community detection algorithms a comparative analysis uncovering the community structure exhibited by real networks is a crucial step toward an understanding of complex systems that goes beyond the local organization of their constituents many algorithms have been proposed so far but none of them has been subjected to strict tests to evaluate their performance most of the sporadic tests performed so far involved small networks with known community structure and or artificial graphs with a simplified structure which is very uncommon in real systems here we test several methods against a recently introduced class of benchmark graphs with heterogeneous distributions of degree and community size the methods are also tested against the benchmark by girvan and newman proc natl acad sci u s a 99 7821 2002 and on random graphs as a result of our analysis three recent algorithms introduced by rosvall and bergstrom proc natl acad sci u s a 104 7327 2007 proc natl acad sci u s a 105 1118 2008 blondel et al j stat mech theory exp 2008 p10008 and ronhovde and nussinov phys rev e 80 016109 2009 have an excellent performance with the additional advantage of low computational complexity which enables one to analyze large systems

community detection in graphs using singular value decomposition a spectral algorithm for community detection is presented the algorithm consists of three stages 1 matrix factorization of two matrix forms square signless laplacian for unipartite graphs and rectangular adjacency matrix for bipartite graphs using singular value decompostion svd 2 dimensionality reduction using an optimal linear approximation and 3 clustering vertices using dot products in reduced dimensional space the algorithm reveals communities in graphs without placing any restriction on the input network type or the output community type it is applicable on unipartite or bipartite unweighted or weighted networks it places no requirement on strict community membership and automatically reveals the number of clusters their respective sizes and overlaps and hierarchical modular organization by representing vertices as vectors in real space expressed as linear combinations of the orthogonal bases described by svd orthogonality becomes the metric for classifying vertices into communities results on several test and real world networks are presented including cases where a mix of disjointed overlapping or hierarchical communities are known to exist in the network

benchmarks for testing community detection algorithms on directed and weighted graphs with overlapping communities many complex networks display a mesoscopic structure with groups of nodes sharing many links with the other nodes in their group and comparatively few with nodes of different groups this feature is known as community structure and encodes precious information about the organization and the function of the nodes many algorithms have been proposed but it is not yet clear how they should be tested recently we have proposed a general class of undirected and unweighted benchmark graphs with heterogeneous distributions of node degree and community size an increasing attention has been recently devoted to develop algorithms able to consider the direction and the weight of the links which require suitable benchmark graphs for testing in this paper we extend the basic ideas behind our previous benchmark to generate directed and weighted networks with built in community structure we also consider the possibility that nodes belong to more communities a feature occurring in real systems such as social networks as a practical application we show how modularity optimization performs on our benchmark

fuzzy communities and the concept of bridgeness in complex networks we consider the problem of fuzzy community detection in networks which complements and expands the concept of overlapping community structure our approach allows each vertex of the graph to belong to multiple communities at the same time determined by exact numerical membership degrees even in the presence of uncertainty in the data being analyzed we create an algorithm for determining the optimal membership degrees with respect to a given goal function based on the membership degrees we introduce a measure that is able to identify outlier vertices that do not belong to any of the communities bridge vertices that have significant membership in more than one single community and regular vertices that fundamentally restrict their interactions within their own community while also being able to quantify the centrality of a vertex with respect to its dominant community the method can also be used for prediction in case of uncertainty in the data set analyzed the number of communities can be given in advance or determined by the algorithm itself using a fuzzified variant of the modularity function the technique is able to discover the fuzzy community structure of different real world networks including but not limited to social networks scientific collaboration networks and cortical networks with high confidence

detect overlapping and hierarchical community structure in networks abstract 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

nonstandard fuzzy sets abstract given a set x we take into consideration the lattice f x 0 1 of the nonstandard fuzzy subsets of x that is the l subsets with l equal to the unitary interval 0 1 of a nonstandard model of analysis to show the appropriateness of such a concept we give two examples of vague concepts positive divergence for functions and vagueness for fuzzy sets that are representable by suitable nonstandard fuzzy sets one proves that they are not representable by zadeh s fuzzy sets also we observe that the same operations and relations defined for fuzzy sets are definable for nonstandard fuzzy sets in particular the complementation operation and the sharpening relation finally one proves that every l subset with l totally ordered is a nonstandard fuzzy set

structural and functional discovery in dynamic networks with non negative matrix factorization time series of graphs are increasingly prevalent in modern data and pose unique challenges to visual exploration and pattern extraction this paper describes the development and application of matrix factorizations for exploration and time varying community detection in time evolving graph sequences the matrix factorization model allows the user to home in on and display interesting underlying structure and its evolution over time the methods are scalable to weighted networks with a large number of time points or nodes and can accommodate sudden changes to graph topology our techniques are demonstrated with several dynamic graph series from both synthetic and real world data including citation and trade networks these examples illustrate how users can steer the techniques and combine them with existing methods to discover and display meaningful patterns in sizable graphs over many time points

