Cluster Synchronization of Kuramoto Oscillators and Brain Functional Connectivity

The recent progress of functional magnetic resonance imaging techniques has unveiled that human brains exhibit clustered correlation patterns of their spontaneous activities. It is important to understand the mechanism of cluster synchronization phenomena since it may reflect the underlying brain functions and brain diseases. In this paper, we investigate cluster synchronization conditions for networks of Kuramoto oscillators. The key analytical tool that we use is the method of averaging, and we provide a unified framework of stability analysis for cluster synchronization. The main results show that cluster synchronization is achieved if (i) the inter-cluster coupling strengths are sufficiently weak and/or (ii) the natural frequencies are largely different among clusters. Moreover, we apply our theoretical findings to empirical brain networks. Discussions on how to understand brain functional connectivity and further directions to investigate neuroscientific questions are provided

Critical Coupling and Synchronized Clusters in Arbitrary Networks of Kuramoto Oscillators

abstract: The Kuramoto model is an archetypal model for studying synchronization in groups of nonidentical oscillators where oscillators are imbued with their own frequency and coupled with other oscillators though a network of interactions. As the coupling strength increases, there is a bifurcation to complete synchronization where all oscillators move with the same frequency and show a collective rhythm. Kuramoto-like dynamics are considered a relevant model for instabilities of the AC-power grid which operates in synchrony under standard conditions but exhibits, in a state of failure, segmentation of the grid into desynchronized clusters. In this dissertation the minimum coupling strength required to ensure total frequency synchronization in a Kuramoto system, called the critical coupling, is investigated. For coupling strength below the critical coupling, clusters of oscillators form where oscillators within a cluster are on average oscillating with the same long-term frequency. A unified order parameter based approach is developed to create approximations of the critical coupling. Some of the new approximations provide strict lower bounds for the critical coupling. In addition, these approximations allow for predictions of the partially synchronized clusters that emerge in the bifurcation from the synchronized state. Merging the order parameter approach with graph theoretical concepts leads to a characterization of this bifurcation as a weighted graph partitioning problem on an arbitrary networks which then leads to an optimization problem that can efficiently estimate the partially synchronized clusters. Numerical experiments on random Kuramoto systems show the high accuracy of these methods. An interpretation of the methods in the context of power systems is provided.Dissertation/ThesisDoctoral Dissertation Applied Mathematics 201

Reduction of Second-Order Network Systems with Structure Preservation

This paper proposes a general framework for structure-preserving model reduction of a secondorder network system based on graph clustering. In this approach, vertex dynamics are captured by the transfer functions from inputs to individual states, and the dissimilarities of vertices are quantified by the H2-norms of the transfer function discrepancies. A greedy hierarchical clustering algorithm is proposed to place those vertices with similar dynamics into same clusters. Then, the reduced-order model is generated by the Petrov-Galerkin method, where the projection is formed by the characteristic matrix of the resulting network clustering. It is shown that the simplified system preserves an interconnection structure, i.e., it can be again interpreted as a second-order system evolving over a reduced graph. Furthermore, this paper generalizes the definition of network controllability Gramian to second-order network systems. Based on it, we develop an efficient method to compute H2-norms and derive the approximation error between the full-order and reduced-order models. Finally, the approach is illustrated by the example of a small-world network

Hierarchical community structure in networks

Modular and hierarchical structures are pervasive in real-world complex systems. A great deal of effort has gone into trying to detect and study these structures. Important theoretical advances in the detection of modular, or "community", structures have included identifying fundamental limits of detectability by formally defining community structure using probabilistic generative models. Detecting hierarchical community structure introduces additional challenges alongside those inherited from community detection. Here we present a theoretical study on hierarchical community structure in networks, which has thus far not received the same rigorous attention. We address the following questions: 1)~How should we define a valid hierarchy of communities? 2)~How should we determine if a hierarchical structure exists in a network? and 3)~how can we detect hierarchical structure efficiently? We approach these questions by introducing a definition of hierarchy based on the concept of stochastic externally equitable partitions and their relation to probabilistic models, such as the popular stochastic block model. We enumerate the challenges involved in detecting hierarchies and, by studying the spectral properties of hierarchical structure, present an efficient and principled method for detecting them.Comment: 22 pages, 12 figure

Node dynamics on graphs: dynamical heterogeneity in consensus, synchronisation and final value approximation for complex networks

Here we consider a range of Laplacian-based dynamics on graphs such as dynamical invariance and coarse-graining, and node-specific properties such as convergence, observability and consensus-value prediction. Firstly, using the intrinsic relationship between the external equitable partition (EEP) and the spectral properties of the graph Laplacian, we characterise convergence and observability properties of consensus dynamics on networks. In particular, we establish the relationship between the original consensus dynamics and the associated consensus of the quotient graph under varied initial conditions. We show that the EEP with respect to a node can reveal nodes in the graph with increased rate of asymptotic convergence to the consensus value as characterised by the second smallest eigenvalue of the quotient Laplacian. Secondly, we extend this characterisation of the relationship between the EEP and Laplacian based dynamics to study the synchronisation of coupled oscillator dynamics on networks. We show that the existence of a non-trivial EEP describes partial synchronisation dynamics for nodes within cells of the partition. Considering linearised stability analysis, the existence of a nontrivial EEP with respect to an individual node can imply an increased rate of asymptotic convergence to the synchronisation manifold, or a decreased rate of de-synchronisation, analogous to the linear consensus case. We show that high degree 'hub' nodes in large complex networks such as Erdős-Rényi, scale free and entangled graphs are more likely to exhibit such dynamical heterogeneity under both linear consensus and non-linear coupled oscillator dynamics. Finally, we consider a separate but related problem concerning the ability of a node to compute the final value for discrete consensus dynamics given only a finite number of its own state values. We develop an algorithm to compute an approximation to the consensus value by individual nodes that is ϵ close to the true consensus value, and show that in most cases this is possible for substantially less steps than required for true convergence of the system dynamics. Again considering a variety of complex networks we show that, on average, high degree nodes, and nodes belonging to graphs with fast asymptotic convergence, approximate the consensus value employing fewer steps.Open Acces

Data-driven Selection of Coarse-Grained Models of Coupled Oscillators

Systematic discovery of reduced-order closure models for multi-scale processes remains an important open problem in complex dynamical systems. Even when an effective lower-dimensional representation exists, reduced models are difficult to obtain using solely analytical methods. Rigorous methodologies for finding such coarse-grained representations of multi-scale phenomena would enable accelerated computational simulations and provide fundamental insights into the complex dynamics of interest. We focus on a heterogeneous population of oscillators of Kuramoto type as a canonical model of complex dynamics, and develop a data-driven approach for inferring its coarse-grained description. Our method is based on a numerical optimization of the coefficients in a general equation of motion informed by analytical derivations in the thermodynamic limit. We show that certain assumptions are required to obtain an autonomous coarse-grained equation of motion. However, optimizing coefficient values enables coarse-grained models with conceptually disparate functional forms, yet comparable quality of representation, to provide accurate reduced-order descriptions of the underlying system.Comment: 13 pages, 10 figure