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Complex network theory is introduced to solve the islanding problem in an emergency of distribution networks. In this study, the authors put forward an intentional islanding method based on community detection. In this method, a new index has been defined called electrical edge betweenness, on the strength of edge betweenness in complex networks, which fuses electrical characteristics with topological features of actual power lines. Based on the index, the Girvan–Newman algorithm is employed to detect the community structure of distribution networks. Through referring to the modularity value (function Q) and coherent generator groups, they can get a reasonable amount and regions of communities. Then the whole distribution network can be partitioned into several self-sustainable islands meeting the stable operation constraints. The effectiveness of the authors’ proposed method is tested on a standard IEEE 118-bus system.

Betweenness centrality quantifies the importance of nodes in a graph in many applications, including network analysis, community detection and identification of influential users. Typically, graphs in such applications evolve over time. Thus, the computation of betweenness centrality should be performed incrementally. This is challenging because updating even a single edge may trigger the computation of all-pairs shortest paths in the entire graph. Existing approaches cannot scale to large graphs: they either require excessive memory (i.e., quadratic to the size of the input graph) or perform unnecessary computations rendering them prohibitively slow. We propose $i$ Central ; a novel incremental algorithm for computing betweenness centrality in evolving graphs. We decompose the graph into biconnected components and prove that processing can be localized within the affected components. $i$ Central is the first algorithm to support incremental betweeness centrality computation within a graph component. This is done efficiently, in linear space; consequently, $i$ Central scales to large graphs. We demonstrate with real datasets that the serial implementation of $i$ Central is up to 3.7 times faster than existing serial methods. Our parallel implementation that scales to large graphs, is an order of magnitude faster than the state-of-the-art parallel algorithm, while using an order of magnitude less computational resources.

China has extensively implemented the Joint Prevention and Control of Atmospheric Pollution (JPCAP) strategy to address regional air pollution challenges; however, the lack of accurate scope and regional priority control sequence has resulted in reduced pollution control efficiency and high air quality improvement costs. We proposed the following new solutions to these problems: (i) constructing inter-regional network correlation models based on three key network topology parameters and (ii) subdividing large regions into sub-regions using the Girvan Newman (GN) algorithm. To test these methods, we applied them to a case study of China's provincial-level JPCAP areas using hourly air resource endowment (ARE) data obtained from 2013 to 2017. We found that the modularity (Q) values of ARE network partitioning in different seasons were approximately 0.5, indicating the well-structured association of the network constructed in this study. Seven areas were identified based on annual values, while eight, five, four, and six areas were classified for spring, summer, autumn, and winter, respectively. The strength of the interconnectedness of the provinces in the region, as measured by the clustering coefficient ( C ˜ ) value, is ordered as follows: Autumn > winter > spring > summer. For a specific JPCAP region, the priority control level can be dynamically adjusted based on the Pearson correlation coefficient (r-value) between the target city or province and the surrounding region. The regional division results, priority control sequences, and key control seasons identified in this study can provide technical and methodological reference for delineating and optimizing China's JPCAP policy.

Community detection is a critical task for complex network analysis. It helps us to understand the properties of the system that a complex network represents and has significance to a wide range of applications. Nowadays, the challenges faced by community detection algorithms include overlapping community structure detection, large scale network analysis, dynamic changing of analyzed network topology and many more. In this paper a self-organizing community detection algorithm, based on the idea of swarm intelligence, was proposed and its parallel algorithm was designed on Giraph++ which is a semi-asynchronous parallel graph computation framework running on distributed environment. In the algorithm, a network of large size is firstly divided into a number of small sub-networks. Then, each sub-network is modeled as a self-evolving swarm intelligence sub-system, while each vertex within the sub-network acts iteratively to join into or leave from communities based on a set of predefined vertex action rules. Meanwhile, the local communities of a sub-network are sent to other sub-networks to make their members have a chance to join into, therefore connecting these self-evolving swarm intelligence sub-systems together as a whole, large and evolving, system. The vertex actions during evolution of a sub-network are sent as well to keep multiple community replicas being consistent. Thus network communication efficiency has a great impact on the algorithm' performance. While there is no vertex changing in its belonging communities anymore, an optimal community structure of the whole network will have emerged as a result. In the algorithm it is natural that a vertex can join into multiple communities simultaneously, thus can be used for overlapping community detection. The algorithm deals with vertex and edge adding or deleting in the same way as the algorithm running, therefore inherently supports dynamic network analysis. The algorithm can be used for the analysis of large scale networks with its parallel version running on distributed environment. A variety of experiments conducted on synthesized networks have shown that the proposed algorithm can effectively detect community structures and its performance is much better than certain popular community detection algorithms.