performance of modularity maximization in practical contexts although widely used in practice the behavior and accuracy of the popular module identification technique called modularity maximization is not well understood in practical contexts here we present a broad characterization of its performance in such situations first we revisit and clarify the resolution limit phenomenon for modularity maximization second we show that the modularity function q exhibits extreme degeneracies it typically admits an exponential number of distinct high scoring solutions and typically lacks a clear global maximum third we derive the limiting behavior of the maximum modularity q max for one model of infinitely modular networks showing that it depends strongly both on the size of the network and on the number of modules it contains finally using three real world metabolic networks as examples we show that the degenerate solutions can fundamentally disagree on many but not all partition properties such as the composition of the largest modules and the distribution of module sizes these results imply that the output of any modularity maximization procedure should be interpreted cautiously in scientific contexts they also explain why many heuristics are often successful at finding high scoring partitions in practice and why different heuristics can disagree on the modular structure of the same network we conclude by discussing avenues for mitigating some of these behaviors such as combining information from many degenerate solutions or using generative models

the similarity of weights on edges and discovering of community structure abstract in this paper we propose a weighted modularity q w based on the similarity of weights on edges and a threshold coefficient to evaluate the equivalence of edge weights simulations on benchmark networks and real networks show that optimization on the modularity enable us to obtain groups of nodes within which the edge weights are distributed uniformly but at random between them the communities can reveal the uniform connections stable relationships measured by the similarity of weights on edges between nodes or some similarity between nodes functions furthermore with the dynamical moving of we observe that optimization on the q w allows for the discovering of a special hierarchical organization which reveals different levels of uniform connections between nodes in networks the substructures revealed by the hierarchical organization enable us to obtain more information of networks and give a potential way for partly remedying the intrinsic resolution problem of modularity

line graphs link partitions and overlapping communities in this paper we use a partition of the links of a network in order to uncover its community structure this approach allows for communities to overlap at nodes so that nodes may be in more than one community we do this by making a node partition of the line graph of the original network in this way we show that any algorithm that produces a partition of nodes can be used to produce a partition of links we discuss the role of the degree heterogeneity and propose a weighted version of the line graph in order to account for this

a model framework for the enhancement of community detection in complex networks abstract community detection is an important data analysis problem in many different areas and how to enhance the quality of community detection in complicated real applications is still a challenge current community detection enhancement methods often take the enhancement as a preprocess of community detection they mainly focus on how to design the suitable topological similarity of nodes to adjust the original network but did not consider how to make use of this topological similarity more effectively in order to better utilize the similarity information we propose a model framework which integrates the enhancement into the whole community detection procedure first we calculate the structural similarity of nodes based on network topology second we present a stochastic model to describe the community memberships of nodes we then model the strong constraint based on structural similarity i e we make each node have the same community membership distribution with its most similar neighbors and then we model the weak constraint i e if two nodes have a high similarity we will make their community membership distributions close otherwise we will make them not close finally we present a nonnegative matrix factorization approach to learn the model parameters we evaluate our method on both synthetic and real world networks with ground truths and compare it with five comparable methods the experimental results demonstrate the superior performance of our new method over the competing ones for community detection and enhancement

two stage algorithm using influence coefficient for detecting the hierarchical non overlapping and overlapping community structure abstract community detection is one of the most important problems in complex networks many algorithms have been proposed in the last decade and most of them focus on the non overlapping community structures in the early days overlapping and hierarchical structures are another two important properties in complex networks which have attracted researchers extensive concern in recent years in this paper we proposed a two stage method which can detect the hierarchical non overlapping and overlapping community structures in complex networks in this method the cnm algorithm a fast hierarchical agglomerative algorithm proposed by clauset newman and moore is used in the first stage in the second stage a new evaluation function named as influence coefficient based on the local community structure is proposed which can get the overlapping community structures at different overlapping levels by adjusting a tunable parameter besides the proposed evaluation function can detect the wrongly classified nodes in the partition of the first stage and correct them finally the computational complexity of the algorithm is low the experimental results on both synthetic and real world network datasets show the efficiency of our method

stochastic blockmodels and community structure in networks stochastic blockmodels have been proposed as a tool for detecting community structure in networks as well as for generating synthetic networks for use as benchmarks most blockmodels however ignore variation in vertex degree making them unsuitable for applications to real world networks which typically display broad degree distributions that can significantly affect the results here we demonstrate how the generalization of blockmodels to incorporate this missing element leads to an improved objective function for community detection in complex networks we also propose a heuristic algorithm for community detection using this objective function or its non degree corrected counterpart and show that the degree corrected version dramatically outperforms the uncorrected one in both real world and synthetic networks

developments in the theory of randomized shortest paths with a comparison of graph node distances there have lately been several suggestions for parametrized distances on a graph that generalize the shortest path distance and the commute time or resistance distance the need for developing such distances has risen from the observation that the above mentioned common distances in many situations fail to take into account the global structure of the graph in this article we develop the theory of one family of graph node distances known as the randomized shortest path dissimilarity which has its foundation in statistical physics we show that the randomized shortest path dissimilarity can be easily computed in closed form for all pairs of nodes of a graph moreover we come up with a new definition of a distance measure that we call the free energy distance the free energy distance can be seen as an upgrade of the randomized shortest path dissimilarity as it defines a metric in addition to which it satisfies the graph geodetic property the derivation and computation of the free energy distance are also straightforward we then make a comparison between a set of generalized distances that interpolate between the shortest path distance and the commute time or resistance distance this comparison focuses on the applicability of the distances in graph node clustering and classification the comparison in general shows that the parametrized distances perform well in the tasks in particular we see that the results obtained with the free energy distance are among the best in all the experiments

