Modular synchronization in complex networks with a gauge Kuramoto model

We modify the Kuramoto model for synchronization on complex networks by introducing a gauge term that depends on the edge betweenness centrality (BC). The gauge term introduces additional phase difference between two vertices from 0 to $\pi$ as the BC on the edge between them increases from the minimum to the maximum in the network. When the network has a modular structure, the model generates the phase synchronization within each module, however, not over the entire system. Based on this feature, we can distinguish modules in complex networks, with relatively little computational time of $\mathcal{O}(NL)$, where $N$ and $L$ are the number of vertices and edges in the system, respectively. We also examine the synchronization of the modified Kuramoto model and compare it with that of the original Kuramoto model in several complex networks.Comment: 10 pages, 7 figure

Power-Law Distributions of Dynamic Cascade Failures in Power-Grid Models

Power-law distributed cascade failures are well known in power-grid systems. Understanding this phenomena has been done by various DC threshold models, self-tuned at their critical point. Here we attempt to describe it using an AC threshold model, with a second-order Kuramoto type equation of motion of the power-flow. We have focused on the exploration of network heterogeneity effects, starting from homogeneous 2D lattices to the US power-grid, possessing identical nodes and links, to a realistic electric power-grid obtained from the Hungarian electrical database. The last one exhibits node dependent parameters, topologically marginally on the verge of robust networks. We show that too weak quenched heterogeneity, coming solely from the probabilistic self-frequencies of nodes (2D lattice) is not sufficient to find power-law distributed cascades. On the other hand too strong heterogeneity destroys the synchronization of the system. We found agreement with the empirically observed power-law failure size distributions on the US grid, as well as on the Hungarian networks near the synchronization transition point. We have also investigated the consequence of replacing the usual Gaussian self-frequencies to exponential distributed ones, describing renewable energy sources. We found a drop in the steady state synchronization averages, but the cascade size distribution both for the US and Hungarian systems remained insensitive and have kept the universal tails, characterized by the exponent $\tau\simeq 1.8$. We have also investigated the effect of an instantaneous feedback mechanism in case of the Hungarian power-grid.Comment: Extended version with minor changes, accepted in Entropy 22 pages, 13 figure