In this paper, we present a novel data clustering framework for big sensory data produced by IoT applications. Based on a network representation of the relations among multi-dimensional data, data clustering is mapped to node clustering over the produced data graphs. To address the potential very large scale of such datasets/graphs that test the limits of state-of-the-art approaches, we map the problem of data clustering to a community detection one over the corresponding data graphs. Specifically, we propose a novel computational approach for enhancing the traditional Girvan–Newman (GN) community detection algorithm via hyperbolic network embedding. The data dependency graph is embedded in the hyperbolic space via Rigel embedding, allowing more efficient computation of edge-betweenness centrality needed in the GN algorithm. This allows for more efficient clustering of the nodes of the data graph in terms of modularity, without sacrificing considerable accuracy. In order to study the operation of our approach with respect to enhancing GN community detection, we employ various representative types of artificial complex networks, such as scale-free, small-world and random geometric topologies, and frequently-employed benchmark datasets for demonstrating its efficacy in terms of data clustering via community detection. Furthermore, we provide a proof-of-concept evaluation by applying the proposed framework over multi-dimensional datasets obtained from an operational smart-city/building IoT infrastructure provided by the Federated Interoperable Semantic IoT/cloud Testbeds and Applications (FIESTA-IoT) testbed federation. It is shown that the proposed framework can be indeed used for community detection/data clustering and exploited in various other IoT applications, such as performing more energy-efficient smart-city/building sensing.

Most of the real-world networks exhibit community structure, a property that reveals the existence of natural vertex clusters whose inter-edge density is lower than intra-edge density between various groups. Despite providing a better understanding of network structure and characteristics, community detection has many practical applications in diverse domains. Communities obtained from the telephone network provides many useful information that can be used for churn prediction, budget control in organizations etc. Detecting communities is a fundamental need in the area of networks, yet challenging. In this paper, we propose an extension to the Girvan-Newman algorithm for finding the betweenness using the transitive closure property and the greedy technique in Dijkstra's single source shortest path method.

Devise an algorithm for updating betweenness centrality in fully dynamic graphs.Recalculate centrality without computing all pairs shortest paths in the entire graph.Adapt a highest centrality edge finding algorithm based on the proposed algorithm.Adapt a community detection algorithm using the proposed algorithms.Experimental results show the proposed algorithm outperforms existing algorithms. Betweenness centrality of a vertex (edge) in a graph is a measure for the relative participation of the vertex (edge) in the shortest paths in the graph. Betweenness centrality is widely used in various areas such as biology, transportation, and social networks. In this paper, we study the update problem of betweenness centrality in fully dynamic graphs. The proposed update algorithm substantially reduces the number of shortest paths which should be re-computed when a graph is changed. In addition, we adapt a community detection algorithm using the proposed algorithm to show how much benefit can be obtained from the proposed algorithm in a practical application. Experimental results on real graphs show that the proposed algorithm efficiently update betweenness centrality and detect communities in a graph.

Betweenness centrality is a centrality measure that is widely used, with applications across several disciplines. It is a measure which quantifies the importance of a vertex based on its occurrence in shortest paths between all possible pairs of vertices in a graph. This is a global measure, and in order to find the betweenness centrality of a node, one is supposed to have complete information about the graph. Most of the algorithms that are used to find betwenness centrality assume the constancy of the graph and are not efficient for dynamic networks. We propose a technique to update betweenness centrality of a graph when nodes are added or deleted. Our algorithm experimentally speeds up the calculation of betweenness centrality (after updation) from 7 to 412 times, for real graphs, in comparison to the currently best known technique to find betweenness centrality.

There is a surge of community detection study on complex network analysis in recent years, since communities often play important roles in network systems. However, many real networks have more complex overlapping community structures. This paper proposes a novel algorithm to discover overlapping communities. Different from conventional algorithms based on node clustering, the proposed algorithm is based on link clustering. Since links usually represent unique relations among nodes, the link clustering will discover groups of links that have the same characteristics. Thus nodes naturally belong to multiple communities. The algorithm applies genetic operation to cluster on links. An effective encoding schema is designed and the number of communities can be automatically determined. Experiments on both artificial networks and real networks validate the effectiveness and efficiency of the proposed algorithm.