finding community structure in networks using the eigenvectors of matrices we consider the problem of detecting communities or modules in networks groups of vertices with a higher than average density of edges connecting them previous work indicates that a robust approach to this problem is the maximization of the benefit function known as modularity over possible divisions of a network here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix which plays a role in community detection similar to that played by the graph laplacian in graph partitioning calculations this result leads us to a number of possible algorithms for detecting community structure as well as several other results including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong the algorithms and measures proposed are illustrated with applications to a variety of real world complex networks

efficient modularity density heuristics for large graphs abstract modularity density maximization is a community detection optimization problem which improves the resolution limit degeneracy of modularity maximization this paper presents seven scalable heuristics for modularity density and compares them with literature results from exact mixed integer linear programming and gaod imeme net hain and bmd heuristics the results are also compared with cnm and louvain which are scalable heuristics for modularity maximization the results suggest that our seven heuristics are faster than gaod imeme net hain and bmd modularity density heuristics our experiments also show that some of our heuristics surpassed the objective function value reported by imeme net hain and bmd for some real graphs our seven heuristics were tested with real graphs from the stanford large network dataset collection and the experiments show that they are scalable this feature was confirmed by an amortized complexity analysis which reveals average linear time for three of our heuristics hypothesis tests suggest that four proposed heuristics are state of the art since they are scalable for hundreds of thousands of nodes for the modularity density problem and they find the high objective value partitions for the largest instances ground truth experiments in artificial random graphs were performed and suggest that our heuristics lead to better cluster detection than both cnm and louvain

overlapping community detection using neighborhood ratio matrix abstract the participation of a node in more than one community is a common phenomenon in complex networks however most existing methods fail to identify nodes with multiple community affiliation correctly in this paper a unique method to define overlapping community in complex networks is proposed using the overlapping neighborhood ratio to represent relations between nodes matrix factorization is then utilized to assign nodes into their corresponding community structures moreover the proposed method demonstrates the use of perron clusters to estimate the number of overlapping communities in a network experimental results in real and artificial networks show with great accuracy that the proposed method succeeds to recover most of the overlapping communities existing in the network

detecting communities in clustered networks based on group action on set abstract in this paper we propose a well targeted algorithm gas algorithm for detecting communities in high clustered networks by presenting group action technology on community division during the processing of this algorithm the underlying community structure of a clustered network emerges simultaneously as the corresponding partition of orbits by the permutation groups acting on the node set are achieved as the derivation of the orbit partition an algebraic structure r cycle can be considered as the origin of the community to be a priori estimation for the community structure of the algorithm the community separability is introduced to indicate whether a network has distinct community structure by executing the algorithm on several typical networks and the lfr benchmark it shows that this gas algorithm can detect communities accurately and effectively in high clustered networks furthermore we compare the gas algorithm and the clique percolation algorithm on the lfr benchmark it is shown that the gas algorithm is more accurate at detecting non overlapping communities in clustered networks it is suggested that algebraic techniques can uncover fresh light on detecting communities in complex networks

finding communities in directed networks by pagerank random walk induced network embedding abstract community structure has been found to exist ubiquitously in many different kinds of real world complex networks most of the previous literature ignores edge directions and applies methods designed for community finding in undirected networks to find communities here we address the problem of finding communities in directed networks our proposed method uses pagerank random walk induced network embedding to transform a directed network into an undirected one where the information on edge directions is effectively incorporated into the edge weights starting from this new undirected weighted network previously developed methods for undirected network community finding can be used without any modification moreover our method improves on recent work in terms of community definition and meaning we provide two simulated examples a real social network and different sets of power law benchmark networks to illustrate how our method can correctly detect communities in directed networks

exploring triad rich substructures by graph theoretic characterizations in complex networks abstract one of the most important problems in complex networks is how to detect communities accurately the main challenge lies in the fact that traditional definition about communities does not always capture the intrinsic features of communities motivated by the observation that communities in ppi networks tend to consist of an abundance of interacting triad motifs we define a 2 club substructure with diameter 2 possessing triad rich property to describe a community based on the triad rich substructure we design a div ision a lgorithm using our proposed edge n iche c entrality divanc to detect communities effectively in complex networks we also extend divanc to detect overlapping communities by proposing a simple 2 hop overlapping strategy to verify the effectiveness of triad rich substructures we compare divanc with existing algorithms on ppi networks lfr synthetic networks and football networks the experimental results show that divanc outperforms most other algorithms significantly and in particular can detect sparse communities

