Synchronizability on complex networks via pinning control

It is proved that the maximum eigenvalue sequence of the principal submatrices of coupling matrix is decreasing. The method of calculating the number of pinning nodes is given based on this theory. The findings reveal the relationship between the decreasing speed of maximum eigenvalue sequence of the principal submatrices for coupling matrix and the synchronizability on complex networks via pinning control. We discuss the synchronizability on some networks, such as scale-free networks and small-world networks. Numerical simulations show that different pinning strategies have different pinning synchronizability on the same complex network, and the consistence between the synchronizability with pinning control and one without pinning control in various complex networks.     

Enhancing the network synchronizability

The structural and dynamical properties, particularly the small-world effect and scale-free feature, of complex networks have attracted tremendous interest and attention in recent years. This article offers a brief review of one focal issue concerning the structural and dynamical behaviors of complex network synchronization. In the presentation, the notions of synchronization of dynamical systems on networks, stability of dynamical networks, and relationships between network structure and synchronizability, will be first introduced. Then, various technical methods for enhancing the network synchronizability will be discussed, which are roughly divided into two classes: Structural Modification and Coupling-Pattern Regulation, where the former includes three typical methods—dividing hub nodes, shortening average distances, and deleting overload edges, while the latter mainly is a method of strengthening the hub-nodes’ influence on the network.      

Synchronizability of small-world networks generated from ring networks with equal-distance edge additions

This paper investigates the impact of edge-adding number m and edge-adding distance d on both synchronizability and average path length of NW small-world networks generated from ring networks via random edge-adding. It is found that the synchronizability of the network as a function of the distance d is fluctuant and there exist some d that have almost no impact on the synchronizability and may only scarcely shorten the average path length of the network. Numerical simulations on a network of Lorenz oscillators confirm the above results. This phenomenon shows that the contributions of randomly added edges to both the synchronizability and the average path length are not uniform nor monotone in building an NW small-world network with equal-distance edge additions, implying that only if appropriately adding edges when building up the NW small-word network can help enhance the synchronizability and/or reduce the average path length of the resultant network. Finally, it is shown that this NW small-world network has worse synchronizability and longer average path length, when compared with the conventional NW small-world network, with random-distance edge additions. This may be due to the fact that with equal-distance edge additions, there is only one shortcut distance for better information exchange among nodes and for shortening the average path length, while with random-distance edge additions, there exist many different distances for doing so.     

Self-similarity, small-world, scale-free scaling, disassortativity, and robustness in hierarchical lattices

In this paper, firstly, we study analytically the topological features of a family of hierarchical lattices (HLs) from the view point of complex networks. We derive some basic properties of HLs controlled by a parameter q: scale-free degree distribution with exponent γ=2+ln 2/(ln q), null clustering coefficient, power-law behavior of grid coefficient, exponential growth of average path length (non-small-world), fractal scaling with dimension d_B=ln (2q)/(ln 2), and disassortativity. Our results show that scale-free networks are not always small-world, and support the conjecture that self-similar scale-free networks are not assortative. Secondly, we define a deterministic family of graphs called small-world hierarchical lattices (SWHLs). Our construction preserves the structure of hierarchical lattices, including its degree distribution, fractal architecture, clustering coefficient, while the small-world phenomenon arises. Finally, the dynamical processes of intentional attacks and collective synchronization are studied and the comparisons between HLs and Barabási-Albert (BA) networks as well as SWHLs are shown. We find that the self-similar property of HLs and SWHLs significantly increases the robustness of such networks against targeted damage on hubs, as compared to the very vulnerable non fractal BA networks, and that HLs have poorer synchronizability than their counterparts SWHLs and BA networks. We show that degree distribution of scale-free networks does not suffice to characterize their synchronizability, and that networks with smaller average path length are not always easier to synchronize.     

Effects of Some Topological Ingredients on the Evolutionary Ultimatum Game

This study aims at figuring out the crucial topological ingredients which affect the outcomes of the ultimatum game located on different networks, encompassing the regular network, the random network, the small-world network, and the scale-free network. With the aid of random interchanging algorithm, we investigate the relations between the outcomes of the ultimatum game and some topological ingredients, including the average range, the clustering coefficient and the heterogeneity, and so forth. It is found that for the regular, random and small-work networks, the average range and the clustering coefficient have evident impacts on the ultimatum game, while for the scale-free network, the original degree heterogeneity and the underlying rich-club characterizations are the mainly important topological ingredients that influence the outcomes of ultimatum game substantially.     

Synchronizability of Highly Clustered Scale-Free Networks

We consider the effect of clustering coefficient on the synchronizability of coupled oscillators located on scale-free networks. The analytic result for the value of clustering coefficient aiming at a highly clustered scale-free network model, the Holme-Kim model is obtained, and the relationship between network synchronizability and clustering coefficient is reported. The simulation results strongly suggest that the more clustered the network, the poorer the synchronizability.     

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.     