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 동적 축척 이론

학위논문 (박사)-- 서울대학교 대학원 : 물리·천문학부 물리학 전공, 2013. 8. 강병남.서로 영향을 주고 받는 많은 개체들로 이루어진 계가 집단적인 행동을 하는 경우가 자주 있다. 집단적인 행동의 대표적인 예로 바로 동기화를 들 수 있다. 매우 간단한 경우로 각 개체를 위상을 갖고 있는 진동자로 생각하자. 진동자 마다 고유한 성질, 즉 고유 진동수를 갖고 있기 때문에 상호작용이 없을 때에 는 위상이 서로 다르다. 이제 진동자들이 강하게 상호작용을 하면, 시간이 갈 수록 진동자들은 점점 위상을 맞추어 가게 되고, 결국 거의 모든 진동자들이 평균 위상에 가까운 위상을 갖게 된다. 이를 위상 동기화가 일어났다고 한다. 결합된 진동자들의 위상 동기화를 모사하는 간단하면서도 대표적인 모 형으로 구라모토 모형이 있다. 이미 지난 수 십년 간 구라모토 모형에 관한 많은 연구들이 활발히 진행되었으며, 그 덕분에 위상 동기화에 대한 흥미로 운 연구 결과들도 많이 알려져 있다. 1장 도입부에서는 동기화와 구라모토 모형을 소개하고, 선행 연구들이 밝혀 낸 연구 결과들을 간략히 소개하였다. 이 학위 논문은 본인이 물리학 박사 과정 동안 연구한 동기화와 관련된 주제 두 가지를 다룬다. 첫 번째 연구는 구라모토 모형의 동적 축척 현상에 대한 연구다. 지금까지의 선행 연구들 대부분은 동기화를 보이는 계의 정상 상태에 초점을 맞추었으나, 우리는 계가 정상 상태에 들어가기 전 동적 상태 에 있을 때에도 초점을 맞추었다. 그 결과 결합 상수가 임계점 근처일 때, 구 라모토 모형의 질서도가 시간에 따라 멱함수 꼴로 변화하는 것을 발견하였고, 이에 맞는 동적 축적 함수를 제안하였다. 계의 크기를 변화시키면서 시간에 따른 질서도의 변화를 구하고 동적 축척 관계를 이용하면, 여러 크기의 계로 그린 그래프들이 하나로 겹쳐지는 것을 확인할 수 있다. 더 나아가서 여기에 사용된 동적 축척 지수들은 유한 크기 축척 이론에서 이미 알려진 정적 축척 지수로 설명될 수 있음을 보였다. 완전히 연결되어 있는 네트워크, n차원 정방형격자, 무작위로 연결된 네트워크, 그리고 축척 없는 네트워크에서 구라 모토 모형의 질서도에 대한 동적 축척 관계를 확인하였다. 각 진동자의 고유 진동수들의 분포로는 정규분포를 사용하였는데, 고유 진동수들이 똑같이 이 분포를 따르더라도 어떻게 생성되었느냐에 따라 질서도의 시간적 변화 양상 이 달라지는 것을 발견하였다. 2장에서는 동적 축척 관계를 해석하는 데 바탕 이 되는 유한 크기 축척 이론에 대하여 다루고, 3장에서는 동적 축척 관계에 대하여 실었다. 결합 상수가 임계점 근처에서 동적 축척 관계가 존재한다는 사실 자체 로도 의미가 있으나, 이것이 중요한 이유가 한 가지 더 있다. 진동자 수가 아주 많은 계에서 임계 결합 상수를 컴퓨터 수치 계산으로 찾으려면, 계산 시 간이 오래 걸린다. 구라모토 모형이 기술하는 미분 방정식으로 dt 시간 후의 위상들을 한 번 계산하는 데 걸리는 시간은 최적화를 하더라도 계의 크기에 비례하여 증가하며, 현실적으로 dt 시간 후의 위상들을 구하는 작업을 무한히 반복할 수도 없다. 동적 축척 관계에 대한 지식이 있으면, 계가 정상 상태에 도달할 때까지 계산할 필요 없이 동적 상태에 있을 때의 질서도의 변화를 분석하여 임계 결합 상수를 구할 수 있으므로 계산 시간을 크게 줄일 수 있 게 된다. 임계점 근처에서 질서도가 멱함수 법칙을 따르는 현상을 이용하여 상대적 오차가 천분의 일 보다 작을 정도로 임계 결합 상수를 구별해 낼 수 있었다. 두 번째 주제는 구라모토 모형을 클러스터링 문제에 응용한 연구로 4 장에서 다루었다. 클러스터링은 복잡계 네트워크에서 모듈 구조를 찾는 문제 인데, 구라모토 모형의 상호작용 함수에 링크 중앙성에 의존하는 새 항을 도 입하여 클러스터링에 활용하였다. 이 항이 도입되면서 링크 중앙성의 크기에 따라 링크 양 끝에 위치한 진자 사이에 0에서 까지의 위상차가 생기게 된 다. 변형된 구라모토 모형을 모듈 구조를 갖는 복잡계 네트워크에 적용하면, 각각의 모듈 안에서는 위상 동기화가 일어나는 반면, 전체 네트워크는 계속 무질서한 상태에 남아있게 된다. 이 현상을 바탕으로 기존의 다른 클러스터 링 방법들과 비교하여 상대적으로 계산 시간은 적으면서도 정확도는 비슷한 새로운 클러스터링 알고리즘을 제안하였다. 마지막 5장에서 본 학위 논문 내용을 요약하고, 결론을 내리고 마무리 하였다. 부록에는 구라모토 모형을 직접 수치 계산하는 데 있어서 도움이 될 만한 정보 몇 가지를 실었다.Systems with many elements interacting with each other often show collective behaviors. Synchronization is a typical example of the collective behaviors. At initial time, phases are different from each other because of the unique characteristics of each element (oscillator). As time goes on, interacting with others strongly, the oscillators adjust their phases and finally set the phases to almost mean phase of their neighbors. The Kuramoto model is a simple and representative model for synchronization of coupled oscillators. For past decades many studies on the Kuramoto model has discovered much interesting results on synchronization of coupled oscillators. In the introduction, a little amount of these results of the previous studies on the Kuramoto model is introduced. This dissertation has two main studies related to synchronization. The first part is a study on dynamic scaling for the Kuramoto model. Until now, most of the previous studies has focused on the steady states. We also focus on temporal behavior of order parameter before the system goes to the steady state. We found that the order parameter of Kuramoto model evolves following a power-law of t at critical coupling strength Kc and conjectured dynamic scaling form of the Kuramoto model at the criticality. With several system sizes, we show that the r v.s. t graphs collapse into one single curve by using this scaling relation. And furthermore, the scaling exponents are explained by already-known exponents from finite-size scaling. All-toall network, n-dimensional square lattices, classical random (Erdös-Rényi, ER) networks and scale-free (SF) networks are tested with natural frequencies chosen from Gaussian distribution. We also found that how to generate the natural frequencies from Gaussian distribution also affects temporal behaviors of the order parameter. In chapter 2, the finite-size scaling theory is described and confirmed numerically that is a base of dynamic scaling relation. And in chapter 3, we study dynamic scaling extensively. The existence of dynamic scaling relation at critical point is an end in itself, and besides there is another reason why it is important practically. One may want to measure order parameter for very long simulation time to find a critical coupling strength with large system size N. But considering the computational cost of numerical integration for Kuramoto dynamics, there is always a practical limit of system size N or allowed simulation time. Thanks to the knowledge of dynamic scaling relation, one can find Kc in short simulation time even though the system does not enter to the steady state yet. The power-law behavior of order parameter distinguishes Kc very well and catches Kc clearly with a resolution of (K−Kc)/Kc ~ O(10−3). The second part of this dissertation is a kind of application of Kuramoto model to clustering problems and chapter 4 treats it. Clustering is to find modular structures of complex networks. We modify the Kuramoto model by introducing an additional term depending on the link betweenness centrality (BC) in interaction function. The additional term induces phase difference (2 [0,pi]) between connected two oscillators as the BC on the link connecting them increases from the minimum to the maximum in the network. When this modified Kuramoto model is applied to networks with modular structure, the model drives each module to ordered state (phase synchronization), however the entire system remains in disordered state. Based on this phenomenon, we proposed a new clustering algorithm that shows quite good performance similar to other clustering algorithms and needs relatively little computational cost compared to other synchronization-based algorithms. Finally, chapter 5 is devoted to the conclusion of this study. And some information which is helpful to simulate the Kuramoto model is in Appendices.Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii 1. Introduction to Synchronization and Kuramoto Model . 1 1.1 What is synchronization? . . . . . . . . . . . . . . . . . . . . . 1 1.2 Coupled oscillators . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Kuramoto model . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Natural frequency . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4.1 Gaussian distribution . . . . . . . . . . . . . . . . . . . 4 1.4.2 Cauchy distribution . . . . . . . . . . . . . . . . . . . . 5 1.4.3 Uniform distribution . . . . . . . . . . . . . . . . . . . . 5 1.4.4 Double delta peaks . . . . . . . . . . . . . . . . . . . . . 5 1.5 Sampling of natural frequency . . . . . . . . . . . . . . . . . . 6 1.5.1 Random sampling . . . . . . . . . . . . . . . . . . . . . 6 1.5.2 Regular sampling . . . . . . . . . . . . . . . . . . . . . 6 1.6 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6.1 Complex order parameter . . . . . . . . . . . . . . . . . 9 1.6.2 Susceptibility, x . . . . . . . . . . . . . . . . . . . . . . 11 1.6.3 Standard deviation, . . . . . . . . . . . . . . . . . . . 12 1.6.4 Binders cumulant, U4 . . . . . . . . . . . . . . . . . . . 12 1.7 Phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.8 Variants of Kuramoto model . . . . . . . . . . . . . . . . . . . 14 1.8.1 Kuramoto model with random noise . . . . . . . . . . . 14 1.8.2 Kuramoto model with periodic driving force . . . . . . 15 1.8.3 Kuramoto model with inertia . . . . . . . . . . . . . . . 15 1.8.4 Kuramoto model with inertia and external periodic force 16 1.8.5 Kuramoto model with inertia and random noise . . . . 16 1.8.6 Kuramoto model on complex networks . . . . . . . . . . 17 1.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2. Finite-Size Scaling . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1 Critical exponents . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Finite-size effect and scaling function . . . . . . . . . . . . . . 22 2.3 Numerical results of finite-size scaling . . . . . . . . . . . . . . 24 2.3.1 Order parameter . . . . . . . . . . . . . . . . . . . . . . 25 2.3.2 Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.3 Standard deviation . . . . . . . . . . . . . . . . . . . . . 27 2.3.4 Binders cumulant . . . . . . . . . . . . . . . . . . . . . 27 2.3.5 Effect of frequency-disorder fluctuation . . . . . . . . . 31 2.4 Analytic approach: Self consistency equation of order parameter 32 2.4.1 Case of random sampling . . . . . . . . . . . . . . . . . 35 2.4.2 Case of regular sampling . . . . . . . . . . . . . . . . . 39 2.5 n-dimensional square lattice . . . . . . . . . . . . . . . . . . . 46 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3. Dynamic Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.1 Motivation for dynamic scaling . . . . . . . . . . . . . . . . . . 51 3.2 Temporal behavior of order parameter near critical coupling strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.3 What is dynamic scaling? . . . . . . . . . . . . . . . . . . . . . 54 3.4 Dynamic scaling with random sampling . . . . . . . . . . . . . 55 3.4.1 Starting from ordered state, r(0) = 1 . . . . . . . . . . . 55 3.4.2 Starting from disordered state, r(0) = O(N−1/2) . . . . 57 3.5 Dynamic scaling with regular sampling . . . . . . . . . . . . . 60 3.5.1 Starting from ordered state, r(0) = 1 . . . . . . . . . . . 61 3.5.2 Starting from disordered state, r(0) = O(N−1/2) . . . . 62 3.6 Dynamic scaling with thermal noise . . . . . . . . . . . . . . . 65 3.7 Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.7.1 6 dimension . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.7.2 5 dimension . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.8 ER network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.9 Scale-free network . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.9.1 g=6.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.9.2 g=4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.9.3 g=3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.10 Temporal behavior of order parameter in systems exhibiting first-order phase transition . . . . . . . . . . . . . . . . . . . . 76 3.11 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4. Modular Synchronization and Its Application . . . . . . . 79 4.1 Module detection in modular networks . . . . . . . . . . . . . 79 4.2 How to detect modular structure in modular networks? . . . . 80 4.3 Modular synchronization and Kuramoto model with gauge term 81 4.4 Clustering algorithm . . . . . . . . . . . . . . . . . . . . . . . 83 4.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Appendix A. Numerical Simulation Method . . . . . . . . . . 93 A.1 Kahans summation . . . . . . . . . . . . . . . . . . . . . . . . 93 A.2 Computational cost: running time . . . . . . . . . . . . . . . . 94 A.3 How to determine rsat and tsat . . . . . . . . . . . . . . . . . . 94 Appendix B. s/K and time step dt . . . . . . . . . . . . . . . . 97 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Abstract in Korean . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Docto