Distributed, networked learning processes are widely touted as a basis for superior performance. Yet we know relatively little about how learning networks operate in the aggregate. We explore this issue by utilizing a unique data set on learning among Swedish municipalities. The data indicate that geographic proximity and county are the basic structuring properties of the global network. Municipalities learn from their near neighbors, especially from neighbors in the same county, and these two principles produce a high degree of local clustering in the municipal learning networks. At the same time, we also find evidence that Swedish municipalities are a small world linked together on a national basis. Two mechanisms knit the Swedish municipalities together. First, county seats serve as hubs that link local clusters together. Second, local clusters aggregate into regional clusters. Despite a high degree of local clustering, hubs and regions provide a structural basis for the national diffusion of policy ideas and practices among Swedish municipalities.

The progressively scale of online social network leads to the difficulty of traditional algorithms on detecting communities. We introduce an efficient and fast algorithm to detect community structure in social networks. Instead of using the eigenvectors in spectral clustering algorithms, we construct a target function for detecting communities. The whole social network communities will be partitioned by this target function. We also analyze and estimate the generalization error of the algorithm. The performance of the algorithm is compared with the standard spectral clustering algorithm, which is applied to different well-known instances of social networks with a community structure, both computer generated and from the real world. The experimental results demonstrate the effectiveness of the algorithm.

Network model recently becomes a popular tool for studying complex systems. Detecting meaningful communities in complex networks, as an important task in network modeling and analysis, has attracted great interests in various research areas. This paper proposes a genetic algorithm with a special encoding schema for community detection in complex networks. The algorithm employs a metric, named modularity Q as the fitness function and applies a special locus-based adjacency encoding schema to represent the community partitions. The encoding schema enables the algorithm to determine the number of communities adaptively and automatically, which provides great flexibility to the detection process. In addition, the schema also significantly reduces the search space. Extensive experiments demonstrate the effectiveness of the proposed algorithm.

Nowadays finding patterns in large social network datasets is a growing challenge and an important subject of interest. One of current problems in this field is identifying clusters within social networks with large number of nodes. Social network clusters are not necessarily disjoint sets; rather they may overlap and have common nodes, in which case it is more appropriate to designate them as communities. Although many clustering algorithms handle small datasets well, they are usually extremely inefficient on large datasets. This paper shows comparative analysis of frequently used classic graph clustering algorithms and well-known Girvan-Newman algorithm that is used for identification of communities in graphs, which is especially optimized for large datasets. The goal of the paper is to show which of the algorithms give best performances on given dataset. The paper presents real problem of data clustering, algorithms that can be used for its solution, methodology of analysis, results that were achieved and conclusions that were derived. This work is licensed under aCreative Commons Attribution-NonCommercial 4.0 International License.

An important problem in the analysis of network data is the detection of groups of densely interconnected nodes also called modules or communities. Community structure reveals functions and organizations of networks. Currently used algorithms for community detection in large-scale real-world networks are computationally expensive or require a priori information such as the number or sizes of communities or are not able to give the same resulting partition in multiple runs. In this paper we investigate a simple and fast algorithm that uses the network structure alone and requires neither optimization of pre-defined objective function nor information about number of communities. We propose a bottom up community detection algorithm in which starting from communities consisting of adjacent pairs of nodes and their maximal similar neighbors we find real communities. We show that the overall advantage of the proposed algorithm compared to the other community detection algorithms is its simple nature, low computational cost and its very high accuracy in detection communities of different sizes also in networks with blurred modularity structure consisting of poorly separated communities. All communities identified by the proposed method for facebook network and E-Coli transcriptional regulatory network have strong structural and functional coherence.

