Self-organized explosive synchronization in complex networks: Emergence of synchronization bombs

We introduce the concept of synchronization bombs as large networks of coupled heterogeneous oscillators that operate in a bistable regime and abruptly transit from incoherence to phase-locking (or vice-versa) by adding (or removing) one or a few links. Here we build a self-organized and stochastic version of these bombs, by optimizing global synchrony with decentralized information in a competitive link-percolation process driven by a local rule. We find explosive fingerprints on the emerging network structure, including frequency-degree correlations, disassortative patterns and a delayed percolation threshold. We show that these bomb-like transitions can be designed both in systems of Kuramoto -- periodic -- and R\"ossler -- chaotic -- oscillators and in a model of cardiac pacemaker cells. We analytically characterize the transitions in the Kuramoto case by combining a precise collective coordinates approach and the Ott-Antonsen ansatz. Furthermore, we study the robustness of the phenomena under changes in the main parameters and the unexpected effect of optimal noise in our model. Our results propose a minimal self-organized mechanism of network growth to understand and control explosive synchronization in adaptive biological systems like the brain and engineered ones like power-grids or electronic circuits. From a theoretical standpoint, the emergence of synchronization explosions and bistability induced by localized structural perturbations -- without any fine-tuning of global parameters -- joins explosive synchronization and percolation under the same mechanistic framework.Comment: 17 pages, 9 figure

Synchronization in Complex Networks Under Uncertainty

La sincronització en xarxes és la música dels sistemes complexes. Els ritmes col·lectius que emergeixen de molts oscil·ladors acoblats expliquen el batec constant del cor, els patrons recurrents d'activitat neuronal i la sincronia descentralitzada a les xarxes elèctriques. Els models matemàtics són sòlids i han avançat significativament, especialment en el problema del camp mitjà, on tots els oscil·ladors estan connectats mútuament. Tanmateix, les xarxes reals tenen interaccions complexes que dificulten el tractament analític. Falta un marc general i les soluciones existents en caixes negres numèriques i espectrals dificulten la interpretació. A més, la informació obtinguda en mesures empíriques sol ser incompleta. Motivats per aquestes limitacions, en aquesta tesi proposem un estudi teòric dels oscil·ladors acoblats en xarxes sota incertesa. Apliquem propagació d'errors per predir com una estructura complexa amplifica el soroll des dels pesos microscòpics fins al punt crític de sincronització, estudiem l'efecte d'equilibrar les interaccions de parelles i d'ordre superior en l'optimització de la sincronia i derivem esquemes d'ajust de pesos per mapejar el comportament de sincronització en xarxes diferents. A més, un desplegament geomètric rigorós de l'estat sincronitzat ens permet abordar escenaris descentralitzats i descobrir regles locals òptimes que indueixen transicions globals abruptes. Finalment, suggerim dreceres espectrals per predir punts crítics amb àlgebra lineal i representacions aproximades de xarxa. En general, proporcionem eines analítiques per tractar les xarxes d'oscil·ladors en condicions sorolloses i demostrem que darrere els supòsits predominants d'informació completa s'amaguen explicacions mecanicistes clares. Troballes rellevants inclouen xarxes particulars que maximitzen el ventall de comportaments i el desplegament exitós del binomi estructura-dinàmica des d'una perspectiva local. Aquesta tesi avança la recerca d'una teoria general de la sincronització en xarxes a partir de principis mecanicistes i geomètrics, una peça clau que manca en l'anàlisi, disseny i control de xarxes neuronals biològiques i artificials i sistemes d'enginyeria complexos.La sincronización en redes es la música de los sistemas complejos. Los ritmos colectivos que emergen de muchos osciladores acoplados explican el latido constante del corazón, los patrones recurrentes de actividad neuronal y la sincronía descentralizada de las redes eléctricas. Los modelos matemáticos son sólidos y han avanzado significativamente, especialmente en el problema del campo medio, donde todos los osciladores están conectados entre sí. Sin embargo, las redes reales tienen interacciones complejas que dificultan el tratamiento analítico. Falta un marco general y las soluciones en cajas negras numéricas y espectrales dificultan la interpretación. Además, las mediciones empíricas suelen ser incompletas. Motivados por estas limitaciones, en esta tesis proponemos un estudio teórico de osciladores acoplados en redes bajo incertidumbre. Aplicamos propagación de errores para predecir cómo una estructura compleja amplifica el ruido desde las conexiones microscópicas hasta puntos críticos macroscópicos, estudiamos el efecto de equilibrar interacciones por pares y de orden superior en la optimización de la sincronía y derivamos esquemas de ajuste de pesos para mapear el comportamiento en estructuras distintas. Una expansión geométrica del estado sincronizado nos permite abordar escenarios descentralizados y descubrir reglas locales que inducen transiciones abruptas globales. Por último, sugerimos atajos espectrales para predecir puntos críticos usando álgebra lineal y representaciones aproximadas de red. En general, proporcionamos herramientas analíticas para manejar redes de osciladores en condiciones ruidosas y demostramos que detrás de las suposiciones predominantes de información completa se ocultaban claras explicaciones mecanicistas. Hallazgos relevantes incluyen redes particulares que maximizan el rango de comportamientos y la explicación del binomio estructura-dinámica desde una perspectiva local. Esta tesis avanza en la búsqueda de una teoría general de sincronización en redes desde principios mecánicos y geométricos, una pieza clave que falta en el análisis, diseño y control de redes neuronales biológicas y artificiales y sistemas de ingeniería complejos.Synchronization in networks is the music of complex systems. Collective rhythms emerging from many interacting oscillators appear across all scales of nature, from the steady heartbeat and the recurrent patterns in neuronal activity to the decentralized synchrony in power-grids. The mathematics behind these processes are solid and have significantly advanced lately, especially in the mean-field problem, where oscillators are all mutually connected. However, real networks have complex interactions that difficult the analytical treatment. A general framework is missing and most existing results rely on numerical and spectral black-boxes that hinder interpretation. Also, the information obtained from measurements is usually incomplete. Motivated by these limitations, in this thesis we propose a theoretical study of network-coupled oscillators under uncertainty. We apply error propagation to predict how a complex structure amplifies noise from the link weights to the synchronization onset, study the effect of balancing pair-wise and higher-order interactions in synchrony optimization, and derive weight-tuning schemes to map the synchronization behavior of different structures. Also, we develop a rigorous geometric unfolding of the synchronized state to tackle decentralized scenarios and to discover optimal local rules that induce global abrupt transitions. Last, we suggest spectral shortcuts to predict critical points using linear algebra and network representations with limited information. Overall, we provide analytical tools to deal with oscillator networks under noisy conditions and prove that mechanistic explanations were hidden behind the prevalent assumptions of complete information. Relevant finding include particular networks that maximize the range of behaviors and the successful unfolding of the structure-dynamics interplay from a local perspective. This thesis advances the quest of a general theory of network synchronization built from mechanistic and geometric principles, a key missing piece in the analysis, design and control of biological and artificial neural networks and complex engineering systems

