Modularity functions maximization with nonnegative relaxation facilitates community detection in networks

We show here that the problem of maximizing a family of quantitative functions, encompassing both the modularity (Q-measure) and modularity density (D-measure), for community detection can be uniformly understood as a combinatoric optimization involving the trace of a matrix called modularity Laplacian. Instead of using traditional spectral relaxation, we apply additional nonnegative constraint into this graph clustering problem and design efficient algorithms to optimize the new objective. With the explicit nonnegative constraint, our solutions are very close to the ideal community indicator matrix and can directly assign nodes into communities. The near-orthogonal columns of the solution can be reformulated as the posterior probability of corresponding node belonging to each community. Therefore, the proposed method can be exploited to identify the fuzzy or overlapping communities and thus facilitates the understanding of the intrinsic structure of networks. Experimental results show that our new algorithm consistently, sometimes significantly, outperforms the traditional spectral relaxation approaches.     

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.     

Detecting overlapping communities in networks via dominant label propagation

Community detection is an important methodology for understanding the intrinsic structure and function of a realworld network.In this paper,we propose an effective and efficient algorithm,called Dominant Label Propagation Algorithm(Abbreviated as DLPA),to detect communities in complex networks.The algorithm simulates a special voting process to detect overlapping and non-overlapping community structure in complex networks simultaneously.Our algorithm is very efficient,since its computational complexity is almost linear to the number of edges in the network.Experimental results on both real-world and synthetic networks show that our algorithm also possesses high accuracies on detecting community structure in networks.     

Community structure discovery method based on the Gaussian kernel similarity matrix

Community structure discovery in complex networks is a popular issue, and overlapping community structure discovery in academic research has become one of the hot spots. Based on the Gaussian kernel similarity matrix and spectral bisection, this paper proposes a new community structure discovery method. First, by adjusting the Gaussian kernel parameter to change the scale of similarity, we can find the corresponding non-overlapping community structure when the value of the modularity is the largest relatively. Second, the changes of the Gaussian kernel parameter would lead to the unstable nodes jumping off, so with a slight change in method of non-overlapping community discovery, we can find the overlapping community nodes. Finally, synthetic data, karate club and political books datasets are used to test the proposed method, comparing with some other community discovery methods, to demonstrate the feasibility and effectiveness of this method.     

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 overlapping communities of weighted networks via a local algorithm

Identification of communities is significant in understanding the structures and functions of networks. Since some nodes naturally belong to several communities, the study of overlapping communities has attracted increasing attention recently, and many algorithms have been designed to detect overlapping communities. In this paper, an overlapping communities detecting algorithm is proposed whose main strategies are finding an initial partial community from a node with maximal node strength and adding tight nodes to expand the partial community. Seven real-world complex networks and one synthetic network are used to evaluate the algorithm. Experimental results demonstrate that the algorithm proposed is efficient for detecting overlapping communities in weighted networks.     

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.     

Adjusting from disjoint to overlapping community detection of complex networks

The investigation of community structures is one of the most important problems in the field of complex networks and has countless applications in different disciplines: biology, computer, social sciences, etc. Many community detection algorithms have been developed in various fields recently. The vast majority of these algorithms only find disjoint communities; however, in many real-world networks communities often overlap to some extent. In this paper, we propose an efficient method for adjusting these classical algorithms to match the requirement for discovering overlapping communities in complex networks, which is based on a local definition of community strength. The method can in principle be applied with any clustering algorithm. Tests on a set of computer generated and real-world networks give excellent results. In particular, we show that the method can also allow one to availably analyze the problem of unstable nodes in community detection, which is very helpful for understanding the structural properties of the networks correctly and comprehensively.     

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.     

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.     

Overlapping community detection using weighted consensus clustering

Many overlapping community detection algorithms have been proposed. Most of them are unstable and behave non-deterministically. In this paper, we use weighted consensus clustering for combining multiple base covers obtained by classic non-deterministic algorithms to improve the quality of the results. We first evaluate a reliability measure for each community in all base covers and assign a proportional weight to each one. Then we redefine the consensus matrix that takes into account not only the common membership of nodes, but also the reliability of the communities. Experimental results on both artificial and real-world networks show that our algorithm can find overlapping communities accurately.     

Detection of node group membership in networks with group overlap

Most networks found in social and biochemical systems have modular structures. An important question prompted by the modularity of these networks is whether nodes can be said to belong to a single group. If they cannot, we would need to consider the role of “overlapping communities.” Despite some efforts in this direction, the problem of detecting overlapping groups remains unsolved because there is neither a formal definition of overlapping community, nor an ensemble of networks with which to test the performance of group detection algorithms when nodes can belong to more than one group. Here, we introduce an ensemble of networks with overlapping groups. We then apply three group identification methods – modularity maximization, k-clique percolation, and modularity-landscape surveying – to these networks. We find that the modularity-landscape surveying method is the only one able to detect heterogeneities in node memberships, and that those heterogeneities are only detectable when the overlap is small. Surprisingly, we find that the k-clique percolation method is unable to detect node membership for the overlapping case.     

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.     

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.     

EAMCD: an efficient algorithm based on minimum coupling distance for community identification in complex networks

Community structure is an important feature in many real-world networks, which can help us understand structure and function in complex networks better. In recent years, there have been many algorithms proposed to detect community structure in complex networks. In this paper, we try to detect potential community beams whose link strengths are greater than surrounding links and propose the minimum coupling distance (MCD) between community beams. Based on MCD, we put forward an optimization heuristic algorithm (EAMCD) for modularity density function to welded these community beams into community frames which are seen as a core part of community. Using the principle of random walk, we regard the remaining nodes into the community frame to form a community. At last, we merge several small community frame fragments using local greedy strategy for the modularity density general function. Real-world and synthetic networks are used to demonstrate the effectiveness of our algorithm in detecting communities in complex networks.     

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.     

LPA-MNI: An Improved Label Propagation Algorithm Based on Modularity and Node Importance for Community Detection

Community detection is of great significance in understanding the structure of the network. Label propagation algorithm (LPA) is a classical and effective method, but it has the problems of randomness and instability. An improved label propagation algorithm named LPA-MNI is proposed in this study by combining the modularity function and node importance with the original LPA. LPA-MNI first identify the initial communities according to the value of modularity. Subsequently, the label propagation is used to cluster the remaining nodes that have not been assigned to initial communities. Meanwhile, node importance is used to improve the node order of label updating and the mechanism of label selecting when multiple labels are contained by the maximum number of nodes. Extensive experiments are performed on twelve real-world networks and eight groups of synthetic networks, and the results show that LPA-MNI has better accuracy, higher modularity, and more reasonable community numbers when compared with other six algorithms. In addition, LPA-MNI is shown to be more robust than the traditional LPA algorithm.     

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.     

Graph spectra and the detectability of community structure in networks

We study networks that display community structure--groups of nodes within which connections are unusually dense. Using methods from random matrix theory, we calculate the spectra of such networks in the limit of large size, and hence demonstrate the presence of a phase transition in matrix methods for community detection, such as the popular modularity maximization method. The transition separates a regime in which such methods successfully detect the community structure from one in which the structure is present but is not detected. By comparing these results with recent analyses of maximum-likelihood methods, we are able to show that spectral modularity maximization is an optimal detection method in the sense that no other method will succeed in the regime where the modularity method fails.