International audience; Due to the development and popularization of Internet, there is more and more research focusing on complex networks. Research shows that there exists community structure in complex networks. Finding out community structure helps to extract useful information in complex networks, so the research on community detection is becoming a hotspot in recent years. There are two remarkable problems in detecting communities. Firstly, the detection accuracy is normally not very high; Secondly, the assessment criteria are not very effective when real communities are unknown. This paper proposes an algorithm for community detection based on hierarchical clustering (CDHC Algorithm). CDHC Algorithm firstly creates initial communities from global central nodes, then expands the initial communities layer by layer according to the link strength between nodes and communities, and at last merges some very small communities into large communities. This paper also proposes the concept of extensive modularity, overcoming some weakness of modularity. The extensive modularity can better evaluate the effectiveness of algorithms for community detection. This paper verifies the advantage of extensive modularity through experiments and compares CDHC Algorithm and some other representative algorithms for community detection on some frequently used datasets, so as to verify the effectiveness and advantages of CDHC Algorithm.

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 of. 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 proteins, industrial sectors and groups of people) 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., Comment: The free academic research software, CFinder, used for the publication is available at the website of the publication: http://angel.elte.hu/clustering

There has been considerable recent interest in algorithms for finding communities in networks— groups of vertices within which connections are dense, but between which connections are sparser. Here we review the progress that has been made towards this end. We begin by describing some traditional methods of community detection, such as spectral bisection, the Kernighan-Lin algorithm and hierarchical clustering based on similarity measures. None of these methods, however, is ideal for the types of real-world network data with which current research is concerned, such as Internet and web data and biological and social networks. We describe a number of more recent algorithms that appear to work well with these data, including algorithms based on edge betweenness scores, on counts of short loops in networks and on voltage differences in resistor networks.

With the development of community detection algorithm, traditional clustering methods are hardly capable enough to meet our needs. This paper proposes a new algorithm based on spectral method that can identify the overlapping and hierarchical communities in complex networks. Experiments on artificial and real datasets are conducted to verify the algorithm by comparing it with other algorithms. The experimental results show that our algorithm can efficiently identify overlapping communities in complex networks and find hierarchies of the communities.

This article presents an efficient hierarchical clustering algorithm that solves the problem of core community detection. It is a variant of the standard community detection problem in which we are particularly interested in the connected core of communities. To provide a solution to this problem, we question standard definitions on communities and provide alternatives. We also propose a function called compactness, designed to assess the quality of a solution to this problem. Our algorithm is based on a graph traversal algorithm, the LexDFS. The time complexity of our method is in $O(n\times log(n))$. Experiments show that our algorithm creates highly compact clusters., 8 pages, 8 figures

Betweenness centrality is an important centrality measure widely used in social network analysis, route planning etc. However, even for mid-size networks, it is practically intractable to compute exact betweenness scores. In this paper, we propose a generic randomized framework for unbiased approximation of betweenness centrality. The proposed framework can be adapted with different sampling techniques and give diverse methods. We discuss the conditions a promising sampling technique should satisfy to minimize the approximation error and present a sampling method partially satisfying the conditions. We perform extensive experiments and show the high efficiency and accuracy of the proposed method.

The betweenness centrality index is essential in the analysis of social networks, but costly to compute. Currently, the fastest known algorithms require Theta(n3) time and Theta(n2) space, where n is the number of actors in the network. Motivated by the fast-growing need to compute centrality indices on large, yet very sparse, networks, new algorithms for betweenness are introduced in this paper. They require O(n + m) space and run in O(nm) and O(nm + n2 log n) time on unweighted and weighted networks, respectively, where m is the number of links. Experimental evidence is provided that this substantially increases the range of networks for which centrality analysis is feasible.

Real world networks often have community structure. It is characteristic that the groups of nodes are connected denser within themselves and rarely with each other. The Girvan-Newman method for the detection and analysis of community structure is based on the iterative elimination of edges with the highest number of the shortest paths that go through them. By eliminating edges the network breaks down into smaller networks, i.e. communities. This paper introduces improved Girvan- Newman method where multi-edge removal is allowed, and presents the results of the application of both methods to the existing real social network (Zachary karate club), the computergenerated network and the tumor genes and their mutations network. The improved algorithm in practice reduces the number of operations, but retains the same computational complexity, so it cannot be applied to networks with a very large number of nodes. The most important feature of our improvement is that the result is graph- theoretical invariant, while original algorithm depends on the vertex labeling.

