Community structure and modularity in networks of correlated brain activity

Functional connectivity patterns derived from neuroimaging data may be represented as graphs or networks, with individual image voxels or anatomically-defined structures representing the nodes, and a measure of correlation between the responses in each pair of nodes determining the edges. This explicit network representation allows network-analysis approaches to be applied to the characterization of functional connections within the brain. Much recent research in complex networks has focused on methods to identify community structure, i.e. cohesive clusters of strongly interconnected nodes. One class of such algorithms determines a partition of a network into 'sub-networks' based on the optimization of a modularity parameter, thus also providing a measure of the degree of segregation versus integration in the full network. Here, we demonstrate that a community structure algorithm based on the maximization of modularity, applied to a functional connectivity network calculated from the responses to acute fluoxetine challenge in the rat, can identify communities whose distributions correspond to anatomically meaningful structures and include compelling functional subdivisions in the brain. We also discuss the biological interpretation of the modularity parameter in terms of segregation and integration of brain function.     

Detecting overlapping communities by seed community in weighted complex networks

Detection of community structures in the weighted complex networks is significant to understand the network structures and analysis of the network properties. We present a unique algorithm to detect overlapping communities in the weighted complex networks with considerable accuracy. For a given weighted network, all the seed communities are first extracted. Then to each seed community, more community members are absorbed using the absorbing degree function. In addition, our algorithm successfully finds common nodes between communities. The experiments using some real-world networks show that the performance of our algorithm is satisfactory.     

Detecting overlapping communities based on vital nodes in complex networks

Detection of community structures in the complex networks is significant to understand the network structures and analyze the network properties. However, it is still a problem on how to select initial seeds as well as to determine the number of communities. In this paper, we proposed the detecting overlapping communities based on vital nodes algorithm(DOCBVA), an algorithm based on vital nodes and initial seeds to detect overlapping communities. First, through some screening method, we find the vital nodes and then the seed communities through the pretreatment of vital nodes. This process differs from most existing methods, and the speed is faster. Then the seeds will be extended. We also adopt a new parameter of attribution degree to extend the seeds and find the overlapping communities. Finally, the remaining nodes that have not been processed in the first two steps will be reprocessed. The number of communities is likely to change until the end of algorithm. The experimental results using some real-world network data and artificial network data are satisfactory and can prove the superiority of the DOCBVA algorithm.     

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 on overlapping community network

In this paper, we propose a simple random network model with overlapping communities controlled by several parameters, and investigate the influence of the overlapping community structure on the synchronization behavior under different parameters. It is found that the synchronizability of the network is mainly influenced by the overlapping size of the communities and the connectivity density of the overlapped group to the other interrelated communities, and has nothing to do with the intra-connectivity of the overlapped group. In addition, it is found that the highly interconnected communities can be almost synchronized in a given time scale, whereas the overlapped group is far from synchronization. Furthermore, the instantaneous frequencies of the nodes in the communities and their overlapped group are also investigated, which show that the nodes in the overlapped group will exhibit a remarkable oscillation with a weighted mean frequency of the other correlative communities.     

Detecting communities by the core-vertex and intimate degree in complex networks

In this paper, we present a new approach to extract communities in the complex networks with considerable accuracy. We introduce the core-vertex and the intimate degree between the community and its neighboring vertices. First, we find the core-vertices as the initial community. These core-vertices are then expanded using intimate degree function during extracting community structure from the given network. In addition, our algorithm successfully finds common nodes between communities. Experimental results using some real-world networks data shows that the performance of our algorithm is satisfactory.     

基于节点间依赖度的社团结构划分方法

本文提出了一种基于节点间依赖度的在复杂网络中划分社团结构的算法,定义了节点对其邻居的依赖度以及节点对社团的依赖度和条件依赖度.算法的基本要点是优先将最大依赖度不小于其他节点且有惟一依赖节点的节点划分到社团,并将对社团的依赖度或条件依赖度达到一定值的节点吸收进社团,直到所有节点都得到准确的社团划分.本算法在几个实际网络的测试上,都成功地划分出了满足条件的社团,并且对社团结构已知的网络的划分结果符合实际情况.     

