Semi-supervised clustering algorithm for community structure detection in complex networks

Discovering a community structure is fundamental for uncovering the links between structure and function in complex networks. In this paper, we discuss an equivalence of the objective functions of the symmetric nonnegative matrix factorization (SNMF) and the maximum optimization of modularity density. Based on this equivalence, we develop a new algorithm, named the so-called SNMF-SS, by combining SNMF and a semi-supervised clustering approach. Previous NMF-based algorithms often suffer from the restriction of measuring network topology from only one perspective, but our algorithm uses a semi-supervised mechanism to get rid of the restriction. The algorithm is illustrated and compared with spectral clustering and NMF by using artificial examples and other classic real world networks. Experimental results show the significance of the proposed approach, particularly, in the cases when community structure is obscure.     

Detecting the community structure in complex networks based on quantum mechanics

In this paper, we develop a novel method to detect the community structure in complex networks. This approach is based on the combination of kernel-based clustering using quantum mechanics, the spectral clustering technique and the concept of the Bayesian information criterion. We test the proposed algorithm on Zachary’s karate club network and the world of American college football. Experimental results indicate that our algorithm is efficient and effective at finding both the optimal number of clusters, and the best clustering of community structures.     

Quantum inspired evolutionary algorithm for community detection in complex networks

Community structure is indispensable to discover the potential property of complex network systems. In this paper we propose two algorithms (QIEA-net and iQIEA-net) to discover communities in social networks by optimizing modularity. Unlike many existing methods, the proposed algorithms adopt quantum inspired evolutionary algorithm (QIEA) to optimize a population of solutions and do not need to give the number of community beforehand, which is determined by optimizing the value of modularity function and needs no human intervention. In order to accelerate the convergence speed, in iQIEA-net, we apply the result of classical partitioning algorithm as a guiding quantum individual, which can instruct other quantum individuals' evolution. We demonstrate the potential of two algorithms on five real social networks. The results of comparison with other community detection algorithms prove our approaches have very competitive performance.     

Community detection in complex networks by density-based clustering

We proposed a method to find the community structure in a complex network by density-based clustering. Physical topological distance is introduced in density-based clustering for determining a distance function of specific influence functions. According to the distribution of the data, the community structures are uncovered. The method keeps a better connection mode of the community structure than the existing algorithms in terms of modularity, which can be viewed as a basic characteristic of community detection in the future. Moreover, experimental results indicate that the proposed method is efficient and effective to be used for community detection of medium and large networks.     

Fuzzy analysis of community detection in complex networks

A snowball algorithm is proposed to find community structures in complex networks by introducing the definition of community core and some quantitative conditions. A community core is first constructed, and then its neighbors, satisfying the quantitative conditions, will be tied to this core until no node can be added. Subsequently, one by one, all communities in the network are obtained by repeating this process. The use of the local information in the proposed algorithm directly leads to the reduction of complexity. The algorithm runs in O(n+m) time for a general network and O(n) for a sparse network, where n is the number of vertices and m is the number of edges in a network. The algorithm fast produces the desired results when applied to search for communities in a benchmark and five classical real-world networks, which are widely used to test algorithms of community detection in the complex network. Furthermore, unlike existing methods, neither global modularity nor local modularity is utilized in the proposal. By converting the considered problem into a graph, the proposed algorithm can also be applied to solve other cluster problems in data mining.     

Computing communities in complex networks using the Dirichlet processing Gaussian mixture model with spectral clustering

Community detection becomes a significant tool for the complex network analysis. The study of the community detection algorithms has received an enormous amount of attention. It is still an open question whether a highly accurate and efficient algorithm is found in most data sets. We propose the Dirichlet Processing Gaussian Mixture Model with Spectral Clustering algorithm for detecting the community structures. The combination of traditional spectral algorithm and new non-parametric Bayesian model provides high accuracy and quality. We compare the proposed algorithm with other existing community detecting algorithms using different real-world data sets and computer-generated synthetic data sets. We show that the proposed algorithm results in high modularity, and better accuracy in a wide range of networks. We find that the proposed algorithm works best for the large size of the data sets.     

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.     

A temporal ant colony optimization approach to the shortest path problem in dynamic scale-free networks

A large number of networks in the real world have a scale-free structure, and the parameters of the networks change stochastically with time. Searching for the shortest paths in a scale-free dynamic and stochastic network is not only necessary for the estimation of the statistical characteristics such as the average shortest path length of the network, but also challenges the traditional concepts related to the “shortest path” of a network and the design of path searching strategies. In this paper, the concept of shortest path is defined on the basis of a scale-free dynamic and stochastic network model, and a temporal ant colony optimization (TACO) algorithm is proposed for searching for the shortest paths in the network. The convergence and the setup for some important parameters of the TACO algorithm are discussed through theoretical analysis and computer simulations, validating the effectiveness of the proposed 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.     