Identification of communities in complex networks is an important topic and issue in many fields such as sociology, biology, and computer science. Communities are often defined as groups of related nodes or links that correspond to functional subunits in the corresponding complex systems. While most conventional approaches have focused on discovering communities of nodes, some recent studies start partitioning links to find overlapping communities straightforwardly. In this paper, we propose a new quantity function for link community identification in complex networks. Based on this quantity function we formulate the link community partition problem into an integer programming model which allows us to partition a complex network into overlapping communities. We further propose a genetic algorithm for link community detection which can partition a network into overlapping communities without knowing the number of communities. We test our model and algorithm on both artificial networks and real-world networks. The results demonstrate that the model and algorithm are efficient in detecting overlapping community structure in complex networks.

The increasing availability of dynamically growing digital data that can be used for extracting social networks has led to an upsurge of interest in the analysis of dynamic social networks. One key aspect of social network analysis is to understand the central nodes in a network. However, dynamic calculation of centrality values for rapidly growing networks might be unfeasibly expensive, especially if it involves recalculation from scratch for each time period. This paper proposes an incremental algorithm that effectively updates betweenness centralities of nodes in dynamic social networks while avoiding re-computations by exploiting information from earlier computations. Our performance results suggest that our incremental betweenness algorithm can achieve substantial performance speedup, on the order of thousands of times, over the state of the art, including the best-performing non-incremental betweenness algorithm and a recently proposed betweenness update algorithm.

The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of the same cluster and comparatively few edges joining vertices of different clusters. Such clusters, or communities, can be considered as fairly independent compartments of a graph, playing a similar role like, e. g., the tissues or the organs in the human body. Detecting communities is of great importance in sociology, biology and computer science, disciplines where systems are often represented as graphs. This problem is very hard and not yet satisfactorily solved, despite the huge effort of a large interdisciplinary community of scientists working on it over the past few years. We will attempt a thorough exposition of the topic, from the definition of the main elements of the problem, to the presentation of most methods developed, with a special focus on techniques designed by statistical physicists, from the discussion of crucial issues like the significance of clustering and how methods should be tested and compared against each other, to the description of applications to real networks., Comment: Review article. 103 pages, 42 figures, 2 tables. Two sections expanded + minor modifications. Three figures + one table + references added. Final version published in Physics Reports

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 method 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 2.6 million customers and by analyzing 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. ., 6 pages, 5 figures, 1 table; new version with new figures in order to clarify our method, where we look more carefully at the role played by the ordering of the nodes and where we compare our method with that of Wakita and Tsurumi

Many networks possess a community structure, such that vertices form densely connected groups which are more sparsely linked to other groups. In some cases these groups overlap, with some vertices shared between two or more communities. Discovering communities in networks is a computationally challenging task, especially if they overlap. In previous work we proposed an algorithm, CONGA, that could detect overlapping communities using the new concept of split betweenness. Here we present an improved algorithm based on a localform of betweenness, which yields good results but is much faster. It is especially effective in discovering small-diameter communities in large networks, and has a time complexity of only O(nlog n) for sparse networks.

Community structure is one of the most important features of real networks and reveals the internal organization of the nodes. Many algorithms have been proposed but the crucial issue of testing, i.e. the question of how good an algorithm is, with respect to others, is still open. Standard tests include the analysis of simple artificial graphs with a built-in community structure, that the algorithm has to recover. However, the special graphs adopted in actual tests have a structure that does not reflect the real properties of nodes and communities found in real networks. Here we introduce a new class of benchmark graphs, that account for the heterogeneity in the distributions of node degrees and of community sizes. We use this new benchmark to test two popular methods of community detection, modularity optimization and Potts model clustering. The results show that the new benchmark poses a much more severe test to algorithms than standard benchmarks, revealing limits that may not be apparent at a first analysis., Comment: 6 pages, 8 figures. Extended version published on Physical Review E. The code to build the new benchmark graphs can be downloaded from http://santo.fortunato.googlepages.com/inthepress2

Recent years have seen the development of many graph clustering algorithms, which can identify community structure in networks. The vast majority of these only find disjoint communities, but in many real-world networks communities overlap to some extent. We present a new algorithm for discovering overlapping communities in networks, by extending Girvan and Newman's well-known algorithm based on the betweennesscentrality measure. Like the original algorithm, ours performs hierarchical clustering -- partitioning a network into any desired number of clusters -- but allows them to overlap. Experiments confirm good performance on randomly generated networks based on a known overlapping community structure, and interesting results have also been obtained on a range of real-world networks.

