Rewiring dynamical networks with prescribed degree distribution for enhancing synchronizability

In this paper, we present an algorithm for enhancing synchronizability of dynamical networks with prescribed degree distribution. The algorithm takes an unweighted and undirected network as input and outputs a network with the same node-degree distribution and enhanced synchronization properties. The rewirings are based on the properties of the Laplacian of the connection graph, i.e., the eigenvectors corresponding to the second smallest and the largest eigenvalues of the Laplacian. A term proportional to the eigenvectors is adopted to choose potential edges for rewiring, provided that the node-degree distribution is preserved. The algorithm can be implemented on networks of any sizes as long as their eigenvalues and eigenvectors can be calculated with standard algorithms. The effectiveness of the proposed algorithm in enhancing the network synchronizability is revealed by numerical simulation on a number of sample networks including scale-free, Watts-Strogatz, and Erdo?s-Re?nyi graphs. Furthermore, a number of network's structural parameters such as node betweenness centrality, edge betweenness centrality, average path length, clustering coefficient, and degree assortativity are tracked as a function of optimization steps.     

Enhancing synchronizability by rewiring networks

<正>According to different forms of synchronized region,complex networks are divided into typeⅠ(unbounded synchronization region) and typeⅡ(bounded synchronization region) networks.This paper presents a rewiring algorithm to enhance the synchronizability of typeⅠand typeⅡnetworks.By utilizing the algorithm for an unweighted and undirected network,a better synchronizability of network with the same number of nodes and edges can be obtained. Numerical simulations on several different network models are used to support the proposed procedure.The relationship between different topological properties of the networks and the number of rewirings are shown.It finds that the final optimized network is independent of the initial network,and becomes homogeneous.In addition the optimized networks have similar structural properties in the sense of degree,and node and edge betweenness centralities.However,they do not have similar cluster coefficients for typeⅡnetworks.The research may be useful for designing more synchronizable networks and understanding the synchronization behaviour of networks.     

Synchronizability of stochastic network ensembles in a model of interacting dynamical units

Synchronization is an important phenomenon which occurs in the dynamics of complex systems. Synchronized states emerge both from an adjustment of the system parameters and from an application of a proper external stimulus. In the present paper we study synchronized activity in a neural network model whose dynamics is driven by an external activation. In this context we are interested in its synchronizability, i.e. the existence of inputs causing the model system to synchronize. Furthermore, we investigate global synchronizability properties of stochastic network structure ensembles (instead of single realizations of a network architecture). We study the small world network, a model of preferential linking structure, and the classical Erd?s-Renyi random graph as particular examples of network topologies. Their synchronizability properties are investigated by analytical arguments and numerical simulations. Our analysis shows the emergence of synchronizable states of network ensembles for a wide range of the parameter values. In addition we observe and study the transition behaviour from synchronizability to nonsynchronizability.     

两层星形网络的特征值谱及同步能力

多层网络是当今网络科学研究的一个前沿方向.本文深入研究了两层星形网络的特征值谱及其同步能力的问题.通过严格导出的两层星形网络特征值的解析表达式,分析了网络的同步能力与节点数、层间耦合强度和层内耦合强度的关系.当同步域无界时,网络的同步能力只与叶子节点之间的层间耦合强度和网络的层内耦合强度有关;当叶子节点之间的层间耦合强度比较弱时,同步能力仅依赖于叶子节点之间的层间耦合强度;而当层内耦合强度比较弱时,同步能力依赖于层内耦合强度;当同步域有界时,节点数、层间耦合强度和层内耦合强度对网络的同步能力都有影响.当叶子节点之间的层间耦合强度比较弱时,增大叶子节点之间的层间耦合强度会增强网络的同步能力,而节点数、中心节点之间的层间耦合强度和层内耦合强度的增大反而会减弱网络的同步能力;而当层内耦合强度比较弱时,增大层内耦合强度会增强网络的同步能力,而节点数、层间耦合强度的增大会减弱网络的同步能力.进一步,在层间和层内耦合强度都相同的基础上,讨论了如何改变耦合强度更有利于同步.最后,对两层BA无标度网络进行数值仿真,得到了与两层星形网络非常类似的结论.     