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 small-world systems

We quantify the dynamical implications of the small-world phenomenon by considering the generic synchronization of oscillator networks of arbitrary topology. The linear stability of the synchronous state is linked to an algebraic condition of the Laplacian matrix of the network. Through numerics and analysis, we show how the addition of random shortcuts translates into improved network synchronizability. Applied to networks of low redundancy, the small-world route produces synchronizability more efficiently than standard deterministic graphs, purely random graphs, and ideal constructive schemes. However, the small-world property does not guarantee synchronizability: the synchronization threshold lies within the boundaries, but linked to the end of the small-world region.     

Deterministic scale-free small-world networks of arbitrary order

In many real-life networks, both the scale-free distribution of degree and small-world behavior are important features. There are many random or deterministic models of networks to simulate these features separately. However, there are few models that combine the scale-free effect and small-world behavior, especially in terms of deterministic versions. What is more, all the existing deterministic algorithms running in the iterative mode generate networks with only several discrete numbers of nodes. This contradicts the purpose of creating a deterministic network model on which we can simulate some dynamical processes as widely as possible. According to these facts, this paper proposes a deterministic network generation algorithm, which can not only generate deterministic networks following a scale-free distribution of degree and small-world behavior, but also produce networks with arbitrary number of nodes. Our scheme is based on a complete binary tree, and each newly generated leaf node is further linked to its full brother and one of its direct ancestors. Analytical computation and simulation results show that the average degree of such a proposed network is less than 5, the average clustering coefficient is high (larger than 0.5, even for a network of size 2 million) and the average shortest path length increases much more slowly than logarithmic growth for the majority of small-world network models.     

Synchronization in complex clustered networks

Synchronization in complex networks has been an active area of research in recent years. While much effort has been devoted to networks with the small-world and scale-free topology, structurally they are often assumed to have a single, densely connected component. Recently it has also become apparent that many networks in social, biological, and technological systems are clustered, as characterized by a number (or a hierarchy) of sparsely linked clusters, each with dense and complex internal connections. Synchronization is fundamental to the dynamics and functions of complex clustered networks, but this problem has just begun to be addressed. This paper reviews some progress in this direction by focusing on the interplay between the clustered topology and network synchronizability. In particular, there are two parameters characterizing a clustered network: the intra-cluster and the inter-cluster link density. Our goal is to clarify the roles of these parameters in shaping network synchronizability. By using theoretical analysis and direct numerical simulations of oscillator networks, it is demonstrated that clustered networks with random inter-cluster links are more synchronizable, and synchronization can be optimized when inter-cluster and intra-cluster links match. The latter result has one counterintuitive implication: more links, if placed improperly, can actually lead to destruction of synchronization, even though such links tend to decrease the average network distance. It is hoped that this review will help attract attention to the fundamental problem of clustered structures/synchronization in network science.      

Monomer-dimer model on a scale-free small-world network

The explicit determination of the number of monomer-dimer arrangements on a network is a theoretical challenge, and exact solutions to monomer-dimer problem are available only for few limiting graphs with a single monomer on the boundary, e.g., rectangular lattice and quartic lattice; however, analytical research (even numerical result) for monomer-dimer problem on scale-free small-world networks is still missing despite the fact that a vast variety of real systems display simultaneously scale-free and small-world structures. In this paper, we address the monomer-dimer problem defined on a scale-free small-world network and obtain the exact formula for the number of all possible monomer-dimer arrangements on the network, based on which we also determine the asymptotic growth constant of the number of monomer-dimer arrangements in the network. We show that the obtained asymptotic growth constant is much less than its counterparts corresponding to two-dimensional lattice and Sierpinski fractal having the same average degree as the studied network, which indicates from another aspect that scale-free networks have a fundamentally distinct architecture as opposed to regular lattices and fractals without power-law behavior.     

Synchronizability of chaotic logistic maps in delayed complex networks

We study a network of coupled logistic maps whose interactions occur with a certain distribution of delay times. The local dynamics is chaotic in the absence of coupling and thus the network is a paradigm of a complex system. There are two regimes of synchronization, depending on the distribution of delays: when the delays are sufficiently heterogeneous the network synchronizes on a steady-state (that is unstable for the uncoupled maps); when the delays are homogeneous, it synchronizes in a time-dependent state (that is either periodic or chaotic). Using two global indicators we quantify the synchronizability on the two regimes, focusing on the roles of the network connectivity and the topology. The connectivity is measured in terms of the average number of links per node, and we consider various topologies (scale-free, small-world, star, and nearest-neighbor with and without a central hub). With weak connectivity and weak coupling strength, the network displays an irregular oscillatory dynamics that is largely independent of the topology and of the delay distribution. With heterogeneous delays, we find a threshold connectivity level below which the network does not synchronize, regardless of the network size. This minimum average number of neighbors seems to be independent of the delay distribution. We also analyze the effect of self-feedback loops and find that they have an impact on the synchronizability of small networks with large coupling strengths. The influence of feedback, enhancing or degrading synchronization, depends on the topology and on the distribution of delays.     