Detecting community structure is fundamental to clarify the link between structure and function in complex networks and is used for practical applications in many disciplines. A successful method relies on the optimization of a quantity called modularity [Newman and Girvan, Phys. Rev. E 69, 026113 (2004)], which is a quality index of a partition of a network into communities. We find that modularity optimization may fail to identify modules smaller than a scale which depends on the total number L of links of the network and on the degree of interconnectedness of the modules, even in cases where modules are unambiguously defined. The probability that a module conceals well-defined substructures is the highest if the number of links internal to the module is of the order of \sqrt{2L} or smaller. We discuss the practical consequences of this result by analyzing partitions obtained through modularity optimization in artificial and real networks., 8 pages, 3 figures. Clarification of definition of community in Section II + minor revisions

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 networks. Specifically, we demonstrate that one can (i) find functional modules in complex networks, and (ii) 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 networks are among the most challenging biological networks and, arguably, the ones with more potential for immediate applicability. We use our method to analyze the metabolic networks of twelve organisms from three different super-kingdoms. 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 low-degree metabolites that connect different modules are more conserved than hubs whose links are mostly within a single module., 17 pages, 4 figures. Go to http://amaral.northwestern.edu for the PDF file of the reprint

We propose a novel method to find the community structure in complex networks based on an extremal optimization of the value of modularity. The method outperforms the optimal modularity found by the existing algorithms in the literature. We present the results of the algorithm for computer simulated and real networks and compare them with other approaches. The efficiency and accuracy of the method make it feasible to be used for the accurate identification of community structure in large complex networks., Comment: 4 pages, 4 figures

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 (md log n) where d is the depth of the dendrogram describing the community structure. Many real-world networks are sparse and hierarchical, with m approximately n and d approximately 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 on-line 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 x 10(6) edges. We show that our algorithm can extract meaningful communities from this network, revealing large-scale patterns present in the purchasing habits of customers.

It has been found that many networks display community structure -- groups of vertices within which connections are dense but between which they are sparser -- and highly sensitive computer algorithms have in recent years been developed for detecting such structure. These algorithms however are computationally demanding, which limits their application to small networks. Here we describe a new 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 50000 physicists., Comment: 5 pages, 4 figures

The investigation of community structures in networks is an important issue in many domains and disciplines. This problem is relevant for social tasks (objective analysis of relationships on the web), biological inquiries (functional studies in metabolic, cellular or protein networks) or technological problems (optimization of large infrastructures). Several types of algorithm exist for revealing the community structure in networks, but a general and quantitative definition of community is still lacking, leading to an intrinsic difficulty in the interpretation of the results of the algorithms without any additional non-topological information. In this paper we face this problem by introducing two quantitative definitions of community and by showing how they are implemented in practice in the existing algorithms. In this way the algorithms for the identification of the community structure become fully self-contained. Furthermore, we propose a new local algorithm to detect communities which outperforms the existing algorithms with respect to the computational cost, keeping the same level of reliability. The new algorithm is tested on artificial and real-world graphs. In particular we show the application of the new algorithm to a network of scientific collaborations, which, for its size, can not be attacked with the usual methods. This new class of local algorithms could open the way to applications to large-scale technological and biological applications., Revtex, final form, 14 pages, 6 figures

We propose a procedure for analyzing and characterizing complex networks. We apply this to the social network as constructed from email communications within a medium sized university with about 1700 employees. Email networks provide an accurate and nonintrusive description of the flow of information within human organizations. Our results reveal the self-organization of the network into a state where the distribution of community sizes is self-similar. This suggests that a universal mechanism, responsible for emergence of scaling in other self-organized complex systems, as, for instance, river networks, could also be the underlying driving force in the formation and evolution of social networks.

We propose and study a set of algorithms for discovering community structure in networks -- natural divisions of network nodes into densely connected subgroups. Our algorithms all share two definitive features: first, they involve iterative removal of edges from the network to split it into communities, the edges removed being identified using one of a number of possible "betweenness" measures, and second, these measures are, crucially, recalculated after each removal. We also propose a measure for the strength of the community structure found by our algorithms, which gives us an objective metric for choosing the number of communities into which a network should be divided. We demonstrate that our algorithms are highly effective at discovering community structure in both computer-generated and real-world network data, and show how they can be used to shed light on the sometimes dauntingly complex structure of networked systems., Comment: 16 pages, 13 figures