Error and attack tolerance of synchronization in Hindmarsh–Rose neural networks with community structure

Synchronization is one of the most important features observed in large-scale complex networks of interacting dynamical systems. As is well known, there is a close relation between the network topology and the network synchronizability. Using the coupled Hindmarsh–Rose neurons with community structure as a model network, in this paper we explore how failures of the nodes due to random errors or intentional attacks affect the synchronizability of community networks. The intentional attacks are realized by removing a fraction of the nodes with high values in some centrality measure such as the centralities of degree, eigenvector, betweenness and closeness. According to the master stability function method, we employ the algebraic connectivity of the considered community network as an indicator to examine the network synchronizability. Numerical evidences show that the node failure strategy based on the betweenness centrality has the most influence on the synchronizability of community networks. With this node failure strategy for a given network with a fixed number of communities, we find that the larger the degree of communities, the worse the network synchronizability; however, for a given network with a fixed degree of communities, we observe that the more the number of communities, the better the network synchronizability.     

Synchronization optimal networks obtained using local structure information

In this paper, the networks with optimal synchronizability are obtained using the local structure information. In scale-free networks, a node will be coupled by its neighbors with maximal degree among the neighbors if and only if the maximal degree is larger than its own degree. If the obtained coupled networks are connected, they are synchronization optimal networks. The connection probability of coupled networks is greatly affected by the average degree which usually increases with the average degree. This method could be further generalized by taking into account the degree of next-nearest neighbors, which will sharply increase the connection probability. Compared to the other proposed methods that obtain synchronization optimal networks, our method uses only local structure information and can hold the structure properties of the original scale-free networks to some extent. Our method may present a useful way to manipulate the synchronizability of real-world scale-free networks.     

Two Novel Methods to Enhance Network Synchronizability

In this paper, we propose two methods to enhance the synchronizability of a class of complex networks which do not hold the positive correlation between betweenness centrality （BC） and degree of a node, and observe other topology characteristics of the network affected by the methods. Numerical simulations show that both methods can effectively enhance the synchronizability of this kind of networks. Furthermore, we show that the maximal BC of all edges is an important factor to affect the network synchronizability, although it is not the unique factor.     

Concentric circular arrays of Josephson junctions as dressed models of granular superconductors

In this paper networks that optimize a combined measure of local and global synchronizability are evolved. It is shown that for low coupling improvements in the local synchronizability dominate network evolution. This leads to an expressed grouping of elements with similar native frequency into cliques, allowing for an early onset of synchronization, but rendering full synchronization hard to achieve. In contrast, for large coupling the network evolution is governed by improvements towards full synchronization, preventing any expressed community structure. Such networks exhibit strong coupling between dissimilar oscillators. Albeit a rapid transition to full synchronization is achieved, the onset of synchronization is delayed in comparison to the first type of networks. The paper illustrates that an early onset of synchronization (which relates to clustering) and global synchronization are conflicting demands on network topology.     

加权方式对网络同步能力的影响

针对真实网络中权值与端点度的相关特性,提出了一种与始点和终点的度 都相关的非对称加权方式.在不同的网络结构下研究加权方式对同步能力的影响. 研究发现网络异质性越强时,通过调节网络权值改变网络同步能力的效果越显著, 而网络越匀质时,调节权值的方式改变网络同步能力的效果越不明显. 仿真实验显示无论在小世界网络还是无标度网络中,网络都是在节点的输入强度为1处获得最优的同步能力.     

Synchronization in power-law networks

We consider realistic power-law graphs, for which the power-law holds only for a certain range of degrees. We show that synchronizability of such networks depends on the expected average and expected maximum degree. In particular, we find that networks with realistic power-law graphs are less synchronizable than classical random networks. Finally, we consider hybrid graphs, which consist of two parts: a global graph and a local graph. We show that hybrid networks, for which the number of global edges is proportional to the number of total edges, almost surely synchronize.     