Phase transition model for community detection

Motivated by social and biological interactions, a novel type of phase transition model is provided in order to investigate the emergence of the clustering phenomenon in networks. The model has two types of interactions: one is attractive and the other is repulsive. In each iteration, the phase of a node (or an agent) moves toward the average phase of its neighbors and moves away from the average phase of its non-neighbors. The velocities of the two types of phase transition are controlled by two parameters, respectively. It is found that the phase transition phenomenon is closely related to the topological structure of the underlying network, and thus can be applied to identify its communities and overlapping groups. By giving each node of the network a randomly generated initial phase and updating these phases by the phase transition model until they reach stability, one or two communities will be detected according to the nodes’ stable phases, confusable nodes are moved into a set named _Of$O_f$. By removing the detected communities and the nodes in _Of$O_f$, another one or two communities will be detected by an iteration of the algorithm, …. In this way, all communities and the overlapping nodes are detected. Simulations on both real-world networks and the LFR benchmark graphs have verified the efficiency of the proposed scheme.     

A fast and efficient heuristic algorithm for detecting community structures in complex networks

Community structure is an important property of complex networks. How to detect the communities is significant for understanding the network structure and to analyze the network properties. Many algorithms, such as K-L and GN, have been proposed to detect community structures in complex networks. According to daily experience, a community should have many nodes and connections. Based on these principles and existing researches, a fast and efficient algorithm for detecting community structures in complex networks is proposed in this paper. The key strategy of the algorithm is to mine a node with the closest relations with the community and assign it to this community. Four real-world networks are used to test the performance of the algorithm. Experimental results demonstrate that the algorithm proposed is rather efficient for detecting community structures in complex 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.     

Detecting community structure in complex networks based on a measure of information discrepancy

Properties of complex networks, such as small-world property, power-law degree distribution, network transitivity, and network- community structure which seem to be common to many real-world networks have attracted great interest among researchers. In this study, global information of the networks is considered by defining the profile of any node based on the shortest paths between it and all the other nodes in the network; then a useful iterative procedure for community detection based on a measure of information discrepancy and the popular modular function Q is presented. The new iterative method does not need any prior knowledge about the community structure and can detect an appropriate number of communities, which can be hub communities or non-hub communities. The computational results of the method on real networks confirm its capability.     

Modelling of weighted evolving networks with community structures

Many social and biological networks consist of communities–groups of nodes within which links are dense but among which links are sparse. It turns out that most of these networks are best described by weighted networks, whose properties and dynamics depend not only on their structures but also on the link weights among their nodes. Recently, there are considerable interests in the study of properties as well as modelling of such networks with community structures. To our knowledge, however, no study of any weighted network model with such a community structure has been presented in the literature to date. In this paper, we propose a weighted evolving network model with a community structure. The new network model is based on the inner-community and inter-community preferential attachments and preferential strengthening mechanism. Simulation results indicate that this network model indeed reflect the intrinsic community structure, with various power-law distributions of the node degrees, link weights, and node strengths.     

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.     

Minimum spanning trees for community detection

A simple deterministic algorithm for community detection is provided by using two rounds of minimum spanning trees. By comparing the first round minimum spanning tree (1st-MST) with the second round spanning tree (2nd-MST) of the network, communities are detected and their overlapping nodes are also identified. To generate the two MSTs, a distance matrix is defined and computed from the adjacent matrix of the network. Compared with the resistance matrix or the communicability matrix used in community detection in the literature, the proposed distance matrix is very simple in computation. The proposed algorithm is tested on real world social networks, graphs which are failed by the modularity maximization, and the LFR benchmark graphs for community detection.     

Growth curve based label propagation algorithm for community detection

How to better and faster identify the community structure is a hot issue in complex networks. During the past decades, various attempts have been made to solve this issue. Amongst them, without doubt, label propagation algorithm (LPA) is one of the most satisfying answers, especially for large-scale networks. However, it has one major flaw that when the community structure is not clear enough, a monster community tends to form. To address this issue, we set a growth curve for communities, gradually increasing from a low capacity to a higher capacity over time. Further, we improve the mechanism of label choosing for small communities to escape from local maximum. The experimental results on both synthetic and real networks demonstrate that our algorithm not only enhances the detection ability of the traditional label propagation algorithm, but also improves the quality of the identified communities.     

Detecting the optimal number of communities in complex networks

To obtain the optimal number of communities is an important problem in detecting community structures. In this paper, we use the extended measurement of community detecting algorithms to find the optimal community number. Based on the normalized mutual information index, which has been used as a measure for similarity of communities, a statistic Ω(c) is proposed to detect the optimal number of communities. In general, when Ω(c) reaches its local maximum, especially the first one, the corresponding number of communities c is likely to be optimal in community detection. Moreover, the statistic Ω(c) can also measure the significance of community structures in complex networks, which has been paid more attention recently. Numerical and empirical results show that the index Ω(c) is effective in both artificial and real world 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.     

Community detection in networks by using multiobjective evolutionary algorithm with decomposition

Community structure is an important property of complex networks. Most optimization-based community detection algorithms employ single optimization criteria. In this study, the community detection is solved as a multiobjective optimization problem by using the multiobjective evolutionary algorithm based on decomposition. The proposed algorithm maximizes the density of internal degrees, and minimizes the density of external degrees simultaneously. It can produce a set of solutions which can represent various divisions to the networks at different hierarchical levels. The number of communities is automatically determined by the non-dominated individuals resulting from our algorithm. Experiments on both synthetic and real-world network datasets verify that our algorithm is highly efficient at discovering quality community structure.