Iterative decomposition of Barabasi-Albert scale-free networks

We study a decomposition process where all nodes with a targeted degree are removed from the network. Each removal step results in changes in the degrees of the remaining nodes, and other nodes may attain the targeted degree. The processes continue iteratively until no more nodes with the targeted degree are present in the decomposed network. The network model used in our study is the well known Barabasi-Albert network, that is built with an iterative growth based on preferential attachment. Our results show an exponential decay of the number of nodes removed at each step. The total number of nodes removed in the whole process depends on the targeted degree and decay with a power law controlled by the same exponent as the degree distribution of the network.     

确定度分布条件下可变同配系数的算法构造与影响分析

复杂网络是现实中大量节点和边的抽象拓扑, 如何揭示网络内部拓扑对网络连通性、脆弱性等特征的影响是当前研究的热点. 本文在确定度分布的条件下, 根据Newman提出的同配系数的定义分析其影响因素. 首先在可变同配系数下分别提出了基于度分布的确定算法和基于概率分布的不确定算法, 并分别在三种不同类型的网络(Erdös-Rényi网络, Barabási-Albert网络, Email真实网络)中验证. 实验结果表明: 当网络规模达到一定程度时, 确定算法优于贪婪算法. 以此为基础, 分析了同配系数改变时聚类系数的变化, 发现两者之间存在关联性, 并从网络的微观结构变化中揭示了聚类系数变化的原因.     

Detecting local community structures in complex networks based on local degree central nodes

Detecting local communities in real-world graphs such as large social networks, web graphs, and biological networks has received a great deal of attention because obtaining complete information from a large network is still difficult and unrealistic nowadays. In this paper, we define the term local degree central node whose degree is greater than or equal to the degree of its neighbor nodes. A new method based on the local degree central node to detect the local community is proposed. In our method, the local community is not discovered from the given starting node, but from the local degree central node that is associated with the given starting node. Experiments show that the local central nodes are key nodes of communities in complex networks and the local communities detected by our method have high accuracy. Our algorithm can discover local communities accurately for more nodes and is an effective method to explore community structures of large networks.     

Density-based shrinkage for revealing hierarchical and overlapping community structure in networks

The investigation of community structure in networks is an important issue in many disciplines, which still remains a challenging task. First, complex networks often show a hierarchical structure with communities embedded within other communities. Moreover, communities in the network may overlap and have noise, e.g., some nodes belonging to multiple communities and some nodes marginally connected with the communities, which are called hub and outlier, respectively. Therefore, a good algorithm is desirable to be able to not only detect hierarchical communities, but also to identify hubs and outliers. In this paper, we propose a parameter-free hierarchical network clustering algorithm DenShrink. By combining the advantages of density-based clustering and modularity optimization methods, our algorithm can reveal the embedded hierarchical community structure efficiently in large-scale weighted undirected networks, and identify hubs and outliers as well. Moreover, it overcomes the resolution limit possessed by other modularity-based methods. Our experiments on the real-world and synthetic datasets show that DenShrink generates more accurate results than the baseline methods.     

Advanced modularity-specialized label propagation algorithm for detecting communities in networks

A modularity-specialized label propagation algorithm (LPAm) for detecting network communities was recently proposed. This promising algorithm offers some desirable qualities. However, LPAm favors community divisions where all communities are similar in total degree and thus it is prone to get stuck in poor local maxima in the modularity space. To escape local maxima, we employ a multistep greedy agglomerative algorithm (MSG) that can merge multiple pairs of communities at a time. Combining LPAm and MSG, we propose an advanced modularity-specialized label propagation algorithm (LPAm+). Experiments show that LPAm+ successfully detects communities with higher modularity values than ever reported in two commonly used real-world networks. Moreover, LPAm+ offers a fair compromise between accuracy and speed.     

Modularity density of network community divisions

The problem of dividing a network into communities is extremely complex and grows very rapidly with the number of nodes and edges that are involved. In order to develop good algorithms to identify optimal community divisions it is extremely beneficial to identify properties that are similar for most networks. We introduce the concept of modularity density, the distribution of modularity values as a function of the number of communities, and find strong indications that the general features of this modularity density are quite similar for different networks. The region of high modularity generally has very low probability density and occurs where the number of communities is small. The properties and shape of the modularity density may give valuable information and aid in the search for efficient algorithms to find community divisions with high modularities.     