Scaling of mean first-passage time as efficiency measure of nodes sending information on scale-free Koch networks

Random walks on complex networks, especially scale-free networks, have attracted considerable interest in the past few years. A lot of previous work showed that the average receiving time (ART), i.e., the average of mean first-passage time (MFPT) for random walks to a given hub node (node with maximum degree) averaged over all starting points in scale-free small-world networks exhibits a sublinear or linear dependence on network order N (number of nodes), which indicates that hub nodes are very efficient in receiving information if one looks upon the random walker as an information messenger. Thus far, the efficiency of a hub node sending information on scale-free small-world networks has not been addressed yet. In this paper, we study random walks on the class of Koch networks with scale-free behavior and small-world effect. We derive some basic properties for random walks on the Koch network family, based on which we calculate analytically the average sending time (AST) defined as the average of MFPTs from a hub node to all other nodes, excluding the hub itself. The obtained closed-form expression displays that in large networks the AST grows with network order as N ln N, which is larger than the linear scaling of ART to the hub from other nodes. On the other hand, we also address the case with the information sender distributed uniformly among the Koch networks, and derive analytically the global mean first-passage time, namely, the average of MFPTs between all couples of nodes, the leading scaling of which is identical to that of AST. From the obtained results, we present that although hub nodes are more efficient for receiving information than other nodes, they display a qualitatively similar speed for sending information as non-hub nodes. Moreover, we show that that AST from a starting point (sender) to all possible targets is not sensitively affected by the sender’s location. The present findings are helpful for better understanding random walks performed on scale-free small-world networks.     

小世界网络与无标度网络的社区结构研究

模块性(modularity)是度量网络社区结构(community structure)的主要参数.探讨了Watts和Strogatz的小世界网络(简称W-S模型)以及Barabàsi 等的B-A无标度网络(简称B-A模型)两类典型复杂网络模块性特点.结果显示，网络模块性受到网络连接稀疏的影响，W-S模型具有显著的社区结构，而B-A模型的社区结构特征不明显.因此，应用中应该分别讨论网络的小世界现象和无标度特性.社区结构不同于小世界现象和无标度特性，并可以利用模块性区别网络类型，因此网络复杂性指标应该包括 关键词： 模块性 社区结构 小世界网络 无标度网络     

Synchronization in evolving snowdrift game model

The interaction between the evolution of the game and the underlying network structure with evolving snowdrift game model is investigated. The constructed network follows a power-law degree distribution typically showing scale-free feature. The topological features of average path length, clustering coefficient, degree-degree correlations and the dynamical feature of synchronizability are studied. The synchronizability of the constructed networks changes by the interaction. It will converge to a certain value when sufficient new nodes are added. It is found that initial payoffs of nodes greatly affect the synchronizability. When initial payoffs for players are equal, low common initial payoffs may lead to more heterogeneity of the network and good synchronizability. When initial payoffs follow certain distributions, better synchronizability is obtained compared to equal initial payoff. The result is also true for phase synchronization of nonidentical oscillators.     

Social network structures and bank runs

This paper investigates the impact of social network structures of depositors on bankruns. The analyzed network structures include random networks, small-world networks andscale-free networks. Simulation results show that the probability of bank run occurrencein random networks is larger than that in small-world networks, but the probability ofbank run occurrence in scale-free networks drops from the highest to the lowest among thethree types of network structures with the increase of the proportion of impatientdepositors. The average degree of depositor networks has a significant impact on bankruns, but this impact is related to the proportion of impatient depositors and theconfidence levels of depositors in banks.     

一类高聚类系数的加权无标度网络及其同步能力分析

针对真实世界中大规模网络都具有明显聚类效应的特点, 提出一类具有高聚类系数的加权无标度网络演化模型, 该模型同时考虑了优先连接、三角结构、随机连接和社团结构等四种演化机制. 在模型演化规则中, 以概率p增加单个节点, 以概率1–p增加一个社团. 与以往研究的不同在于新边的建立, 以概率φ在旧节点之间进行三角连接, 以概率1–φ进行随机连接. 仿真分析表明, 所提出的网络度、强度和权值分布都是服从幂律分布的形式, 且具有高聚类系数的特性, 聚类系数的提高与社团结构和随机连接机制有直接的关系. 最后通过数值仿真分析了网络演化机制对同步动态特性的影响, 数值仿真结果表明, 网络的平均聚类系数越小, 网络的同步能力越强. 关键词： 无标度网络 加权网络 聚类系数 同步能力     

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.     

Developmental time windows for spatial growth generate multiple-cluster small-world networks

Many networks extent in space, may it be metric (e.g. geographic) or non-metric (ordinal). Spatial network growth, which depends on the distance between nodes, can generate a wide range of topologies from small-world to linear scale-free networks. However, networks often lacked multiple clusters or communities. Multiple clusters can be generated, however, if there are time windows during development. Time windows ensure that regions of the network develop connections at different points in time. This novel approach could generate small-world but not scale-free networks. The resulting topology depended critically on the overlap of time windows as well as on the position of pioneer nodes.