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.     

Community detection based on modularity and an improved genetic algorithm

Complex networks are widely applied in every aspect of human society, and community detection is a research hotspot in complex networks. Many algorithms use modularity as the objective function, which can simplify the algorithm. In this paper, a community detection method based on modularity and an improved genetic algorithm (MIGA) is put forward. MIGA takes the modularity Q$Q$ as the objective function, which can simplify the algorithm, and uses prior information (the number of community structures), which makes the algorithm more targeted and improves the stability and accuracy of community detection. Meanwhile, MIGA takes the simulated annealing method as the local search method, which can improve the ability of local search by adjusting the parameters. Compared with the state-of-art algorithms, simulation results on computer-generated and four real-world networks reflect the effectiveness of MIGA.     

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.     

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.     

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.     

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.     

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.     

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.     

Maximum modular graphs

Modularity has been explored as an important quantitative metric for community and cluster detection in networks. Finding the maximum modularity of a given graph has been proven to be NP-complete and therefore, several heuristic algorithms have been proposed. We investigate the problem of finding the maximum modularity of classes of graphs that have the same number of links and/or nodes and determine analytical upper bounds. Moreover, from the set of all connected graphs with a fixed number of links and/or number of nodes, we construct graphs that can attain maximum modularity, named maximum modular graphs. The maximum modularity is shown to depend on the residue obtained when the number of links is divided by the number of communities. Two applications in transportation networks and data-centers design that can benefit of maximum modular partitioning are proposed.     

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.     

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.     

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.     

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.     

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.     

A very fast algorithm for detecting community structures in complex networks

In this paper, a new algorithm is proposed, which uses only local information to analyze community structures in complex networks. The algorithm is based on a table that describes a network and a virtual cache similar to the cache in the computer structure. When being tested on some typical computer-generated and real-world networks, this algorithm demonstrates excellent detection results and very fast processing performance, much faster than the existing comparable algorithms of the same kind.     

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.     

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.     

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.     

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.