Phase synchronization on small-world networks with community structure

In this paper, we propose a simple model that can generate small-world network with community structure. The network is introduced as a tunable community organization with parameter r, which is directly measured by the ratio of inter- to intra-community connectivity, and a smaller r corresponds to a stronger community structure. The structure properties, including the degree distribution, clustering, the communication efficiency and modularity are also analysed for the network. In addition, by using the Kuramoto model, we investigated the phase synchronization on this network, and found that increasing the fuzziness of community structure will markedly enhance the network synchronizability; however, in an abnormal region (r ≤ 0.001), the network has even worse synchronizability than the case of isolated communities (r = 0). Furthermore, this network exhibits a remarkable synchronization behaviour in topological scales: the oscillators of high densely interconnected communities synchronize more easily, and more rapidly than the whole network.     

Community detection with consideration of non-topological information

In a network described by a graph, only topological structure information is considered to determine how the nodes are connected by edges. Non-topological information denotes that which cannot be determined directly from topological information. This paper shows, by a simple example where scientists in three research groups and one external group form four communities, that in some real world networks non-topological information (in this example, the research group affiliation) dominates community division. If the information has some influence on the network topological structure, the question arises as to how to find a suitable algorithm to identify the communities based only on the network topology. We show that weighted Newman algorithm may be the best choice for this example. We believe that this idea is general for real-world complex networks.     

Overlapping community detection using a generative model for networks

Detecting overlapping communities is a challenging task in analyzing networks, where nodes may belong to more than one community. Many present methods optimize quality functions to extract the communities from a network. In this paper, we present a probabilistic method for detecting overlapping communities using a generative model. The model describes the probability of generating a network with the model parameters, which reflect the communities in the network. The community memberships of each node are determined based on a probabilistic approach using those model parameters, whose values can be obtained by fitting the model to the network. This method has the advantage that the node participation degrees in each community are also computed. The proposed method is compared with some other community detection methods on both synthetic networks and real-world networks. The experiments show that this method is efficient at detecting overlapping communities and can provide better performance on the networks where a majority of nodes belong to more than one community.     

Detecting hierarchical and overlapping network communities using locally optimal modularity changes

Agglomerative clustering is a well established strategy for identifying communities in networks. Communities are successively merged into larger communities, coarsening a network of actors into a more manageable network of communities. The order in which merges should occur is not in general clear, necessitating heuristics for selecting pairs of communities to merge. We describe a hierarchical clustering algorithm based on a local optimality property. For each edge in the network, we associate the modularity change for merging the communities it links. For each community vertex, we call the preferred edge that edge for which the modularity change is maximal. When an edge is preferred by both vertices that it links, it appears to be the optimal choice from the local viewpoint. We use the locally optimal edges to define the algorithm: simultaneously merge all pairs of communities that are connected by locally optimal edges that would increase the modularity, redetermining the locally optimal edges after each step and continuing so long as the modularity can be further increased. We apply the algorithm to model and empirical networks, demonstrating that it can efficiently produce high-quality community solutions. We relate the performance and implementation details to the structure of the resulting community hierarchies. We additionally consider a complementary local clustering algorithm, describing how to identify overlapping communities based on the local optimality condition.     

两相流流型复杂网络社团结构及其统计特性

利用气液两相流电导波动信号构建了流型复杂网络. 基于K均值聚类的社团探寻算法对该网络的社团结构进行了分析,发现该网络存在分别对应于泡状流、段塞流及混状流的三个社团,并且两个社团间联系紧密的点分别对应于相应的过渡流型. 基于复杂网络理论从全新的角度探讨了两相流流型复杂网络社团结构及统计特性问题,并取得了满意的流型识别效果,与此同时,在对该网络特性进一步分析的基础上,发现了对两相流流动参数变化敏感的相关复杂网络统计量,为更好地理解两相流流型动力学特性提供了参考. 关键词： 两相流流型 复杂网络 社团探寻算法 网络统计特性     

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.     

A validity index approach for network partitions

The validity index has been used to evaluate the fitness of partitions produced by clustering algorithms for points in Euclidean space. In this paper, we propose a new validity index for network partitions, which can provide a measure of goodness for the community structure of networks. It is defined as a product of two factors, and involves the compactness and separation for each partition. The simulated annealing strategy is used to minimize such a validity index function in coordination with our previous k-means algorithm based on the optimal reduction of a random walker Markovian dynamics on the network. It is demonstrated that the algorithm can efficiently find the community structure during the cooling process. The number of communities can be automatically determined without any prior knowledge of the community structure. Moreover, the algorithm is successfully applied to three real-world networks.