Modularity is widely used to effectively measure the strength of the community structure found by community detection algorithms. However, modularity maximization suffers from two opposite yet coexisting problems: in some cases, it tends to favor small communities over large ones while in others, large communities over small ones. The latter tendency is known in the literature as the resolution limit problem. To address them, we propose to modify modularity by subtracting from it the fraction of edges connecting nodes of different communities and by including community density into modularity. We refer to the modified metric as Modularity Density and we demonstrate that it indeed resolves both problems mentioned above. We describe the motivation for introducing this metric by using intuitively clear and simple examples. We also prove that this new metric solves the resolution limit problem. Finally, we discuss the results of applying this metric, modularity, and several other popular community quality metrics to two real dynamic networks. The results imply that Modularity Density is consistent with all the community quality measurements but not modularity, which suggests that Modularity Density is an improved measurement of the community quality compared to modularity.
Community detection in complex networks has received a considerable amount of attention in the last years. Numerous techniques have been developed for both efficient and effective community detection, including Modularity Optimization @cite_26 @cite_8 @cite_35 @cite_3 @cite_22 @cite_11 @cite_25 , Clique Percolation @cite_6 @cite_13 , Local Expansion @cite_28 @cite_5 @cite_0 , Fuzzy Clustering @cite_32 @cite_17 , Link Partitioning @cite_20 , and Label Propagation @cite_19 @cite_9 @cite_21 . The above algorithms are designed to detect communities on static networks. However, networks, such as Internet and online social networks, are usually dynamic, with changes arriving as a stream. Thus, a large number of algorithms were proposed to cope with community detection on dynamically evolving networks, such as LabelRankT @cite_24 and Estrangement @cite_2 . LabelRankT @cite_24 detects communities in large-scale dynamic networks through stabilized label propagation. Estrangement @cite_2 detects temporal communities by maximizing modularity in a snapshot subject to a constraint on the estrangement from the partition in the previous snapshot.
@cite_35 @cite_22 @cite_3 @cite_2 @cite_5 @cite_20 @cite_8 @cite_21 @cite_17 @cite_26 @cite_28 @cite_32 @cite_6 @cite_19 @cite_25 @cite_9 @cite_0 @cite_24 @cite_13 @cite_11
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The discovery and analysis of community structure in networks is a topic of considerable recent interest within the physics community, but most methods proposed so far are unsuitable for very large networks because of their computational cost. Here we present a hierarchical agglomeration algorithm for detecting community structure which is faster than many competing algorithms: its running time on a network with n vertices and m edges is O(m d log n) where d is the depth of the dendrogram describing the community structure. Many real-world networks are sparse and hierarchical, with m n and d log n, in which case our algorithm runs in essentially linear time, O(n log^2 n). As an example of the application of this algorithm we use it to analyze a network of items for sale on the web-site of a large online retailer, items in the network being linked if they are frequently purchased by the same buyer. The network has more than 400,000 vertices and 2 million edges. We show that our algorithm can extract meaningful communities from this network, revealing large-scale patterns present in the purchasing habits of customers. We formulate a spectral graph-partitioning algorithm that uses the two leading eigenvectors of the matrix corresponding to a selected quality function to split a network into three communities in a single step. In so doing, we extend the recursive bipartitioning methods developed by Newman [M. E. J. Newman, Proc. Natl. Acad. Sci. U.S.A. 103, 8577 (2006); Phys. Rev. E 74, 036104 (2006)] to allow one to consider the best available two-way and three-way divisions at each recursive step. We illustrate the method using simple "bucket brigade" examples and then apply the algorithm to examine the community structures of the coauthorship graph of network scientists and of U. S. Congressional networks inferred from roll call voting similarities. We propose a simple method to extract the community structure of large networks. Our method is a heuristic method that is based on modularity optimization. It is shown to outperform all other known community detection methods in terms of computation time. Moreover, the quality of the communities detected is very good, as measured by the so-called modularity. This is shown first by identifying language communities in a Belgian mobile phone network of 2 million customers and by analysing a web graph of 118 million nodes and more than one billion links. The accuracy of our algorithm is also verified on ad hoc modular networks. Temporal communities are the result of a consistent partitioning of nodes across multiple snapshots of an evolving network, and they provide insights into how dense clusters in a network emerge, combine, split and decay over time. To reliably detect temporal communities we need to not