controllability such as centrality (15), energy (16), effect of correlations (17), emergence of bimodality (18), transition and nonlocality =-=(19)-=-, the specific role of individual nodes (20), target control (21), and control ofs is a topic of vital theoretical importance in ing and various other sciences. Motivated by synchronization and consid... |

18 |
Pinning control of spatiotemporal chaos
- Grigoriev, Cross, et al.
- 1997
(Show Context)
Citation Context ...k has made significant advances in understanding structural controllability (13, 14), and significant progress has been made in the development of control mechanisms for networks of nonlinear systems =-=(23, 24, 28)-=-. Nonetheless, because of the problem-sensitive nature of most real-world problems and applications requiring control techniques, further progress in designing and implementing efficient and effective... |

17 |
Theoretical mechanics : Crowd synchrony on the millennium bridge
- Strogatz, Abrams, et al.
(Show Context)
Citation Context ...or which, on average, a fraction nD = 0.05, 0.1, and 0.2 will achieve synchronization as a function of the average degree 〈k〉. We plot tionality of cell circuits (55), stability of pedestrian bridges =-=(56)-=-, and Fig. 3. Driver and free nodes. (A and B) [L w]i versus wi/ki for driver (red crosses) and free (blue circles) nodes in the ER (A) and SF (B) networks used in Fig. 2. R E S EARCH ART I C L Eso ns... |

16 |
R M (2011) Systemic risk in banking ecosystems. Nature 469: 351–355
- Haldane
(Show Context)
Citation Context ...to our lives (1–3). The failure of such large-scale systems to operate in the desired way can thus lead to catastrophic events such as power outages (4, 5), extinctions (6, 7), and economic collapses =-=(8, 9)-=-. Thus, the development and design of efficient and effective control mechanisms for such systems is not only a question of theoretical interest to mathematicians but also has a wide range of importan... |

16 |
Controlling complex networks: How much energy is needed
- Yan, Ren, et al.
(Show Context)
Citation Context ... by Lin (12) and more recently investigated by Liu et al. (13) and Yuan et al. (14). These advances have enabled further progress related to structural controllability such as centrality (15), energy =-=(16)-=-, effect of correlations (17), emergence of bimodality (18), transition and nonlocality (19), the specific role of individual nodes (20), target control (21), and control ofs is a topic of vital theor... |

15 | Spontaneous synchrony in power-grid networks
- Motter, Myers, et al.
(Show Context)
Citation Context ...t our world and host the dynamical processes vital to our lives (1–3). The failure of such large-scale systems to operate in the desired way can thus lead to catastrophic events such as power outages =-=(4, 5)-=-, extinctions (6, 7), and economic collapses (8, 9). Thus, the development and design of efficient and effective control mechanisms for such systems is not only a question of theoretical interest to m... |

13 | Synchronization in complex networks of phase oscillators: A survey.” Automatica
- Dorfler, Bullo
- 2014
(Show Context)
Citation Context ...d and weighted networks. We also assume that the network is connected, that is, irreducible. Over the last few decades, the Kuramoto model has proven to be very useful for modeling real-world systems =-=(36, 40)-=-, uncovering the mechanisms behind emergent collective behavior (41, 42), exploring additional effects such as time delays (43) and community structure (44), and finding optimal network structure (45)... |

12 | Exact controllability of complex networks
- Yuan, Zhao, et al.
- 2013
(Show Context)
Citation Context ...structural controllability,” which is based on the paradigm of linear homogeneous dynamical systems, was first introduced by Lin (12) and more recently investigated by Liu et al. (13) and Yuan et al. =-=(14)-=-. These advances have enabled further progress related to structural controllability such as centrality (15), energy (16), effect of correlations (17), emergence of bimodality (18), transition and non... |

12 |
Large coupled oscillator systems with heterogeneous interaction delays
- Lee, Ott, et al.
- 2009
(Show Context)
Citation Context ...o model has proven to be very useful for modeling real-world systems (36, 40), uncovering the mechanisms behind emergent collective behavior (41, 42), exploring additional effects such as time delays =-=(43)-=- and community structure (44), and finding optimal network structure (45). Depending on the coupling strength K, as well as the frequency vector w and the network topology, the steady-state dynamics o... |

11 |
Control centrality and hierarchical structure in complex networks,” PLoS One 7:e44459
- Liu, Slotine, et al.
- 2012
(Show Context)
Citation Context ...st introduced by Lin (12) and more recently investigated by Liu et al. (13) and Yuan et al. (14). These advances have enabled further progress related to structural controllability such as centrality =-=(15)-=-, energy (16), effect of correlations (17), emergence of bimodality (18), transition and nonlocality (19), the specific role of individual nodes (20), target control (21), and control ofs is a topic o... |

10 | Design of optimal sparse interconnection graphs for synchronization of oscillator networks,” arXiv preprint arXiv:1302.0449
- Fardad, Lin, et al.
- 2013
(Show Context)
Citation Context ... effort is needed to attain a desired state (30)? Motivated by ongoing studies on the stability and function of power grids (31, 32), we study the control of heterogeneous coupled oscillator networks =-=(33, 34)-=-. Recent research into smart grid technolnamical parameters. Although Kuramoto osci our primary system of interest due to both its microgrids and its rich body of literature, we can be applied to a mu... |

8 |
Control Techniques for Complex Networks (Cambridge Univ
- Meyn
- 2008
(Show Context)
Citation Context ..., biology, chemistry, engineering, and the social sciences. The control of complex networks and complex systems is particularly important because, together, they comprise most of the world we live in =-=(51)-=-; however, the nonlinear nature of realistic dynamical processes and the complex network topologies of real networks represent challenges for the scientific community. Building on concepts from classi... |