Optimization of network structure to random failures

Network's resilience to the malfunction of its components has been of great concern. The goal of this work is to determine the network design guidelines, which maximizes the network efficiency while keeping the cost of the network (that is the average connectivity) constant. With a global optimization method, memory tabu search (MTS), we get the optimal network structure with the approximately best efficiency. We analyze the statistical characters of the network and find that a network with a small quantity of hub nodes, high degree of clustering may be much more resilient to perturbations than a random network and the optimal network is one kind of highly heterogeneous networks. The results strongly suggest that networks with higher efficiency are more robust to random failures. In addition, we propose a simple model to describe the statistical properties of the optimal network and investigate the synchronizability of this model.     

Cluster synchronization in networks of distinct groups of maps

In this paper, we study cluster synchronization in general bi-directed networks of nonidentical clusters, where all nodes in the same cluster share an identical map. Based on the transverse stability analysis, we present sufficient conditions for local cluster synchronization of networks. The conditions are composed of two factors: the common inter-cluster coupling, which ensures the existence of an invariant cluster synchronization manifold, and communication between each pair of nodes in the same cluster, which is necessary for chaos synchronization. Consequently, we propose a quantity to measure the cluster synchronizability for a network with respect to the given clusters via a function of the eigenvalues of the Laplacian corresponding to the generalized eigenspace transverse to the cluster synchronization manifold. Then, we discuss the clustering synchronous dynamics and cluster synchronizability for four artificial network models: (i) p-nearest-neighborhood graph; (ii) random clustering graph; (iii) bipartite random graph; (iv) degree-preferred growing clustering network. From these network models, we are to reveal how the intra-cluster and inter-cluster links affect the cluster synchronizability. By numerical examples, we find that for the first model, the cluster synchronizability regularly enhances with the increase of p, yet for the other three models, when the ratio of intra-cluster links and the inter-cluster links reaches certain quantity, the clustering synchronizability reaches maximal.     

多层单向耦合星形网络的特征值谱及同步能力分析

随着复杂网络同步的进一步发展,对复杂网络的研究重点由单层网络转向更加接近实际网络的多层有向网络.本文分别严格推导出三层、多层的单向耦合星形网络的特征值谱,并分析了耦合强度、节点数、层数对网络同步能力的影响,重点分析了层数和层间中心节点之间的耦合强度对多层单向耦合星形网络同步能力的影响,得出了层数对多层网络同步能力的影响至关重要.当同步域无界时,网络的同步能力与耦合强度、层数有关,同步能力随其增大而增强;当同步域有界时,对于叶子节点向中心节点耦合的多层星形网络,当层内耦合强度较弱时,层内耦合强度的增大会使同步能力增强,而层间叶子节点之间的耦合强度、层数的增大反而会使同步能力减弱;当层间中心节点之间的耦合强度较弱时,层间中心节点之间的耦合强度、层数的增大会使同步能力增强,层内耦合强度、层间叶子节点之间的耦合强度的增大反而会使同步能力减弱.对于中心节点向叶子节点耦合的多层星形网络,层间叶子节点之间的耦合强度、层数的增大会使同步能力增强,层内耦合强度、节点数、层间中心节点之间的耦合强度的增大反而会使同步能力减弱.     

Synchronizability of dynamical scale-free networks subject to random errors

In this paper the robustness of network synchronizability against random deletion of nodes, i.e. errors, in dynamical scale-free networks was studied. To this end, two measures of network synchronizability, namely, the eigenratio of the Laplacian and the order parameter quantifying the degree of phase synchrony were adopted, and the synchronizability robustness on preferential attachment scale-free graphs was investigated. The findings revealed that as the network size decreases, the robustness of its synchronizability against random removal of nodes declines, i.e. the more the number of randomly removed nodes from the network, the worse its synchronizability. We also showed that this dependence of the synchronizability on the network size is different with that in the growing scale-free networks. The profile of a number of network properties such as clustering coefficient, efficiency, assortativity, and eccentricity, as a function of the network size was investigated in these two cases, growing scale-free networks and those with randomly removed nodes. The results showed that these processes are also different in terms of these metrics.     

