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.     

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.     

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.     

基于联合矩阵分解的节点多属性网络社团检测

发现复杂网络中的社团结构在社会网络、生物组织网络和在线网络等复杂网络中具备十分重要的意义. 针对社交媒体网络的社团检测通常需要利用两种信息源: 网络拓扑结构特征和节点属性特征, 丰富的节点内容属性信息为社团检测的增加了灵活性和挑战. 传统方法是要么仅针对这两者信息之一进行单独挖掘, 或者将两者信息得到的社团结果进行线性叠加判决, 不能有效进行信息源的融合. 本文将节点的多维属性特征作为社团划分的一种有效协同学习项进行研究, 将两者信息源进行融合分析, 提出了一种基于联合矩阵分解的节点多属性网络社团检测算法CDJMF, 提高了社团检测的有效性和鲁棒性. 实验表明, 本文所提的方法能够有效利用节点的属性信息指导社团检测, 具备更高的社团划分质量.     

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.     

Community detection based on network communicability

We propose a new method for detecting communities based on the concept of communicability between nodes in a complex network. This method, designated as N-ComBa K-means, uses a normalized version of the adjacency matrix to build the communicability matrix and then applies K-means clustering to find the communities in a graph. We analyze how this method performs for some pathological cases found in the analysis of the detection limit of communities and propose some possible solutions on the basis of the analysis of the ratio of local to global densities in graphs. We use four different quality criteria for detecting the best clustering and compare the new approach with the Girvan-Newman algorithm for the analysis of two "classical" networks: karate club and bottlenose dolphins. Finally, we analyze the more challenging case of homogeneous networks with community structure, for which the Girvan-Newman completely fails in detecting any clustering. The N-ComBa K-means approach performs very well in these situations and we applied it to detect the community structure in an international trade network of miscellaneous manufactures of metal having these characteristics. Some final remarks about the general philosophy of community detection are also discussed.     

Community detection based on the “clumpiness” matrix in complex networks

The “clumpiness” matrix of a network is used to develop a method to identify its community structure. A “projection space” is constructed from the eigenvectors of the clumpiness matrix and a border line is defined using some kind of angular distance in this space. The community structure of the network is identified using this borderline and/or hierarchical clustering methods. The performance of our algorithm is tested on some computer-generated and real-world networks. The accuracy of the results is checked using normalized mutual information. The effect of community size heterogeneity on the accuracy of the method is also discussed.     

加权网络权重自相似评判函数及其社团结构检测

提出了权重自相似性加权网络社团结构评判函数,并基于该函数提出一种谱分析算法检测社团结构,结果表明算法能将加权网络划分为同一社团内边权值分布均匀,而社团间边权值分布随机的社团结构.通过建立具有社团结构的加权随机网络分析了该算法的准确性,与WEO和WGN算法相比,在评判权重自相似的阈值系数取较小时,该算法具有较高的准确性.对于一个具有n个节点和c个社团的加权网络,社团结构检测的复杂度为O(cn2/2).通过设置评判权重自相似的阈值系数,可检测出能反映节点联系稳定性的层化性社团结构.这与传统意义上只将加权网络划分为社团中边权值较大而社团间边权值较小的标准不同,从另一个角度更好地提取了加权网络的结构信息.     

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.     

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.     

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.     

Modular and hierarchical structure of social contact networks

Social contact networks exhibit overlapping qualities of communities, hierarchical structure and spatial-correlated nature. We propose a mixing pattern of modular and growing hierarchical structures to reconstruct social contact networks by using an individual’s geospatial distribution information in the real world. The hierarchical structure of social contact networks is defined based on the spatial distance between individuals, and edges among individuals are added in turn from the modular layer to the highest layer. It is a gradual process to construct the hierarchical structure: from the basic modular model up to the global network. The proposed model not only shows hierarchically increasing degree distribution and large clustering coefficients in communities, but also exhibits spatial clustering features of individual distributions. As an evaluation of the method, we reconstruct a hierarchical contact network based on the investigation data of a university. Transmission experiments of influenza H1N1 are carried out on the generated social contact networks, and results show that the constructed network is efficient to reproduce the dynamic process of an outbreak and evaluate interventions. The reproduced spread process exhibits that the spatial clustering of infection is accordant with the clustering of network topology. Moreover, the effect of individual topological character on the spread of influenza is analyzed, and the experiment results indicate that the spread is limited by individual daily contact patterns and local clustering topology rather than individual degree.     