detecting the optimal number of communities in complex networks abstract 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 c is proposed to detect the optimal number of communities in general when 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 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 c is effective in both artificial and real world networks

asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications in this paper we extend our previous work on the stochastic block model a commonly used generative model for social and biological networks and the problem of inferring functional groups or communities from the topology of the network we use the cavity method of statistical physics to obtain an asymptotically exact analysis of the phase diagram we describe in detail properties of the detectability undetectability phase transition and the easy hard phase transition for the community detection problem our analysis translates naturally into a belief propagation algorithm for inferring the group memberships of the nodes in an optimal way i e that maximizes the overlap with the underlying group memberships and learning the underlying parameters of the block model finally we apply the algorithm to two examples of real world networks and discuss its performance

benchmark graphs for testing community detection algorithms 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 class of benchmark graphs that account for the heterogeneity in the distributions of node degrees and of community sizes we use this benchmark to test two popular methods of community detection modularity optimization and potts model clustering the results show that the benchmark poses a much more severe test to algorithms than standard benchmarks revealing limits that may not be apparent at a first analysis

finding communities in networks in the strong and almost strong sense finding communities or clusters or modules in networks can be done by optimizing an objective function defined globally and or by specifying conditions which must be satisfied by all communities radicchi et al proc natl acad sci usa 101 2658 2004 define a susbset of vertices of a network to be a community in the strong sense if each vertex of that subset has a larger inner degree than its outer degree a partition in the strong sense has only strong communities in this paper we first define an enumerative algorithm to list all partitions in the strong sense of a network of moderate size the results of this algorithm are given for the zachary karate club data set which is solved by hand as well as for several well known real world problems of the literature moreover this algorithm is slightly modified in order to apply it to larger networks keeping only partitions with the largest number of communities it is shown that some of the partitions obtained are informative although they often have only a few communities while they fail to give any information in other cases having only one community it appears that degree 2 vertices play a big role in forcing large inhomogeneous communities therefore a weakening of the strong condition is proposed and explored we define a partition in the almost strong sense by substituting a nonstrict inequality to a strict one in the definition of strong community for all vertices of degree 2 results for the same set of problems as before then give partitions with a larger number of communities and are more informative

proof of the achievability conjectures for the general stochastic block model in a paper that initiated the modern study of the stochastic block model sbm decelle krzakala moore and zdeborova backed by mossel neeman and sly conjectured that detecting clusters in the symmetric sbm in polynomial time is always possible above the kesten stigum ks threshold while it is possible to detect clusters information theoretically i e not necessarily in polynomial time below the ks threshold when the number of clusters k is at least 4 massoulie mossel et al and bordenave lelarge and massoulie proved that the ks threshold is in fact efficiently achievable for k 2 while mossel et al proved that it cannot be crossed at k 2 the above conjecture remained open for k 3 this paper proves the two parts of the conjecture further extending the results to general sbms for the efficient part an approximate acyclic belief propagation abp algorithm is developed and proved to detect communities for any k down to the ks threshold in quasi linear time achieving this requires showing optimality of abp in the presence of cycles a challenge for message passing algorithms the paper further connects abp to a power iteration method on a nonbacktracking operator of generalized order formalizing the interplay between message passing and spectral methods for the information theoretic part a nonefficient algorithm sampling a typical clustering is shown to break down the ks threshold at k 4 the emerging gap is shown to be large in some cases making the sbm a good case study for information computation gaps 2017 wiley periodicals inc

network community detection enhancement by proper weighting in this paper we show how proper assignment of weights to the edges of a complex network can enhance the detection of communities and how it can circumvent the resolution limit and the extreme degeneracy problems associated with modularity our general weighting scheme takes advantage of graph theoretic measures and it introduces two heuristics for tuning its parameters we use this weighting as a preprocessing step for the greedy modularity optimization algorithm of newman to improve its performance the result of the experiments of our approach on computer generated and real world data networks confirm that the proposed approach not only mitigates the problems of modularity but also improves the modularity optimization

on fuzzy convergence abstract we propose and examine two concepts of convergence for sequences of elements of a set x with respect to a fuzzy topology on x such concepts are obtained by interpreting in a multivalued logic two different first order formulas expressing the classical convergence

phase transition model for community detection abstract 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 o f by removing the detected communities and the nodes in 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

map equation for link communities community structure exists in many real world networks and has been reported being related to several functional properties of the networks the conventional approach was partitioning nodes into communities while some recent studies start partitioning links instead of nodes to find overlapping communities of nodes efficiently we extended the map equation method which was originally developed for node communities to find link communities in networks this method is tested on various kinds of networks and compared with the metadata of the networks and the results show that our method can identify the overlapping role of nodes effectively the advantage of this method is that the node community scheme and link community scheme can be compared quantitatively by measuring the unknown information left in the networks besides the community structure it can be used to decide quantitatively whether or not the link community scheme should be used instead of the node community scheme furthermore this method can be easily extended to the directed and weighted networks since it is based on the random walk