From unweighted to weighted networks with local information

In this paper, we analyze an evolving model with local information which can generate a class of networks by choosing different values of the parameter p. The model introduced exhibits the transition from unweighted networks to weighted networks because the distribution of the edge weight can be widely tuned. With the increase in the local information, the degree correlation of the network transforms from assortative to disassortative. We also study the distribution of the degree, strength and edge weight, which all show crossover between exponential and scale-free. Finally, an application of the proposed model to the study of the synchronization is considered. It is concluded that the synchronizability is enhanced when the heterogeneity of the edge weight is reduced.     

Bootstrapping Topological Properties and Systemic Risk of Complex Networks Using the Fitness Model

In this paper we present a novel method to reconstruct global topological properties of a complex network starting from limited information. We assume to know for all the nodes a non-topological quantity that we interpret as fitness. In contrast, we assume to know the degree, i.e. the number of connections, only for a subset of the nodes in the network. We then use a fitness model, calibrated on the subset of nodes for which degrees are known, in order to generate ensembles of networks. Here, we focus on topological properties that are relevant for processes of contagion and distress propagation in networks, i.e. network density and k-core structure, and we study how well these properties can be estimated as a function of the size of the subset of nodes utilized for the calibration. Finally, we also study how well the resilience to distress propagation in the network can be estimated using our method. We perform a first test on ensembles of synthetic networks generated with the Exponential Random Graph model, which allows to apply common tools from statistical mechanics. We then perform a second test on empirical networks taken from economic and financial contexts. In both cases, we find that a subset as small as 10 % of nodes can be enough to estimate the properties of the network along with its resilience with an error of 5 %.     

Heterogeneity in oscillator networks: are smaller worlds easier to synchronize?

Small-world and scale-free networks are known to be more easily synchronized than regular lattices, which is usually attributed to the smaller network distance between oscillators. Surprisingly, we find that networks with a homogeneous distribution of connectivity are more synchronizable than heterogeneous ones, even though the average network distance is larger. We present numerical computations and analytical estimates on synchronizability of the network in terms of its heterogeneity parameters. Our results suggest that some degree of homogeneity is expected in naturally evolved structures, such as neural networks, where synchronizability is desirable.     

Synchronization in complex dynamical networks with nonsymmetric coupling

Based on the work of Nishikawa and Motter, who have extended the well-known master stability framework to include non-diagonalizable cases, we develop another extension of the master stability framework to obtain criteria for global synchronization. Several criteria for global synchronization are provided which generalize some previous results. The Jordan canonical transformation method is used in stead of the matrix diagonalization method. Especially, we show clearly that, the synchronizability of a dynamical network with nonsymmetric coupling is not always characterized by its second-largest eigenvalue, even though all the eigenvalues of the nonsymmetric coupling matrix are real. Furthermore, the effects of the asymmetry of coupling on synchronizability of networks with different structures are analyzed. Numerical simulations are also done to illustrate and verify the theoretical results on networks in which each node is a dynamical limit cycle oscillator consisting of a two-cell cellular neural network.     

最优的Newman-Watts网络与遍历网络的同步

在Hindmarsh-Rose神经元动力系统中研究了Newman-Watts （NW）网络的同步,给出了一些最优同步网络的拓扑结构. 数值结果表明：NW网络的同步能力主要由耦合点在耦合空间的分布决定,耦合点分布均匀的NW网络一般具有较强的同步能力;在给定连边数的情况下,可能存在多个结构不同的最优同步网络,最优同步网络具有最强的同步能力、均匀的度分布和较好的对称性,但是其对称性不一定是最好的. 最优同步网络一般是非规则网络,但在少数情况下,规则网络也有可能是最优同步网络. 提出了一种新的网络——遍历网络,该网络具有最优同步网络的特点和很强的同步能力. 关键词： Newman-Watts网络 对称度 耦合空间 同步