The Kuramoto model in complex networks

181 pages, 48 figures. In Press, Accepted Manuscript, Physics Reports 2015 Acknowledgments We are indebted with B. Sonnenschein, E. R. dos Santos, P. Schultz, C. Grabow, M. Ha and C. Choi for insightful and helpful discussions. T.P. acknowledges FAPESP (No. 2012/22160-7 and No. 2015/02486-3) and IRTG 1740. P.J. thanks founding from the China Scholarship Council (CSC). F.A.R. acknowledges CNPq (Grant No. 305940/2010-4) and FAPESP (Grants No. 2011/50761-2 and No. 2013/26416-9) for financial support. J.K. would like to acknowledge IRTG 1740 (DFG and FAPESP).Peer reviewedPreprin

결합된 진동자들의 동기화에 대한 유효 포텐셜 및 기계학습 접근법

학위논문 (박사) -- 서울대학교 대학원 : 자연과학대학 물리학과, 2021. 2. 강병남.Systems with multiple interacting elements exhibit collective behaviors. As one of examples of collective behaviors, synchronization is a process of coordinating two or more elements to realize the system in unison. It is an omnipresent phenomena in nature, for instance, firefly flashing, cricket chirping, cardiac pacemaker cell, and so on. To understand and describe the mechanism of synchronization phenomena, coupled oscillator system is often adopted as the most conventional and suitable model for interacting system. Each oscillator has own frequency representing each unique characteristics, and its phase is adjusted through the interaction with other oscillators on the system. On the way to phase synchronization, such interactions or connections between oscillators can be expressed as links on the complex network and each element (oscillator) is then denoted by a node. A number of studies for coupled oscillators on complex networks have been progressed over the past two decades. Among the studies for synchronization of coupled oscillator systems, the Kuramoto model has played a crucial role as a simple and representative model for describing such collective behavior. Owing to its rich properties such as chaotic dynamical behavior and synchronization transition, the Kuramoto model is an appropriate model to explore. First, fundamental results of previous studies on synchronization of the coupled oscillator system, especially the Kuramoto model, are introduced. This dissertation is composed of two main studies for the coupled oscillator system by adopting two different approaches, respectively. As the first main study of this dissertation, we examine the Kuramoto model using analytical way, the effective potential approach. The Kuramoto model exhibits different types of synchronization transitions depending on the type of natural frequency distribution. To obtain these results, the Kuramoto self-consistency equation (SCE) approach has been used successfully. However, this approach affords only limited understanding of more detailed properties such as the stability. We here extend the SCE approach by introducing an effective potential, that is, an integral version of the SCE. We examine the landscape of this effective potential for second-order, first-order, and hybrid synchronization transitions in the thermodynamic limit. In particular, for the hybrid transition, we find that the minimum of effective potential displays a plateau