finding overlapping communities using seed set abstract the local optimization algorithm using seed set to find overlapping communities has become more and more a significant method but it is a great challenge how to choose a good seed set in this paper a new method is proposed to achieve the choice of candidate seed sets and yields a new algorithm to find overlapping communities in complex networks by testing in real world networks and synthetic networks this method can successfully detect overlapping communities and outperform other state of the art overlapping community detection methods

hyperbolicity measures democracy in real world networks in this work we analyze the hyperbolicity of real world networks a geometric quantity that measures if a space is negatively curved we provide two improvements in our understanding of this quantity first of all in our interpretation a hyperbolic network is aristocratic since few elements connect the system while a non hyperbolic network has a more democratic structure with a larger number of crucial elements the second contribution is the introduction of the average hyperbolicity of the neighbors of a given node through this definition we outline an influence area for the vertices in the graph we show that in real networks the influence area of the highest degree vertex is small in what we define local networks i e social or peer to peer networks and large in global networks i e power grid metabolic networks or autonomous system networks

ganc greedy agglomerative normalized cut for graph clustering this paper describes a graph clustering algorithm that aims to minimize the normalized cut criterion and has a model order selection procedure the performance of the proposed algorithm is comparable to spectral approaches in terms of minimizing normalized cut however unlike spectral approaches the proposed algorithm scales to graphs with millions of nodes and edges the algorithm consists of three components that are processed sequentially a greedy agglomerative hierarchical clustering procedure model order selection and a local refinement for a graph of n nodes and o n edges the computational complexity of the algorithm is o nlog 2n a major improvement over the o n 3 complexity of spectral methods experiments are performed on real and synthetic networks to demonstrate the scalability of the proposed approach the effectiveness of the model order selection procedure and the performance of the proposed algorithm in terms of minimizing the normalized cut metric

identification of overlapping and non overlapping community structure by fuzzy clustering in complex networks this paper proposes a novel method based on fuzzy clustering to detect community structure in complex networks in contrast to previous studies our method does not focus on a graph model but rather on a fuzzy relation model which uses the operations of fuzzy relation to replace a traversal search of the graph for identifying community structure in our method we first use a fuzzy relation to describe the relation between vertices as well as the similarity in network topology to determine the membership grade of the relation then we transform this fuzzy relation into a fuzzy equivalence relation finally we map the non overlapping communities as equivalence classes that satisfy a certain equivalence relation because most real world networks are made of overlapping communities e g in social networks people may belong to multiple communities we can consider the equivalence classes above as the skeletons of overlapping communities and extend our method by adding vertices to the skeletons to identify overlapping communities we evaluated our method on artificial networks with built in communities and real world networks with known and unknown communities the experimental results show that our method works well for detecting these communities and gives a new understanding of network division and community formation

overlapping community detection in complex networks using symmetric binary matrix factorization discovering overlapping community structures is a crucial step to understanding the structure and dynamics of many networks in this paper we develop a symmetric binary matrix factorization model to identify overlapping communities our model allows us not only to assign community memberships explicitly to nodes but also to distinguish outliers from overlapping nodes in addition we propose a modified partition density to evaluate the quality of community structures we use this to determine the most appropriate number of communities we evaluate our methods using both synthetic benchmarks and real world networks demonstrating the effectiveness of our approach

finding community structure in very large networks 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 phantom rule 0 2em 0ex mathrm log phantom rule 0 2em 0ex n where d is the depth of the dendrogram describing the community structure many real world networks are sparse and hierarchical with m ensuremath sim n and d ensuremath sim mathrm log phantom rule 0 2em 0ex n in which case our algorithm runs in essentially linear time o n phantom rule 0 2em 0ex mathrm log 2 phantom rule 0 2em 0ex 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 ifmmode times else texttimes fi 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

exploring the structural regularities in networks in this paper we consider the problem of exploring structural regularities of networks by dividing the nodes of a network into groups such that the members of each group have similar patterns of connections to other groups specifically we propose a general statistical model to describe network structure in this model a group is viewed as a hidden or unobserved quantity and it is learned by fitting the observed network data using the expectation maximization algorithm compared with existing models the most prominent strength of our model is the high flexibility this strength enables it to possess the advantages of existing models and to overcome their shortcomings in a unified way as a result not only can broad types of structure be detected without prior knowledge of the type of intrinsic regularities existing in the target network but also the type of identified structure can be directly learned from the network moreover by differentiating outgoing edges from incoming edges our model can detect several types of structural regularities beyond competing models tests on a number of real world and artificial networks demonstrate that our model outperforms the state of the art model in shedding light on the structural regularities of networks including the overlapping community structure multipartite structure and several other types of structure which are beyond the capability of existing models

estimating mixed memberships with sharp eigenvector deviations we consider the problem of estimating community memberships of nodes in a network where every node is associated with a vector determining its degree of membership in each community existing prov