Uncovering hidden flows in physical networks

Chengwei Wang is supported by a studentship funded by the College of Physical Sciences, University of Aberdeen.Peer reviewedPostprintPostprin

A unified framework for Simplicial Kuramoto models

Simplicial Kuramoto models have emerged as a diverse and intriguing class of models describing oscillators on simplices rather than nodes. In this paper, we present a unified framework to describe different variants of these models, categorized into three main groups: "simple" models, "Hodge-coupled" models, and "order-coupled" (Dirac) models. Our framework is based on topology, discrete differential geometry as well as gradient flows and frustrations, and permits a systematic analysis of their properties. We establish an equivalence between the simple simplicial Kuramoto model and the standard Kuramoto model on pairwise networks under the condition of manifoldness of the simplicial complex. Then, starting from simple models, we describe the notion of simplicial synchronization and derive bounds on the coupling strength necessary or sufficient for achieving it. For some variants, we generalize these results and provide new ones, such as the controllability of equilibrium solutions. Finally, we explore a potential application in the reconstruction of brain functional connectivity from structural connectomes and find that simple edge-based Kuramoto models perform competitively or even outperform complex extensions of node-based models.Comment: 36 pages, 11 figure

Module hierarchy and centralisation in the anatomy and dynamics of human cortex

Systems neuroscience has recently unveiled numerous fundamental features of the macroscopic architecture of the human brain, the connectome, and we are beginning to understand how characteristics of brain dynamics emerge from the underlying anatomical connectivity. The current work utilises complex network analysis on a high-resolution structural connectivity of the human cortex to identify generic organisation principles, such as centralised, modular and hierarchical properties, as well as specific areas that are pivotal in shaping cortical dynamics and function. After confirming its small-world and modular architecture, we characterise the cortex’ multilevel modular hierarchy, which appears to be reasonably centralised towards the brain’s strong global structural core. The potential functional importance of the core and hub regions is assessed by various complex network metrics, such as integration measures, network vulnerability and motif spectrum analysis. Dynamics facilitated by the large-scale cortical topology is explored by simulating coupled oscillators on the anatomical connectivity. The results indicate that cortical connectivity appears to favour high dynamical complexity over high synchronizability. Taking the ability to entrain other brain regions as a proxy for the threat posed by a potential epileptic focus in a given region, we also show that epileptic foci in topologically more central areas should pose a higher epileptic threat than foci in more peripheral areas. To assess the influence of macroscopic brain anatomy in shaping global resting state dynamics on slower time scales, we compare empirically obtained functional connectivity data with data from simulating dynamics on the structural connectivity. Despite considerable micro-scale variability between the two functional connectivities, our simulations are able to approximate the profile of the empirical functional connectivity. Our results outline the combined characteristics a hierarchically modular and reasonably centralised macroscopic architecture of the human cerebral cortex, which, through these topological attributes, appears to facilitate highly complex dynamics and fundamentally shape brain function

Biological Lattice Gauge Theory as Modeling of Quantum Neural Networks

Based on quantum biology and biological gauge field theory, we propose the biological lattice gauge theory as modeling of quantum neural networks. This method applies completely the same lattice theory in quantum field, but, whose two anomaly problems may just describe the double helical structure of DNA and violated chiral symmetry in biology. Further, we discuss the model of Neural Networks (NN) and the quantum neutral networks, which are related with biological loop quantum theory. Finally, we research some possible developments on described methods of networks by the extensive graph theory and their new mathematical forms