across the region in which the order parameter jumps. This result suggests that the effective potential can be used to determine a type of synchronization transition. In the second study for the coupled oscillator systems, we applied the machine learning approach to investigate the system based on data-driven analysis and to figure out whether the methodology can be extended to the real world system. With growing interest in the machine learning, recent works on physical systems has demonstrated successful progresses by adopting the machine learning approaches for tasks of classification and generation. We here perform various machine learning approaches to the Kuramoto system which is basic model for synchronization phenomena and exhibits complicated chaotic behavior. As the system displays rich properties such as synchronization transition and nonlinearity with varying parameters, we applied machine learning for finding the value of the coupling strength and the critical value. Considering the finite size scaling, we confirm that results follow the critical behavior of the Kuramoto system. By focusing on the phase dynamics of all oscillators, we applied the performance of the artificial neural network for predicting future behaviors of all oscillators and detecting underlying real brain network topology. As the Kuramoto model offers support for the application on real-world systems exhibiting synchronization phenomena or nonlinear behaviors, our work has potential for utilizing the machine learning approaches to such systems.서로 간의 상호작용이 있는 다수의 개체로 구성된 계는 집단적인 행동을 보인다는 것이 잘 알려져있다. 그러한 집단적인 행동의 대표적인 예로써, 동기화 현상은 두 개 이상의 개체가 상호작용을 통해 모두 동일한 상태에 이르게 되는 과정을 뜻한다. 반딧불의 깜빡임, 귀뚜라미의 울음소리, 심장박동원세포 등 자연에는 동기화 현상의 수많은 예들이 있다. 동기화 현상을 이해하고 묘사하기 위한 가장 대표적이고 적합한 모형으로, 결합된 진동자들로 이루어진 시스템을 생각해 볼 수 있다. 시스템에 있는 각각의 진동자들은 각자의 특성을 나타내는 고유 진동수(natural frequency)를 갖고 있으며, 각각의 위상(phase)들은 시스템의 다른 진동자들과의 상호작용을 통해 시간이 지남에 따라 점차 맞추어 나가게 된다. 이 때, 이러한 위상 동기화가 일어나는 과정에서 진동자들 사이의 연결 또는 상호작용들은 복잡계 네트워크 위의 링크(link)로 표현될 수 있으며, 각각의 개체 혹은 진동자들은 노드(node)로 표현된다. 이러한 결합된 진동자들에 대한 수많은 연구들이 지난 20여년간 이루어져 왔다. 집단현상을 묘사하는 간단하면서도 대표적인 모형인 구라모토 모형을 차용하여 결합된 진동자들의 동기화 현상에 대한 많은 연구들이 진행되어왔다. 구라모토 모형은 카오스 동역학, 동기화 상전이 등의 다양한 특성을 나타내는만큼, 흥미로운 연구들이 많이 이루어져 왔는데, 먼저, 구라모토 모형에서 나타나는 동기화 현상에 대한 선행연구들에서 밝혀진 중요한 결과 및 배경들을 이 학위 논문의 앞부분에서 소개하였다. 그리고 각각을 주요한 연구주제로써, 결합된 진동자들의 시스템에 대한 두 가지 방법론을 사용하여 논문을 구성을 하였다. 첫번째 연구에서는, 유효 포텐셜(effective potential)을 이용한 방법론을 도입하여 해석적인 방법으로 구라모토 모형을 분석하였다. 구라모토 모형에서는 고유 진동수의 분포형태가 변함에 따라 동기화 상전이의 유형또한 변하게 되는데, 구라모토 방정식으로부터 유도한 자기일관성 방정식(self-consistency equation)을 사용하여 이러한 결과를 해석적으로 분석할 수 있다. 하지만, 이러한 방법은 시스템의 안정성과 같은 상세한 특징을 파악하는 데에는 어려움이 있다. 이 연구에서는, 자기일관성 방정식을 적분하여 유도한 유효포텐셜 방법론을 도입하여, 열역학적 극한에 있는 시스템에 대하여 1차 상전이, 2차 상전이 뿐만 아니라 하이브리드 동기화 상전이가 나타날 때의 포텐셜 경관(potential landscape)을 파악하였으며, 특히, 하이브리드 상전이에서는 유효 포텐셜의 최솟값이 임계점에서 평평한 형태를 보인다는 것을 확인하였다. 이러한 결과들은 동기화 상전이의 형태를 파악하는 데에 있어서 유효 포텐셜이 주요한 역할을 해줄 수 있음을 의미한다. 두번째 연구에서는, 데이터 기반 방법론인 기계학습을 사용하여 결합된 진동자들의 시스템을 파악하고, 이러한 방법이 실제의 시스템에 대해서도 확장이 가능한지에 대하여 연구하였다. 최근, 과학분야 뿐만아니라 여러 다양한 분야에서 기계학습에 대한 관심이 높아져 왔는데, 물리적 계에 대해서도 기계학습을 이용한 분류 및 생성 작업을 통해 많은 발전이 이루어져왔다. 본 연구에서는, 여러 기계학습의 모형들을 이용해 구라모토 모형에서 보이는 동기화 상전이 및 비선형, 카오스 동역학을 분석하였다. 