entropy of stochastic blockmodel ensembles stochastic blockmodels are generative network models where the vertices are separated into discrete groups and the probability of an edge existing between two vertices is determined solely by their group membership in this paper we derive expressions for the entropy of stochastic blockmodel ensembles we consider several ensemble variants including the traditional model as well as the newly introduced degree corrected version karrer et al phys rev e 83 016107 2011 which imposes a degree sequence on the vertices in addition to the block structure the imposed degree sequence is implemented both as soft constraints where only the expected degrees are imposed and as hard constraints where they are required to be the same on all samples of the ensemble we also consider generalizations to multigraphs and directed graphs we illustrate one of many applications of this measure by directly deriving a log likelihood function from the entropy expression and using it to infer latent block structure in observed data due to the general nature of the ensembles considered the method works well for ensembles with intrinsic degree correlations i e with entropic origin as well as extrinsic degree correlations which go beyond the block structure

community mining with new node similarity by incorporating both global and local topological knowledge in a constrained random walk abstract detection of community is a crucial step to understand the structure and dynamics of complex networks most of conventional community detection methods focus on optimizing a certain objective function or on clustering nodes based on their similarities which leads to a phenomenon that they have preference for specific types of networks but are not general using constrained random walk we exploit global and local topology structures of network to propose a modified transition matrix and further to define a new similarity metric named isim between two nodes in contrast to the existing similarities isim does not work directly on the observed data but in a convergent stable space this feature makes isim robust to the observed noisy data in real world networks isim not only measures node s distance but also captures node s topology structure in network experiments on synthetic and real world networks demonstrate that isim can be successfully applied to community detection in broader types of networks and outperforms other community detection methods

communities and bottlenecks trees and treelike networks have high modularity much effort has gone into understanding the modular nature of complex networks communities also known as clusters or modules are typically considered to be densely interconnected groups of nodes that are only sparsely connected to other groups in the network discovering high quality communities is a difficult and important problem in a number of areas the most popular approach is the objective function known as modularity used both to discover communities and to measure their strength to understand the modular structure of networks it is then crucial to know how such functions evaluate different topologies what features they account for and what implicit assumptions they may make we show that trees and treelike networks can have unexpectedly and often arbitrarily high values of modularity this is surprising since trees are maximally sparse connected graphs and are not typically considered to possess modular structure yet the nonlocal null model used by modularity assigns low probabilities and thus high significance to the densities of these sparse tree communities we further study the practical performance of popular methods on model trees and on a genealogical data set and find that the discovered communities also have very high modularity often approaching its maximum value statistical tests reveal the communities in trees to be significant in contrast with known results for partitions of sparse random graphs

on equivalence of likelihood maximization of stochastic block model and constrained nonnegative matrix factorization abstract community structures detection in complex network is important for understanding not only the topological structures of the network but also the functions of it stochastic block model and nonnegative matrix factorization are two widely used methods for community detection which are proposed from different perspectives in this paper the relations between them are studied the logarithm of likelihood function for stochastic block model can be reformulated under the framework of nonnegative matrix factorization despite model equivalence the algorithms employed by the two methods are different preliminary numerical experiments are carried out to compare the behaviors of the algorithms demonstrating that the multiplicative update rules for nmf are more effective

discriminative metric learning for multi view graph partitioning we propose discriminative metric learning for multi view graph partitioning we envision the multi view graph as an adaptive dynamic system the intra view connections and the inter view couplings are interplayed extensive experiments have been conducted to show the effectiveness in recent years multi view graph partitioning has attracted more and more attention but most efforts have been made to develop graph partitioning approaches directly in the original topological structure in many real world applications graph may contain noisy links and the distance metric may not be so discriminative for revealing cluster structure this paper addresses the problem of discriminative metric learning for multi view graph partitioning in particular we propose a novel method called multi view dml abbr of multi view discriminative metric learning to transform the metric space in the original graph into a more discriminative metric space in which better graph partitioning results will be obtained we envision the multi view graph as an adaptive dynamic system where both the intra view connections and the inter view couplings are interplayed to gradually update the relation metric among nodes on the one hand the inter view coupling will be influenced by the intra view connection between two nodes on the other hand the inter view coupling will also affect the intra view connection such interplay eventually makes the whole graph reach a steady state which has a stronger cluster structure than the original graph extensive experiments are conducted on both synthetic and real world graphs to confirm that the proposed method is able to learn more discriminative metric

efficient and principled method for detecting communities in networks a fundamental problem in the analysis of network data is the detection of network communities groups of densely interconnected nodes which may be overlapping or disjoint here we describe a method for finding overlapping communities based on a principled statistical approach using generative network models we show how the method can be implemented using a fast closed form expectation maximization algorithm that allows us to analyze networks of millions of nodes in reasonable running times we test the method both on real world networks and on synthetic benchmarks and find that it gives results competitive with previous methods we also show that the same approach can be used to extract nonoverlapping community divisions via a relaxation method and demonstrate that the algorithm is competitively fast and accurate for the nonoverlapping problem

