Surveying network community structure in the hidden metric space

Most real-world networks from various fields share a universal topological property as community structure. In this paper, we propose a node-similarity based mechanism to explore the formation of modular networks by applying the concept of hidden metric spaces of complex networks. It is demonstrated that network community structure could be formed according to node similarity in the underlying hidden metric space. To clarify this, we generate a set of observed networks using a typical kind of hidden metric space model. By detecting and analyzing corresponding communities both in the observed network and the hidden space, we show that the values of the fitness are rather close, and the assignments of nodes for these two kinds of community structures detected based on the fitness parameter are extremely matching ones. Furthermore, our research also shows that networks with strong clustering tend to display prominent community structures with large values of network modularity and fitness.     

复杂网络中社团结构发现的多分辨率密度模块度

现实中的许多复杂网络呈现出明显的模块性或社团性.模块度是衡量社团结构划分优劣的效益函数, 它也通常被用作社团结构探测的目标函数,但最为广泛使用的Newman-Girvan模块度却存在着分辨率限制问题,多分辨率模块度也不能克服误合并社团和误分裂社团同时存在的缺陷. 本文在网络密度的基础上提出了多分辨率的密度模块度函数, 通过实验和分析证实了该函数能够使社团结构的误划分率显著降低, 而且能够体现出网络社团结构是一个有机整体,不是各个社团的简单相加.     

Refinement of the community detection performance by weighted relationship coupling

The complexity of many community detection algorithms is usually an exponential function with the scale which hard to uncover community structure with high speed. Inspired by the ideas of the famous modularity optimization, in this paper, we proposed a proper weighting scheme utilizing a novel k-strength relationship which naturally represents the coupling distance between two nodes. Community structure detection using a generalized weighted modularity measure is refined based on the weighted k-strength matrix. We apply our algorithm on both the famous benchmark network and the real networks. Theoretical analysis and experiments show that the weighted algorithm can uncover communities fast and accurately and can be easily extended to large-scale real networks.     

Synchronization and modularity in complex networks

We investigate the connection between the dynamics of synchronization and the modularity on complex networks. Simulating the Kuramoto's model in complex networks we determine patterns of meta-stability and calculate the modularity of the partition these patterns provide. The results indicate that the more stable the patterns are, the larger tends to be the modularity of the partition defined by them. This correlation works pretty well in homogeneous networks (all nodes have similar connectivity) but fails when networks contain hubs, mainly because the modularity is never improved where isolated nodes appear, whereas in the synchronization process the characteristic of hubs is to have a large stability when forming its own community.     

Response of scale-free networks with community structure to external stimuli

The response of scale-free networks with community structure to external stimuli is studied. By disturbing some nodes with different strategies, it is shown that the robustness of this kind of network can be enhanced due to the existence of communities in the networks. Some of the response patterns are found to coincide with topological communities. We show that such phenomena also occur in the cat brain network which is an example of a scale-free like network with community structure. Our results provide insights into the relationship between network topology and the functional organization in complex networks from another viewpoint.     

蛋白质相互作用网络特征的理论再现

无标度性、小世界性、功能模块结构及度负关联性是大量生物网络共同的特征. 为了理解生物网络无标度性、小世界性和度负关联性的形成机制, 研究者已经提出了各种各样基于复制和变异的网络增长模型. 在本文中,我们从生物学的角度通过引入偏爱小复制原则及变异和非均匀的异源二聚作用构建了一个简单的蛋白质相互作用网络演化模型.数值模拟结果表明,该演化模型几乎可以再现现在实测结果所公认的蛋白质相互作用网络的性质：无标度性、小世界性、度负关联性和功能模块结构. 我们的演化模型对理解蛋白质相互作用网络演化过程中的可能机制提供了一定的帮助. 关键词： 蛋白质相互作用网络 偏爱小 非均匀的异源二聚作用 功能模块结构     

Fuzzy modularity and fuzzy community structure in networks

To find the fuzzy community structure in a complex network, in which each node has a certain probability of belonging to a certain community, is a hard problem and not yet satisfactorily solved over the past years. In this paper, an extension of modularity, the fuzzy modularity is proposed, which can provide a measure of goodness for the fuzzy community structure in networks. The simulated annealing strategy is used to maximize the fuzzy modularity function, associating with an alternating iteration based on our previous work. The proposed algorithm can efficiently identify the probabilities of each node belonging to different communities with random initial fuzzy partition during the cooling process. An appropriate number of communities can be automatically determined without any prior knowledge about the community structure. The computational results on several artificial and real-world networks confirm the capability of the algorithm.     