질서변수의 시간에 따른 동역학으로부터 진동자들 사이에 내재된 상호작용을 찾고, 진동자들의 위상으로 부터 동기화된 상태와 비동기화된 상태를 구분하여 임계점을 찾는 데에 기계학습 방법을 적용시켜 보았다. 유한 크기 축적 방법(finite-size scaling)을 이용하여 이러한 결과들이 기존의 알려진 구라모토 모형에 대한 눈금 바꿈 행태(scaling behavior)의 결과와 일관성이 있는 것을 확인하였다. 또한, 모든 진동자들의 위상 동역학을 인공 신경망에 입력으로 넣어줌으로써, 진동자들의 이후의 동역학 행태를 파악할 뿐만 아니라, 기저에 깔려 있는 실제 쥐의 시각 피질 네트워크를 알아내는 연구를 진행하였다. 따라서, 동기화 현상 및 비선형 동역학을 보이는 여러 실제의 시스템들에 대한 구라모토 모형의 응용이 가능함에 따라, 본 연구는 그러한 시스템에 대해서도 기계학습 방법을 활용할 수 있는 가능성을 내포한다.1 Introduction 1 1.1 Complex network 1 1.2 Coupled oscillators on complex networks 2 1.3 Machine learning 3 2 Synchronization of coupled oscillators 6 2.1 Synchronization 6 2.2 Coupled oscillators 7 2.3 The Kuramoto model 8 2.4 Natural frequency 9 2.4.1 Gaussian distribution 9 2.4.2 Lorentzian distribution 10 2.4.3 Uniform distribution 10 2.5 Sampling of natural frequency 10 2.5.1 Random sampling 11 2.5.2 Regular sampling 11 2.6 Order parameter 13 2.7 Phase transition 14 2.7.1 Synchronization transition 14 2.7.2 Hybrid phase transition 15 2.7.3 Type of synchronization transition 16 2.8 Finite-size scaling 18 2.8.1 Critical exponents 19 2.8.2 Finite-size effect 20 3 Effective potential approach to synchronization transition 26 3.1 Analytic approaches to the Kuramoto model 29 3.1.1 Self-consistency analysis 29 3.1.2 Ott-Antonsen ansatz 31 3.2 Ad hoc free energy 33 3.3 Second-order synchronization transition 37 3.4 First-order synchronization transition 38 3.4.1 Degree-frequency correlation on scale-free network with 2 < λ < 3 38 3.4.2 Dependence of interaction strength on the frequency 42 3.5 Hybrid synchronization transition 45 3.5.1 Uniform distribution g(ω) 45 3.5.2 Degree-frequency correlation on scale-free networks with λ = 3 49 3.5.3 Flat distribution with exponential tails 50 3.5.4 Flat distribution with power-law tails 51 3.6 Summary 55 4 Machine learning approaches to coupled oscillators 56 4.1 Machine learning models 57 4.1.1 Feed-forward neural network 58 4.1.2 Fully-connected neural network 59 4.1.3 Convolutional neural network 59 4.1.4 Recurrent neural network 59 4.1.5 Reservoir computing 59 4.2 Supervised learning 61 4.3 Finding the coupling strength 62 4.4 Finding the synchronized state 65 4.5 Application I : Prediction of the phase dynamics 68 4.6 Application II : Reconstruction of the network structure 72 4.7 Summary 74 5 Conclusion 76 Appendices 79 Appendix A Numerical simulation method 80 A.1 Runge-Kutta method 80 A.2 Kahan summation 82 A.3 Simulation of the Kuramoto equation 82 Appendix B Asymmetric interaction-frequency correlated model 84 Appendix C Effective potential approaches for finite size systems 87 C.1 Random sampling of frequencies 87 C.2 Regular sampling of frequencies 91 C.3 Trapped at metastable states 92 Bibliography 100 Abstract in Korean 108Docto

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Research supported by FAPESP 2015/50122-0 and DFG-GRTK 1740/2. RP and AR are also part of the Research, Innovation and Dissemination Center for Neuromathematics FAPESP grant (2013/07699-0). RP is supported by a FAPESP scholarship (2013/25667-8). ACR is partially supported by a CNPq fellowship (grant 306251/2014-0)