minimum state unsteady aerodynamic shape sensitivities using sensitivity of optimal solutions to problem parameters parameters involved including the aerodynamic lag roots its solution is unique and convergence of the solution process based on standard mathematical programming techniques allows accurate determination of all unknown variables an improvement of the math programming minimum state fitting is presented here and it is developed further to include sensitivity analysis for the resulting minimum state series free of the nonuniqueness and convergence issues that plagued previous attempts to obtain sensitivities of minimum state fits a variable planform double swept forward wing case is used to demonstrate performance of the fitting technique itself shape sensitivity calculations and taylor series based approximations that can be used to replace detailed full order analysis in the course of multidisciplinary design optimization of flight vehicles nomenclature a0 a1 a2 minimum state approximation matrices corresponding to aerodynamic stiffness damping and inertia respectively a 0 a 1 a 2 elements of the matrices a0 a1 a2 b reference semichord length of wing c structural viscous damping matrix c reference chord length of wing d e minimum state approximation matrices

inversion method for content based networks in this paper we generalize a recently introduced expectation maximization em method for graphs and apply it to content based networks the em method provides a classification of the nodes of a graph and allows one to infer relations between the different classes content based networks are ideal models for graphs displaying any kind of community and or multipartite structure we show both numerically and analytically that the generalized em method is able to recover the process that led to the generation of such networks we also investigate the conditions under which our generalized em method can recover the underlying contentbased structure in the presence of randomness in the connections two entropies sq and sc are defined to measure the quality of the node classification and to what extent the connectivity of a given network is content based sq and sc are also useful in determining the number of classes for which the classification is optimal

spectra of random graphs with community structure and arbitrary degrees using methods from random matrix theory researchers have recently calculated the full spectra of random networks with arbitrary degrees and with community structure both reveal interesting spectral features including deviations from the wigner semicircle distribution and phase transitions in the spectra of community structured networks in this paper we generalize both calculations giving a prescription for calculating the spectrum of a network with both community structure and an arbitrary degree distribution in general the spectrum has two parts a continuous spectral band which can depart strongly from the classic semicircle form and a set of outlying eigenvalues that indicate the presence of communities

identification of overlapping community structure in complex networks using fuzzy c means clustering abstract 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

fuzzy overlapping community detection based on local random walk and multidimensional scaling a b s t r a c t a fuzzy overlapping community is an important kind of overlapping community in which each node belongs to each community to different extents it exists in many real networks but how to identify a fuzzy overlapping community is still a challenging task in this work the concept of local random walk and a new distance metric are introduced based on the new distance measurement the dissimilarity index between each node of a network is calculated firstly then in order to keep the original node distance as much as possible the network structure is mapped into low dimensional space by the multidimensional scaling mds finally the fuzzy c means clustering is employed to find fuzzy communities in a network the experimental results show that the proposed algorithm is effective and efficient to identify the fuzzy overlapping communities in both artificial networks and real world networks

the architecture of complex weighted networks networked structures arise in a wide array of different contexts such as technological and transportation infrastructures social phenomena and biological systems these highly interconnected systems have recently been the focus of a great deal of attention that has uncovered and characterized their topological complexity along with a complex topological structure real networks display a large heterogeneity in the capacity and intensity of the connections these features however have mainly not been considered in past studies where links are usually represented as binary states i e either present or absent here we study the scientific collaboration network and the world wide air transportation network which are representative examples of social and large infrastructure systems respectively in both cases it is possible to assign to each edge of the graph a weight proportional to the intensity or capacity of the connections among the various elements of the network we define appropriate metrics combining weighted and topological observables that enable us to characterize the complex statistical properties and heterogeneity of the actual strength of edges and vertices this information allows us to investigate the correlations among weighted quantities and the underlying topological structure of the network these results provide a better description of the hierarchies and organizational principles at the basis of the architecture of weighted networks

bayesian nonparametric models for community detection we propose a series of bayesian nonparametric statistical models for community detection in graphs we model the probability of the presence or absence of edges within the graph using these models we naturally incorporate uncertainty and variability and take advantage of nonparametric techniques such as the chinese restaurant process and the dirichlet process some of the contributions include a the community structure is directly modeled without specifying the number of communities a priori b the probabilities of edges within or between communities may be modeled as varying by community or pairs of communities c some nodes can be classified as not belonging to any community and d bayesian model diagnostics are used to compare models and help with appropriate model selection we start by fitting an initial model to a well known network dataset and we develop a series of increasingly complex models we propose markov chain monte carlo algorithms to carry out the estimation as well as an app

clustering and outlier detection using isoperimetric number of trees we propose a graph based data clustering algorithm which is based on exact clustering of a minimum spanning tree in terms of a minimum isoperimetry criteria we show that our basic clustering algorithm runs in o nlogn and with post processing in almost o nlogn average case and o n 2 worst case time where n is the size of the data set it is also shown that our generalized graph model which also allows the use of potentials at vertices can be used to extract an extra piece of information related to anomalous data patterns and outliers in this regard we propose an algorithm that extracts outliers in parallel to data clustering we also provide a comparative performance analysis of our algorithms with other related ones and we show that they behave quite effectively on hard synthetic data sets as well as real world benchmarks