only find a good community partition in a given snapshot but also ensure that it bears some similarity to the partition(s) found in the previous snapshot(s), a particularly difficult task given the extreme sensitivity of community structure yielded by current methods to changes in the network structure. Here, motivated by the inertia of inter-node relationships, we present a new measure of partition distance called estrangement, and show that constraining estrangement enables one to find meaningful temporal communities at various degrees of temporal smoothness in diverse real-world datasets. Estrangement confinement thus provides a principled approach to uncovering temporal communities in evolving networks. Many networks in nature, society and technology are characterized by a mesoscopic level of organization, with groups of nodes forming tightly connected units, called communities or modules, that are only weakly linked to each other. Uncovering this community structure is one of the most important problems in the field of complex networks. Networks often show a hierarchical organization, with communities embedded within other communities; moreover, nodes can be shared between different communities. Here, we present the first algorithm that finds both overlapping communities and the hierarchical structure. The method is based on the local optimization of a fitness function. Community structure is revealed by peaks in the fitness histogram. The resolution can be tuned by a parameter enabling different hierarchical levels of organization to be investigated. Tests on real and artificial networks give excellent results. Network theory has become pervasive in all sectors of biology, from biochemical signalling to human societies, but identification of relevant functional communities has been impaired by many nodes belonging to several overlapping groups at once, and by hierarchical structures. These authors offer a radically different viewpoint, focusing on links rather than nodes, which allows them to demonstrate that overlapping communities and network hierarchies are two faces of the same issue. Many networks of interest in the sciences, including social networks, computer networks, and metabolic and regulatory networks, are found to divide naturally into communities or modules. The problem of detecting and characterizing this community structure is one of the outstanding issues in the study of networked systems. One highly effective approach is the optimization of the quality function known as “modularity” over the possible divisions of a network. Here I show that the modularity can be expressed in terms of the eigenvectors of a characteristic matrix for the network, which I call the modularity matrix, and that this expression leads to a spectral algorithm for community detection that returns results of demonstrably higher quality than competing methods in shorter running times. I illustrate the method with applications to several published network data sets. Membership diversity is a characteristic aspect of social networks in which a person may belong to more than one social group. For this reason, discovering overlapping structures is necessary for realistic social analysis. In this paper, we present a fast algorithm, called SLPA, for overlapping community detection in large-scale networks. SLPA spreads labels according to dynamic interaction rules. It can be applied to both unipartite and bipartite networks. It is also able to uncover overlapping nested hierarchy . The time complexity of SLPA scales linearly with the number of edges in the network. Experiments in both synthetic and real-world networks show that SLPA has an excellent performance in identifying both node and community level overlapping structures. Identifying overlapping communities in networks is a challenging task. In this work we present a probabilistic approach to community detection that utilizes a Bayesian non-negative matrix factorization model to extract overlapping modules from a network. The scheme has the advantage of soft-partitioning solutions, assignment of node participation scores to modules, and an intuitive foundation. We present the performance of the method against a variety of benchmark problems and compare and contrast it to several other algorithms for community detection. Many networks display community structure---groups of vertices within which connections are dense but between which they are sparser---and sensitive computer algorithms have in recent years been developed for detecting this structure. These algorithms, however, are computationally demanding, which limits their application to small networks. Here we describe an algorithm which gives excellent results when tested on both computer-generated and real-world networks and is much faster, typically thousands of times faster, than previous algorithms. We give several example applications, including one to a collaboration network of more than 50 000 physicists. In this paper, we present an efficient algorithm for finding overlapping communities in social networks. Our algorithm does not rely on the contents of the messages and uses the communication graph only. The knowledge of the structure of the communities is important for the analysis of social behavior and evolution of the society as a whole, as well as its individual members. This knowledge can be helpful in discovering groups of actors that hide their