相依网络上基于相连边的择优恢复算法

如何有效地应对和控制故障在相依网络上的级联扩散避免系统发生结构性破碎,对于相依网络抗毁性研究具有十分重要的理论价值和现实意义.最新的研究提出一种基于相依网络的恢复模型,该模型的基本思想是通过定义共同边界节点,在每轮恢复阶段找出符合条件的共同边界节点并以一定比例实施恢复.当前的做法是按照随机概率进行选择.这种方法虽然简单直观,却没有考虑现实世界中资源成本的有限性和择优恢复的必然性.为此,针对相依网络的恢复模型,本文利用共同边界节点在极大连通网络内外的连接边数计算边界节点的重要性,提出一种基于相连边的择优恢复算法(preferential recovery based on connectivity link,PRCL)算法.利用渗流理论的随机故障模型,通过ER随机网络和无标度网络构建的不同结构相依网络上的级联仿真结果表明,相比随机方法和度数优先以及局域影响力优先的恢复算法,PRCL算法具备恢复能力强、起效时间早且迭代步数少的优势,能够更有效、更及时地遏制故障在网络间的级联扩散,极大地提高了相依网络遭受随机故障时的恢复能力.     

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.     

A weight’s agglomerative method for detecting communities in weighted networks based on weight’s similarity

This paper proposes the new definition of the community structure of the weighted networks that groups of nodes in which the edge's weights distribute uniformly but at random between them. It can describe the steady connections between nodes or some similarity between nodes' functions effectively. In order to detect the community structure efficiently, a threshold coefficient κ to evaluate the equivalence of edges' weights and a new weighted modularity based on the weight's similarity are proposed. Then, constructing the weighted matrix and using the agglomerative mechanism, it presents a weight's agglomerative method based on optimizing the modularity to detect communities. For a network with n nodes, the algorithm can detect the community structure in time O(n²log₂ⁿ). Simulations on networks show that the algorithm has higher accuracy and precision than the existing techniques. Furthermore, with the change of κ the algorithm discovers a special hierarchical organization which can describe the various steady connections between nodes in groups.     

Efficient overlapping community detection in huge real-world networks

The detection of overlapping community structure in networks can give insight into the structures and functions of many complex systems. In this paper, we propose a simple but efficient overlapping community detection method for very large real-world networks. Taking a high-quality, non-overlapping partition generated by existing, efficient, non-overlapping community detection methods as input, our method identifies overlapping nodes between each pair of connected non-overlapping communities in turn. Through our analysis on modularity, we deduce that, to become an overlapping node without demolishing modularity, nodes should satisfy a specific condition presented in this paper. The proposed algorithm outputs high quality overlapping communities by efficiently identifying overlapping nodes that satisfy the above condition. Experiments on synthetic and real-world networks show that in most cases our method is better than other algorithms either in the quality of results or the computational performance. In some cases, our method is the only one that can produce overlapping communities in the very large real-world networks used in the experiments.     

Edge intensity-based community measurement in complex networks

Community is the dominant structure of complex networks. In recent years, community detection has become a heavily researched issue in network science, and many algorithms have been proposed to solve it. However, how to evaluate these algorithms and measure the strength of community structures is still an open problem. The modularity, as well as many of its variants, is widely used for this purpose, and maximizing such metrics is also a main approach to uncover communities, but this technique has a resolution limit problem in some cases, which means larger structures are favored over smaller ones. In this paper, we define the edge intensity to measure local density of network and propose an intensity-based measurement to support community evaluation; with an additional constraint the proposed measurement would also support multiresolution investigation of the networks. Experimental results on synthetic and real networks illustrate that the maximization of the new metric further reduces the resolution limit problem, and the maximization of the restricted intensity-based measurement provides multiresolution details of the investigated networks.     

Communities in Italian corporate networks

The community structure of two real-world financial networks, namely the board network and the ownership network of the firms of the Italian Stock Exchange, is analyzed by means of the maximum modularity approach. The main result is that both networks exhibit a strong community structure and, moreover, that the two structures overlap significantly. This is due to a number of reasons, including the existence of pyramidal groups and directors serving in several boards. Overall, this means that the “small world” of listed companies is actually split into well identifiable “continents” (i.e., the communities).     

Evaluation of network robustness using a node tearing algorithm

Until recently the study of failure and vulnerability in complex networks focused on the role of high degree nodes, and the relationship between their removal and network connectivity. Recent evidence suggested that in some network configurations, the removal of lower degree nodes can also cause network fragmentation. We present a disassembling algorithm that identifies nodes that are core to network connectivity. The algorithm is based on network tearing in which communities are defined and used to construct a hierarchical structure. Cut-nodes, which are located at the boundaries of the communities, are the key interest. Their importance in the overall network connectivity is characterized by their participation with neighbouring communities in each level of the hierarchy. We examine the impact of these cut-nodes by studying the change in size of the giant component, local and global efficiencies, and how the algorithm can be combined with other community detection methods to reveal the finer internal structure within a community.     