7 |
Hierarchical synchrony of phase oscillators in modular networks
- Skardal, Restrepo
- 2012
(Show Context)
Citation Context ... useful for modeling real-world systems (36, 40), uncovering the mechanisms behind emergent collective behavior (41, 42), exploring additional effects such as time delays (43) and community structure =-=(44)-=-, and finding optimal network structure (45). Depending on the coupling strength K, as well as the frequency vector w and the network topology, the steady-state dynamics of Eq. 1 can attain many diffe... |

6 |
Effect of correlations on network controllability
- Pósfai, Liu, et al.
- 2013
(Show Context)
Citation Context ...y investigated by Liu et al. (13) and Yuan et al. (14). These advances have enabled further progress related to structural controllability such as centrality (15), energy (16), effect of correlations =-=(17)-=-, emergence of bimodality (18), transition and nonlocality (19), the specific role of individual nodes (20), target control (21), and control ofs is a topic of vital theoretical importance in ing and ... |

6 |
Emergence of bimodality in controlling complex networks
- Jia, Liu, et al.
- 2013
(Show Context)
Citation Context ...13) and Yuan et al. (14). These advances have enabled further progress related to structural controllability such as centrality (15), energy (16), effect of correlations (17), emergence of bimodality =-=(18)-=-, transition and nonlocality (19), the specific role of individual nodes (20), target control (21), and control ofs is a topic of vital theoretical importance in ing and various other sciences. Motiva... |

6 |
Nonlinear dynamics of heart rhythm disorders
- Karma, Gilmour
- 2007
(Show Context)
Citation Context ...n the area of power grid technology, synchronization phenomenon plays a vital role in a variety of complex processes that occur in both natural and manmade systems, including healthy cardiac behavior =-=(54)-=-, func-gles, and green squares, respectively). Results for ER and SF networks are plotted with unfilled and filled symbols, respectively, and each curve represents an average of over 100 network reali... |

6 | E.: Low-energy control of electrical turbulence in the heart.
- Luther, Fenton, et al.
- 2011
(Show Context)
Citation Context ... Eq. 7 are of the same form as Eq. 2, and therefore, the control method we present above cantreatments that require minimal shock to knock out fatal asynchronous behavior such as cardiac fibrillation =-=(58)-=- and the promotion of normal brain oscillations (59) while repressing disorders such as Parkinson’s disease, which are associated with abnormal oscillations (60). MATERIALS AND METHODS Steady-state so... |

5 |
Network controllability is determined by the density of low in-degree and out-degree
- Menichetti, Dall’Asta, et al.
- 2014
(Show Context)
Citation Context ...d to structural controllability such as centrality (15), energy (16), effect of correlations (17), emergence of bimodality (18), transition and nonlocality (19), the specific role of individual nodes =-=(20)-=-, target control (21), and control ofs is a topic of vital theoretical importance in ing and various other sciences. Motivated by synchronization and consider the important s—a paradigmatic example th... |

5 | Sparsity-promoting optimal wide-area control of power networks
- Dörfler, Jovanović, et al.
- 2014
(Show Context)
Citation Context ... effort is needed to attain a desired state (30)? Motivated by ongoing studies on the stability and function of power grids (31, 32), we study the control of heterogeneous coupled oscillator networks =-=(33, 34)-=-. Recent research into smart grid technolnamical parameters. Although Kuramoto osci our primary system of interest due to both its microgrids and its rich body of literature, we can be applied to a mu... |

5 |
Transforming the electric infrastructure
- Gellings, Yeagee
- 2004
(Show Context)
Citation Context ...research. Here, we have focused on the control of synchronization (that is, consensus) in coupled oscillator networks. Our primary inspiration has been advances in the research of power grid networks =-=(52, 53)-=-. In particular, recent studies have shown that certain power grids known as microgrids can be treated as Kuramoto oscillator networks (35, 39). Here, we have presented a control method that can easil... |

4 |
Physics of cardiac arrhythmogenesis
- Karma
- 2013
(Show Context)
Citation Context ...ng in connection to a RESULTS The Kuramoto model1 2ogies has shown that certain power grid networks called “microgrids” evolve and can be treated as networks of Kuramoto phase oscillators arrhythmias =-=(37)-=- and treatments for pathological brain dynamics (38).been made in the control of nonlinear systems, for instance, the control of chaotic systems using unstable periodic orbits (23), control via pinnin... |

4 |
Optimal synchronization of complex networks
- Skardal, Taylor, et al.
- 2014
(Show Context)
Citation Context ... 40), uncovering the mechanisms behind emergent collective behavior (41, 42), exploring additional effects such as time delays (43) and community structure (44), and finding optimal network structure =-=(45)-=-. Depending on the coupling strength K, as well as the frequency vector w and the network topology, the steady-state dynamics of Eq. 1 can attain many different states that included complete incoheren... |

4 |
Erosion of synchronization in networks of coupled oscillators
- Skardal, Taylor, et al.
- 2015
(Show Context)
Citation Context ...yield equivalent linearizations and therefore can also be controlled using themethod we provide here. We present an example of such a general system with arbitrary coupling function [for example, see =-=(47)-=-] inMaterials andMethods. The stability of q = q* is dictated by the Jacobian matrix whose entries are defined DFij ∂q̇i =∂qj , and is stable if all the eigenvalues of DF|q* are nonpositive. In ou... |

4 |
Control of coupled oscillator networks with application to microgrid technologies
- Skardal, Arenas
- 2015
(Show Context)
Citation Context ... such systems is not only a question of theoretical interest to mathematicians but also has a wide range of important applications in physics, chemistry, biology, engineering, and the social sciences =-=(10, 11)-=-. The roots of modern linear and nonlinear control reach back several decades, but recently, research in this direction has seen a revival in physics and engineering communities. For instance, the con... |