communications, possibly for malicious reasons. Although the idea of using communication graphs for identifying clusters of actors is not new, most of the traditional approaches, with the exception of the work by , produce disjoint clusters of actors, de facto postulating that an actor is allowed to belong to at most one cluster. Our algorithm is significantly more efficient than the previous algorithm by ; it also produces clusters of a comparable or better quality. Identification of (overlapping) communities clusters in a complex network is a general problem in data mining of network data sets. In this paper, we devise a novel algorithm to identify overlapping communities in complex networks by the combination of a new modularity function based on generalizing NG's Q function, an approximation mapping of network nodes into Euclidean space and fuzzy c-means clustering. Experimental results indicate that the new algorithm is efficient at detecting both good clusterings and the appropriate number of clusters. A network is a network — be it between words (those associated with ‘bright’ in this case) or protein structures. Many complex systems in nature and society can be described in terms of networks capturing the intricate web of connections among the units they are made of1,2,3,4. A key question is how to interpret the global organization of such networks as the coexistence of their structural subunits (communities) associated with more highly interconnected parts. Identifying these a priori unknown building blocks (such as functionally related proteins5,6, industrial sectors7 and groups of people8,9) is crucial to the understanding of the structural and functional properties of networks. The existing deterministic methods used for large networks find separated communities, whereas most of the actual networks are made of highly overlapping cohesive groups of nodes. Here we introduce an approach to analysing the main statistical features of the interwoven sets of overlapping communities that makes a step towards uncovering the modular structure of complex systems. After defining a set of new characteristic quantities for the statistics of communities, we apply an efficient technique for exploring overlapping communities on a large scale. We find that overlaps are significant, and the distributions we introduce reveal universal features of networks. Our studies of collaboration, word-association and protein interaction graphs show that the web of communities has non-trivial correlations and specific scaling properties. Community detection and analysis is an important methodology for understanding the organization of various real-world networks and has applications in problems as diverse as consensus formation in social communities or the identification of functional modules in biochemical networks. Currently used algorithms that identify the community structures in large-scale real-world networks require a priori information such as the number and sizes of communities or are computationally expensive. In this paper we investigate a simple label propagation algorithm that uses the network structure alone as its guide and requires neither optimization of a predefined objective function nor prior information about the communities. In our algorithm every node is initialized with a unique label and at every step each node adopts the label that most of its neighbors currently have. In this iterative process densely connected groups of nodes form a consensus on a unique label to form communities. We validate the algorithm by applying it to networks whose community structures are known. We also demonstrate that the algorithm takes an almost linear time and hence it is computationally less expensive than what was possible so far. High-throughput techniques are leading to an explosive growth in the size of biological databases and creating the opportunity to revolutionize our understanding of life and disease. Interpretation of these data remains, however, a major scientific challenge. Here, we propose a methodology that enables us to extract and display information contained in complex networks1,2,3. Specifically, we demonstrate that we can find functional modules4,5 in complex networks, and classify nodes into universal roles according to their pattern of intra- and inter-module connections. The method thus yields a ‘cartographic representation’ of complex networks. Metabolic networks6,7,8 are among the most challenging biological networks and, arguably, the ones with most potential for immediate applicability9. We use our method to analyse the metabolic networks of twelve organisms from three different superkingdoms. We find that, typically, 80 of the nodes are only connected to other nodes within their respective modules, and that nodes with different roles are affected by different evolutionary constraints and pressures. Remarkably, we find that metabolites that participate in only a few reactions but that connect different modules are more conserved than hubs whose links are mostly within a single module. Studies of community structure and evolution in large social networks require a fast and accurate algorithm for community detection. As the size of analyzed communities grows, complexity of the community detection algorithm needs to be kept close to linear. The Label Propagation Algorithm (LPA) has the benefits of nearly-linear running time and easy implementation, thus