A local fuzzy method based on “p-strong” community for detecting communities in networks

In this paper,we propose a local fuzzy method based on the idea of "p-strong" community to detect the disjoint and overlapping communities in networks.In the method,a refined agglomeration rule is designed for agglomerating nodes into local communities,and the overlapping nodes are detected based on the idea of making each community strong.We propose a contribution coefficient b_v~(ci)to measure the contribution of an overlapping node to each of its belonging communities,and the fuzzy coefficients of the overlapping node can be obtained by normalizing the b_v~(ci) to all its belonging communities.The running time of our method is analyzed and varies linearly with network size.We investigate our method on the computergenerated networks and real networks.The testing results indicate that the accuracy of our method in detecting disjoint communities is higher than those of the existing local methods and our method is efficient for detecting the overlapping nodes with fuzzy coefficients.Furthermore,the local optimizing scheme used in our method allows us to partly solve the resolution problem of the global modularity.     

静息状态下脑功能连接的磁共振成像研究

静息状态下脑功能连接的磁共振成像研究近年来取得了迅猛发展. 通过对fMRI信号低频涨落成分的同步性分析,可以得到大脑静息态任意脑区的功能连接和多套网络系统,其中“默认网络”的发现可能为人脑固有网络的研究提供新的思路. 而静息态网络与解剖连接之间可能存在的对应,以及在神经精神疾病患者脑中性质和连接的异常改变,使其具有重要的研究和临床应用价值. 该文总结了静息状态功能磁共振成像的主要研究成果,对静息状态脑功能网络的发现和发展、研究方法、各网络及其特点以及在临床方面的应用进行简单的介绍和分析.     

The map equation

Many real-world networks are so large that we must simplify their structure before we can extract useful information about the systems they represent. As the tools for doing these simplifications proliferate within the network literature, researchers would benefit from some guidelines about which of the so-called community detection algorithms are most appropriate for the structures they are studying and the questions they are asking. Here we show that different methods highlight different aspects of a network's structure and that the the sort of information that we seek to extract about the system must guide us in our decision. For example, many community detection algorithms, including the popular modularity maximization approach, infer module assignments from an underlying model of the network formation process. However, we are not always as interested in how a system's network structure was formed, as we are in how a network's extant structure influences the system's behavior. To see how structure influences current behavior, we will recognize that links in a network induce movement across the network and result in system-wide interdependence. In doing so, we explicitly acknowledge that most networks carry flow. To highlight and simplify the network structure with respect to this flow, we use the map equation. We present an intuitive derivation of this flow-based and information-theoretic method and provide an interactive on-line application that anyone can use to explore the mechanics of the map equation. The differences between the map equation and the modularity maximization approach are not merely conceptual. Because the map equation attends to patterns of flow on the network and the modularity maximization approach does not, the two methods can yield dramatically different results for some network structures. To illustrate this and build our understanding of each method, we partition several sample networks. We also describe an algorithm and provide source code to efficiently decompose large weighted and directed networks based on the map equation.     

Principle of structural equation modeling for exploring functional interactivity within a putative network of interconnected brain areas

Functional neuroimaging first allowed researchers to describe the functional segregation of regionally activated areas during a variety of experimental tasks. More recently, functional integration studies have described how these functionally specialized areas, interact within a highly distributed neural network. When applied to the field of neurosciences, structural equation modeling (SEM) uses theoretical and/or empirical hypotheses to estimate the effects of an experimental task within a putative network. SEM represents a linear technique for multivariate analysis of neuroimaging data and has been developed to simultaneously examine ratios of multiple causality in an experimental design; the method attempts to explain a covariance structure within an anatomical constrained model. This method, when combined with the concept of effective connectivity, can provide information on the strength and direction of the functional interactions that take place between identified brain regions of a putative network.     

Detecting community structure using biased random merging

Decomposing a network into small modules or communities is beneficial for understanding the structure and dynamics of the network. One of the most prominent approaches is to repeatedly join communities together in pairs with the greatest increase in modularity so that a dendrogram that shows the order of joins is obtained. Then the community structure is acquired by cutting the dendrogram at the levels corresponding to the maximum modularity. However, there tends to be multiple pairs of communities that share the maximum modularity increment and the greedy agglomerative procedure may only merge one of them. Although the modularity function typically admits a lot of high-scoring solutions, the greedy strategy may fail to reach any of them. In this paper we propose an enhanced data structure in order to enable diverse choices of merging operations in community finding procedure. This data structure is actually a max-heap equipped with an extra array that stores the maximum modularity increments; and the corresponding community pairs is merged in the next move. By randomly sampling elements in this head array, additional diverse community structures can be efficiently extracted. The head array is designed to host the community pairs corresponding to the most significant increments in modularity so that the modularity structures obtained out of the sampling exhibit high modularity scores that are, on the average, even greater than what the CNM algorithm produces. Our method is tested on both real-world and computer-generated networks.