Fuzzy overlapping community detection based on local random walk and multidimensional scaling

A fuzzy overlapping community is an important kind of overlapping community in which each node belongs to each community to different extents. It exists in many real networks but how to identify a fuzzy overlapping community is still a challenging task. In this work, the concept of local random walk and a new distance metric are introduced. Based on the new distance measurement, the dissimilarity index between each node of a network is calculated firstly. Then in order to keep the original node distance as much as possible, the network structure is mapped into low-dimensional space by the multidimensional scaling (MDS). Finally, the fuzzy c$c$-means clustering is employed to find fuzzy communities in a network. The experimental results show that the proposed algorithm is effective and efficient to identify the fuzzy overlapping communities in both artificial networks and real-world networks.     

Detection of community overlap according to belief propagation and conflict

Most existing methods for detection of community overlap cannot balance efficiency and accuracy for large and densely overlapping networks. To quickly identify overlapping communities for such networks, we propose a new method that uses belief propagation and conflict (PCB) to occupy communities. We first identify triangles with maximal clustering coefficients as seed nodes and sow a new type of belief to the seed nodes. Then the beliefs explore their territory by occupying nodes with high assent ability. The beliefs propagate their strength along the graph to consolidate their territory, and conflict with each other when they encounter the same node simultaneously. Finally, the node membership is judged from the belief vectors. The PCB time complexity is nearly linear and its space complexity is linear. The algorithm was tested in extensive experiments on three real-world social networks and three computer-generated artificial graphs. The experimental results show that PCB is very fast and highly reliable. Tests on real and artificial networks give excellent results compared with three newly proposed overlapping community detection algorithms.     

Clustering and community detection in directed networks: A survey

Networks (or graphs) appear as dominant structures in diverse domains, including sociology, biology, neuroscience and computer science. In most of the aforementioned cases graphs are directed — in the sense that there is directionality on the edges, making the semantics of the edges nonsymmetric as the source node transmits some property to the target one but not vice versa. An interesting feature that real networks present is the clustering or community structure property, under which the graph topology is organized into modules commonly called communities or clusters. The essence here is that nodes of the same community are highly similar while on the contrary, nodes across communities present low similarity. Revealing the underlying community structure of directed complex networks has become a crucial and interdisciplinary topic with a plethora of relevant application domains. Therefore, naturally there is a recent wealth of research production in the area of mining directed graphs — with clustering being the primary method sought and the primary tool for community detection and evaluation. The goal of this paper is to offer an in-depth comparative review of the methods presented so far for clustering directed networks along with the relevant necessary methodological background and also related applications. The survey commences by offering a concise review of the fundamental concepts and methodological base on which graph clustering algorithms capitalize on. Then we present the relevant work along two orthogonal classifications. The first one is mostly concerned with the methodological principles of the clustering algorithms, while the second one approaches the methods from the viewpoint regarding the properties of a good cluster in a directed network. Further, we present methods and metrics for evaluating graph clustering results, demonstrate interesting application domains and provide promising future research directions.     

Graph kernels,hierarchical clustering,and network community structure: experiments and comparative analysis

There has been a quickly growing interest in properties of complex networks, such as the small world property, power-law degree distribution, network transitivity, and community structure, which seem to be common to many real world networks. In this study, we consider the community property which is also found in many real networks. Based on the diffusion kernels of networks, a hierarchical clustering approach is proposed to uncover the community structure of different extent of complex networks. We test the method on some networks with known community structures and find that it can detect significant community structure in these networks. Comparison with related methods shows the effectiveness of the method.     

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.     

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.     

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.