it forms a good basis for efficient community detection methods. In this paper, we propose new update rule and label propagation criterion in LPA to improve both its computational efficiency and the quality of communities that it detects. The speed is optimized by avoiding unnecessary updates performed by the original algorithm. This change reduces significantly (by order of magnitude for large networks) the number of iterations that the algorithm executes. We also evaluate our generalization of the LPA update rule that takes into account, with varying strength, connections to the neighborhood of a node considering a new label. Experiments on computer generated networks and a wide range of social networks show that our new rule improves the quality of the detected communities compared to those found by the original LPA. The benefit of considering positive neighborhood strength is pronounced especially on real-world networks containing sufficiently large fraction of nodes with high clustering coefficient. Clustering and community structure is crucial for many network systems and the related dynamic processes. It has been shown that communities are usually overlapping and hierarchical. However, previous methods investigate these two properties of community structure separately. This paper proposes an algorithm (EAGLE) to detect both the overlapping and hierarchical properties of complex community structure together. This algorithm deals with the set of maximal cliques and adopts an agglomerative framework. The quality function of modularity is extended to evaluate the goodness of a cover. The examples of application to real world networks give excellent results. An increasingly important challenge in network analysis is efficient detection and tracking of communities in dynamic networks for which changes arrive as a stream. There is a need for algorithms that can incrementally update and monitor communities whose evolution generates huge real-time data streams, such as the Internet or on-line social networks. In this paper, we propose LabelRankT, an on-line distributed algorithm for detection of communities in large-scale dynamic networks through stabilized label propagation. Results of tests on real-world networks demonstrate that LabelRankT has much lower computational costs than other algorithms. It also improves the quality of the detected communities compared to dynamic detection methods and matches the quality achieved by static detection approaches. Unlike most of other algorithms which apply only to binary networks, LabelRankT works on weighted and directed networks, which provides a flexible and promising solution for real-world applications. The inclusion of link weights into the analysis of network properties allows a deeper insight into the (often overlapping) modular structure of real- world webs. We introduce a clustering algorithm clique percolation method with weights (CPMw) for weighted networks based on the concept of percolating k-cliques with high enough intensity. The algorithm allows overlaps between the modules. First, we give detailed analytical and numerical results about the critical point of weighted k-clique percolation on (weighted) Erdý os-Renyi graphs. Then, for a scientist collaboration web and a stock correlation graph we compute three-link weight correlations and with the CPMw the weighted modules. After reshuffling link weights in both networks and computing the same quantities for the randomized control graphs as well, we show that groups of three or more strong links prefer to cluster together in both original graphs. The description of the structure of complex networks has been one of the focus of attention of the physicist’s community in the recent years. The levels of description range from the microscopic (degree, clustering coefficient, centrality measures, etc., of individual nodes) to the macroscopic description in terms of statistical properties of the whole network (degree distribution, total clustering coefficient, degree-degree correlations, etc.) [1, 2, 3, 4]. Between these two extremes there is a ”mesoscopic” description of networks that tries to explain its community structure. The general notion of community structure in complex networks was first pointed out in the physics literature by Girvan and Newman [5], and refers to the fact that nodes in many real networks appear to group in subgraphs in which the density of internal connections is larger than the connections with the rest of nodes in the network. The community structure has been empirically found in many real technological, biological and social networks [6, 7, 8, 9, 10] and its emergence seems to be at the heart of the network formation process [11]. The existing methods intended to devise the community structure in complex networks have been recently reviewed in [10]. All these methods require a definition of community that imposes the limit up to which a group should be considered a community. However, the concept of community itself is qualitative: nodes must be more connected within its community than with the rest of the network, and its quantification is still a subject of debate. Some quantitative definitions that came from sociology have been used in recent studies [12], but in general, the physics community has widely accepted a recent measure for the community structure based on the concept of modularity Q introduced by Newman and Girvan [13]: