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     

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     

Traffic dynamics in scale-free networks with tunable strength of community structure

In this paper we systematically investigate the impact of community structure on traffic dynamics in scale-free networks based on local routing strategy. A growth model is introduced to construct scale-free networks with tunable strength of community structure, and a packet routing strategy with a parameter α is used to deal with the navigation and transportation of packets simultaneously. Simulations show that the maximal network capacity stands at α=−1 in the case of identical vertex capacity and monotonously decreases with the strength of community structure which suggests that the networks with fuzzy community structure (i.e., community strength is weak) are more efficient in delivering packets than those with pronounced community structure. To explain these results, the distribution of packets of each vertex is carefully studied. Our results indicate that the moderate strength of community structure is more convenient for the information transfer of real complex systems.     

Structural preferential attachment: network organization beyond the link

We introduce a mechanism which models the emergence of the universal properties of complex networks, such as scale independence, modularity and self-similarity, and unifies them under a scale-free organization beyond the link. This brings a new perspective on network organization where communities, instead of links, are the fundamental building blocks of complex systems. We show how our simple model can reproduce social and information networks by predicting their community structure and more importantly, how their nodes or communities are interconnected, often in a self-similar manner.     

Synchronization speed of identical oscillators on community networks

Synchronizability of complex oscillators networks has attracted much research interest in recent years. In contrast, in this paper we investigate numerically the synchronization speed, rather than the synchronizability or synchronization stability, of identical oscillators on complex networks with communities. A new weighted community network model is employed here, in which the community strength could be tunable by one parameter δ. The results showed that the synchronization speed of identical oscillators on community networks could reach a maximal value when δ is around 0.1. We argue that this is induced by the competition between the community partition and the scale-free property of the networks. Moreover, we have given the corresponding analysis through the second least eigenvalue λ₂ of the Laplacian matrix of the network which supports the previous result that the synchronization speed is determined by the value of λ₂.     

Community structure of complex software systems: Analysis and applications

Due to notable discoveries in the fast evolving field of complex networks, recent research in software engineering has also focused on representing software systems with networks. Previous work has observed that these networks follow scale-free degree distributions and reveal small-world phenomena, while we here explore another property commonly found in different complex networks, i.e. community structure. We adopt class dependency networks, where nodes represent software classes and edges represent dependencies among them, and show that these networks reveal a significant community structure, characterized by similar properties as observed in other complex networks. However, although intuitive and anticipated by different phenomena, identified communities do not exactly correspond to software packages. We empirically confirm our observations on several networks constructed from Java and various third party libraries, and propose different applications of community detection to software engineering.     

A preliminary investigation on the topology of Chinese and climate networks

Complex networks have been studied across many fields of science in recent years. In this paper, we give a brief introduction of networks, then follow the original works by Tsonis et al (2004, 2006) starting with data of the surface temperature from 160 Chinese weather observations to investigate the topology of Chinese climate networks. Results show that the Chinese climate network exhibits a characteristic of regular, almost fully connected networks, which means that most nodes in this case have the same number of links, and so-called super nodes with a very large number of links do not exist there. In other words, though former results show that nodes in the extratropical region provide a property of scale-free networks, they still have other different local fine structures inside. We also detect the community of the Chinese climate network by using a Bayesian technique; the effective number of communities of the Chinese climate network is about four in this network. More importantly, this technique approaches results in divisions which have connections with physics and dynamics; the division into communities may highlight the aspects of the dynamics of climate variability.     

Detecting community structure in complex networks via node similarity

The detection of the community structure in networks is beneficial to understand the network structure and to analyze the network properties. Based on node similarity, a fast and efficient method for detecting community structure is proposed, which discovers the community structure by iteratively incorporating the community containing a node with the communities that contain the nodes with maximum similarity to this node to form a new community. The presented method has low computational complexity because of requiring only the local information of the network, and it does not need any prior knowledge about the communities and its detection results are robust on the selection of the initial node. Some real-world and computer-generated networks are used to evaluate the performance of the presented method. The simulation results demonstrate that this method is efficient to detect community structure in complex networks, and the ZLZ metrics used in the proposed method is the most suitable one among local indices in community detection.     

Detecting communities in clustered networks based on group action on set

In this paper, we propose a well targeted algorithm (GAS algorithm) for detecting communities in high clustered networks by presenting group action technology on community division. During the processing of this algorithm, the underlying community structure of a clustered network emerges simultaneously as the corresponding partition of orbits by the permutation groups acting on the node set are achieved. As the derivation of the orbit partition, an algebraic structure r-cycle can be considered as the origin of the community. To be a priori estimation for the community structure of the algorithm, the community separability is introduced to indicate whether a network has distinct community structure. By executing the algorithm on several typical networks and the LFR benchmark, it shows that this GAS algorithm can detect communities accurately and effectively in high clustered networks. Furthermore, we compare the GAS algorithm and the clique percolation algorithm on the LFR benchmark. It is shown that the GAS algorithm is more accurate at detecting non-overlapping communities in clustered networks. It is suggested that algebraic techniques can uncover fresh light on detecting communities in complex networks.     

Robustness measurement of scale-free networks based on motif entropy

As a classical complex network model, scale-free network is widely used and studied. And motifs, as a high-order subgraph structure, frequently appear in scale-free networks, and have a great influence on the structural integrity, functional integrity and dynamics of the networks. In order to overcome the shortcomings in the existing work on the robustness of complex networks, only nodes or edges are considered, while the defects of high-order structure in the network are ignored. From the perspective of network motif, we propose an entropy of node degree distribution based on motif to measure the robustness of scale-free networks under random attacks. The effectiveness and superiority of our method are verified and analyzed in the BA scale-free networks.     

Eigenvectors of network complement reveal community structure more accurately

We propose a general spectral method to find communities of a network based on network complement and anti-community concepts. Analytical and numerical results show that the eigenspace of matrices corresponding to a network complement reveals the community structure of a network more accurately than the eigenspace of matrices corresponding to the network itself. It is shown that the Laplacian eigenspace is the best candidate for spectral community detection especially in networks with a heterogeneous community structure. The method is applied to some computer-generated and real-world networks with known community structures.     

Concentric characterization and classification of complex network nodes: Application to an institutional collaboration network

Differently from theoretical scale-free networks, most real networks present multi-scale behavior, with nodes structured in different types of functional groups and communities. While the majority of approaches for classification of nodes in a complex network has relied on local measurements of the topology/connectivity around each node, valuable information about node functionality can be obtained by concentric (or hierarchical) measurements. This paper extends previous methodologies based on concentric measurements, by studying the possibility of using agglomerative clustering methods, in order to obtain a set of functional groups of nodes, considering particular institutional collaboration network nodes, including various known communities (departments of the University of São Paulo). Among the interesting obtained findings, we emphasize the scale-free nature of the network obtained, as well as identification of different patterns of authorship emerging from different areas (e.g. human and exact sciences). Another interesting result concerns the relatively uniform distribution of hubs along concentric levels, contrariwise to the non-uniform pattern found in theoretical scale-free networks such as the BA model.     

有倾向性重连产生的反匹配网络

在具有网络结构的系统中度关联属性对于动力学行为具有重要的影响, 所以产生适当度关联网络的方法对于大量网络系统的研究具有重要的作用. 尽管产生正匹配网络的方法已经得到很好的验证, 但是产生反匹配网络的方法还没有被系统的讨论过. 重新连接网络中的边是产生度关联网络的一个常用方法. 这里我们研究使用重连方法产生反匹配无标度网络的有效性. 我们的研究表明, 有倾向的重连可以增强网络的反匹配属性. 但是有倾向重连不能使皮尔森度相关系数下降到-1, 而是存在一个依赖于网络参数的最小值. 我们研究了网络的主要参数对于网络度相关系数的影响, 包括网络尺寸, 网络的连接密度和网络节点的度差异程度. 研究表明在网络尺寸大的情况下和节点度差异性强的情况下, 重连的效果较差. 我们研究了真实Internet网络, 发现模型产生的网络经过重连不能达到真实网络的度关联系数.     

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.     

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.     

一种强容侵能力的无线传感器网络无标度拓扑模型研究

针对无线传感器网络无标度拓扑容侵能力差的问题,本文借助节点批量到达的Poisson网络模型,提出了一种具有容侵优化特性的无标度拓扑模型,并在构建拓扑时引入剩余能量调节因子和节点度调节因子,得到了一种幂率指数可以在(1,+∞)调节的无标度拓扑结构,并通过网络结构熵优化幂率指数,得出了具有强容侵特性的幂律指数值.实验结果表明:新的拓扑保持了无标度网络的强容错性,增强了无标度网络的容侵性,并具有较好的节能优势.     

The robustness of sparse network under limited attack capacity

The paper studies the robustness of the network in terms of the network structure. We define a strongly dominated relation between nodes and then we use the relation to merge the network. Based on that, we design a dominated clustering algorithm aiming at finding the critical nodes in the network. Furthermore, this merging process is lossless which means the original structure of the network is kept. In order to realize the visulization of the network, we also apply the lossy consolidation to the network based on detection of the community structures. Simulation results show that compared with six existed centrality algorithms, our algorithm performs better when the attack capacity is limited. The simulations also illustrate our algorithm does better in assortative scale-free networks.     

A thermal flux-diffusing model for complex networks and its applications in community structure detection

We introduce a thermal flux-diffusing model for complex networks. Based on this model, we propose a physical method to detect the communities in the complex networks. The method allows us to obtain the temperature distribution of nodes in time that scales linearly with the network size. Then, the local community enclosing a given node can be easily detected for the reason that the dense connections in the local communities lead to the temperatures of nodes in the same community being close to each other. The community structure of a network can be recursively detected by randomly choosing the nodes outside the detected local communities. In the experiments, we apply our method to a set of benchmarking networks with known pre-determined community structures. The experiment results show that our method has higher accuracy and precision than most existing globe methods and is better than the other existing local methods in the selection of the initial node. Finally, several real-world networks are investigated.     

Modelling one-dimensional driven diffusive systems by the Zero-Range Process

In order to characterize networks in the scale-free network class we study the frequency of cycles of length h that indicate the ordering of network structure and the multiplicity of paths connecting two nodes. In particular we focus on the scaling of the number of cycles with the system size in off-equilibrium scale-free networks. We observe that each off-equilibrium network model is characterized by a particular scaling in general not equal to the scaling found in equilibrium scale-free networks. We claim that this anomalous scaling can occur in real systems and we report the case of the Internet at the Autonomous System Level.Received: 15 January 2004, Published online: 14 May 2004PACS: 89.75.-k Complex systems - 89.75.Hc Networks and genealogical trees     

Finding communities in directed networks by PageRank random walk induced network embedding

Community structure has been found to exist ubiquitously in many different kinds of real world complex networks. Most of the previous literature ignores edge directions and applies methods designed for community finding in undirected networks to find communities. Here, we address the problem of finding communities in directed networks. Our proposed method uses PageRank random walk induced network embedding to transform a directed network into an undirected one, where the information on edge directions is effectively incorporated into the edge weights. Starting from this new undirected weighted network, previously developed methods for undirected network community finding can be used without any modification. Moreover, our method improves on recent work in terms of community definition and meaning. We provide two simulated examples, a real social network and different sets of power law benchmark networks, to illustrate how our method can correctly detect communities in directed networks.