nonparametric weighted stochastic block models we present a bayesian formulation of weighted stochastic block models that can be used to infer the large scale modular structure of weighted networks including their hierarchical organization our method is nonparametric and thus does not require the prior knowledge of the number of groups or other dimensions of the model which are instead inferred from data we give a comprehensive treatment of different kinds of edge weights i e continuous or discrete signed or unsigned bounded or unbounded as well as arbitrary weight transformations and describe an unsupervised model selection approach to choose the best network description we illustrate the application of our method to a variety of empirical weighted networks such as global migrations voting patterns in congress and neural connections in the human brain

aeroservoelastic aspects of wing control surface planform shape optimization equivalent plate structural modeling and doublet point lifting surface unsteady aerodynamics are used to obtain analytic sensitivities of aeroelastic and aeroservoelastic response with respect to wing and control surface planform shape parameters rational function approximations for unsteady aerodynamic forces their shape sensitivities and the resulting linear time invariant state space models of aeroservoelastic systems and their shape sensitivities are examined the goal is to develop effective and numerically efficient approximation techniques for wing shape optimization for use with nonlinear programming and approximation concepts as a multidisciplinary optimization strategy effects of structural and unsteady aerodynamic modeling errors are studied examination of approximation accuracy using alternative approximation techniques and the resulting move limits provide insight and experience on the way to realistic wing control surface shape optimization with active controls and aeroservoelastic constraints

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

graph wavelets for multiscale community mining we develop a signal processing approach to the multiscale detection of communities in networks that is of groups of nodes well connected together the method relies on carefully engineered wavelets on graphs to introduce the notion of scale and to obtain a local view of the graph from each node computing the correlations between wavelets centered at different nodes one has access to a notion of similarity between nodes thereby enabling a clustering procedure that groups nodes according to their community at the scale of analysis by using a collection of random vectors to estimate the correlation between the nodes we show that the method is suitable for the analysis of large graphs furthermore we introduce a notion of partition stability and a statistical test allowing us to assess which scales of analysis of the network are relevant the effectiveness of the method is discussed first on multiscale graph benchmarks then on real data of social networks and on models for signal processing on graphs

inferring the mesoscale structure of layered edge valued and time varying networks many network systems are composed of interdependent but distinct types of interactions which cannot be fully understood in isolation these different types of interactions are often represented as layers attributes on the edges or as a time dependence of the network structure although they are crucial for a more comprehensive scientific understanding these representations offer substantial challenges namely it is an open problem how to precisely characterize the large or mesoscale structure of network systems in relation to these additional aspects furthermore the direct incorporation of these features invariably increases the effective dimension of the network description and hence aggravates the problem of overfitting i e the use of overly complex characterizations that mistake purely random fluctuations for actual structure in this work we propose a robust and principled method to tackle these problems by constructing generative models of modular network structure incorporating layered attributed and time varying properties as well as a nonparametric bayesian methodology to infer the parameters from data and select the most appropriate model according to statistical evidence we show that the method is capable of revealing hidden structure in layered edge valued and time varying networks and that the most appropriate level of granularity with respect to the additional dimensions can be reliably identified we illustrate our approach on a variety of empirical systems including a social network of physicians the voting correlations of deputies in the brazilian national congress the global airport network and a proximity network of high school students

minimum state unsteady aerodynamics for aeroservoelastic configuration shape optimization of flight vehicles the minimum state technique for creating linear time invariant state space models of unsteady aerodynamic loads leads to major savings in terms of the dimension of the resulting aeroservoelastic systems compared to alternative techniques the upfront effort of the generation of minimum state approximants however is large and involves an iterative double least squares function fitting process when the overall shape of a configuration varies during the repetitive analysis required for configuration design optimization the cost of creating new minimum state approximations becomes prohibitive this paper presents an efficient method for obtaining shape sensitivity analysis of minimum state approximants the resulting sensitivities are then used based on a single detailed analysis and sensitivity analysis at some reference design to generate approximate aeroelastic poles divergence speed and flutter speed results for configurations that are subject to considerable planform shape changes

criterions for locally dense subgraphs abstract community detection is one of the most investigated problems in the field of complex networks although several methods were proposed there is still no precise definition of communities as a step toward a definition i highlight two necessary properties of communities separation and internal cohesion the latter being a new concept i propose a local method of community detection based on two dimensional local optimization which i tested on common benchmarks and on the word association database

community detection in complex networks by density based clustering abstract 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

a density based link clustering algorithm for overlapping community detection in networks abstract overlapping is an interesting and common characteristic of community structure in networks link clustering method for overlapping community detection has attracted a lot of attention in the area of social networks applications however it may make the clustering result with excessive overlap and cluster bridge edge and border edge mistakenly to adjacent communities to solve this problem a density based link clustering algorithm is proposed to improve the accuracy of detecting overlapping communities in networks in this study it creates a number of clusters containing core edges only based on concept named as core density reachable during the expansion then an updating strategy for unclassified edges is designed to assign them to the closest cluster in addition a similarity measure for computing the similarity between two edges is presented experiments on synthetic networks and real networks have been conducted the experimental results demonstrate that our method performs better than other algorithms on detecting community structure and overlapping nodes it can get nearly 15 higher than the nmi value of other algorithms on some synthetic networks

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