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

the average distances in random graphs with given expected degrees abstract random graph theory is used to examine the small world phenomenon any two strangers are connected through a short chain of mutual acquaintances we will show that for certain families of random graphs with given expected degrees the average distance is almost surely of order log n log d where d is the weighted average of the sum of squares of the expected degrees of particular interest are power law random graphs in which the number of vertices of degree k is proportional to 1 k for some fixed exponent for the case of 3 we prove that the average distance of the power law graphs is almost surely of order log n log d however many internet social and citation networks are power law graphs with exponents in the range 2 3 for which the power law random graphs have average distance almost surely of order log log n but have diameter of order log n provided having some mild constraints for the average distance and maximum degree in particular these graphs contain a dense subgraph which we call the core having nc log log n vertices almost all vertices are within distance log log n of the core although there are vertices at distance log n from the core

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

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

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

statistical mechanics of community detection starting from a general ansatz we show how community detection can be interpreted as finding the ground state of an infinite range spin glass our approach applies to weighted and directed networks alike it contains the ad hoc introduced quality function from j reichardt and s bornholdt phys rev lett 93 218701 2004 and the modularity q as defined by newman and girvan phys rev e 69 026113 2004 as special cases the community structure of the network is interpreted as the spin configuration that minimizes the energy of the spin glass with the spin states being the community indices we elucidate the properties of the ground state configuration to give a concise definition of communities as cohesive subgroups in networks that is adaptive to the specific class of network under study further we show how hierarchies and overlap in the community structure can be detected computationally efficient local update rules for optimization procedures to find the ground state are given we show how the ansatz may be used to discover the community around a given node without detecting all communities in the full network and we give benchmarks for the performance of this extension finally we give expectation values for the modularity of random graphs which can be used in the assessment of statistical significance of community structure

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

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

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

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

mean field theory of the general collective motion algebra gcm 3 the lie algebra gcm 3 is a 15 dimensional semidirect sum that contains the algebra of the general linear group gl 3 r the six dimensional abelian ideal of gcm 3 is generated by the mass quadrupole and monopole tensors the bohr mottelson quantum model of nuclear rotational and vibrational states is a particular irreducible representation of gcm 3 the gcm 3 dual space consists of density matrices which are defined by the expectations of the gcm 3 generators a coadjoint orbit is a level surface in the dual space of the gcm 3 casimir which is the squared length of the conserved kelvin circulation vector this paper develops mean field theory on any coadjoint orbit of gcm 3 densities a lax pair determines the dynamics of gcm 3 densities on each coadjoint orbit this lax equation is equivalent to the riemann ellipsoid equations of motion pacs number 21 60 fw

vector coherent state theory as a theory of induced representations a general definition is given of vector coherent state vcs representation theory it is shown that the theory is more general than suggested by previous applications and that it incorporates the standard theories of induced representations as special cases the associated k matrix theory is also given a fuller treatment than hitherto and shown to provide a rather general algorithm both for projecting vcs representations from larger representations in which they are embedded and for determining the hermitian form with respect to which an isometric equivalent representation is in fact isometric

a reducible weil representation of sp 4 realized by differential operators in the space of smooth functions on h2 s1 this work presents a novel way to obtain the associated romanovski functions rn m x with n m in the three separate regions in terms of n and m we obtain the raising and lowering relations with respect to the both indices simultaneously in the three regions then a reducible weil representation of the real lie algebra sp 4 is realized in the space of complex valued smooth functions on h2 s1 by differential forms for the cartan weyl basis its invariant subspace is the second rare instance of the highest weight irreducible representation of sp 4 all whose weight spaces are one dimensional

spectral and dynamical properties in classes of sparse networks with mesoscopic inhomogeneities we study structure eigenvalue spectra and random walk dynamics in a wide class of networks with subgraphs modules at mesoscopic scale the networks are grown within the model with three parameters controlling the number of modules their internal structure as scale free and correlated subgraphs and the topology of connecting network within the exhaustive spectral analysis for both the adjacency matrix and the normalized laplacian matrix we identify the spectral properties which characterize the mesoscopic structure of sparse cyclic graphs and trees the minimally connected nodes the clustering and the average connectivity affect the central part of the spectrum the number of distinct modules leads to an extra peak at the lower part of the laplacian spectrum in cyclic graphs such a peak does not occur in the case of topologically distinct tree subgraphs connected on a tree whereas the associated eigenvectors remain localized on the subgraphs both in trees and cyclic graphs we also find a characteristic pattern of periodic localization along the chains on the tree for the eigenvector components associated with the largest eigenvalue lambda l 2 of the laplacian further differences between the cyclic modular graphs and trees are found by the statistics of random walks return times and hitting patterns at nodes on these graphs the distribution of first return times averaged over all nodes exhibits a stretched exponential tail with the exponent sigma approximately 1 3 for trees and sigma approximately 2 3 for cyclic graphs which is independent of their mesoscopic and global structure

branching rules for restriction of the weil representations of sp n r to its maximal parabolic subgroup cm n the symplectic group sp n r is the group of linear canonical transformations of a real 2n dimensional phase space and cm n sp n r is a maximal parabolic subgroup the symplectic groups are the fundamental dynamical groups of classical and quantal hamiltonian mechanics in particular sp 3 r is the dynamical group of the spherical harmonic oscillator and its weil harmonic series representations are important for the microscopic shell model description of the collective motions of many particle systems the subgroup cm 3 sp 3 r also appears in the microscopic theory of nuclear collective motion as the dynamical group of a hydrodynamic model of quadrupole vibrations and rotations of a nucleus thus the sp 3 r cm 3 branching rules are needed in finding the embedding of the hydrodynamic collective model in the microscopic shell model some new developments are made in the vector coherent state theory of induced representations

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

discrete signal processing on graphs sampling theory we propose a sampling theory for signals that are supported on either directed or undirected graphs the theory follows the same paradigm as classical sampling theory we show that perfect recovery is possible for graph signals bandlimited under the graph fourier transform the sampled signal coefficients form a new graph signal whose corresponding graph structure preserves the first order difference of the original graph signal for general graphs an optimal sampling operator based on experimentally designed sampling is proposed to guarantee perfect recovery and robustness to noise for graphs whose graph fourier transforms are frames with maximal robustness to erasures as well as for erd s renyi graphs random sampling leads to perfect recovery with high probability we further establish the connection to the sampling theory of finite discrete time signal processing and previous work on signal recovery on graphs to handle full band graph signals we propose a graph filter bank based on sampling theory on graphs finally we apply the proposed sampling theory to semi supervised classification of online blogs and digit images where we achieve similar or better performance with fewer labeled samples compared to previous work

a combination of proper orthogonal decomposition discrete empirical interpolation method pod deim and meshless local rbf dq approach for prevention of groundwater contamination abstract the main presented idea is to reduce the used cpu time for employing the local radial basis functions differential quadrature lrbf dq method to this end the proper orthogonal decomposition discrete empirical interpolation method pod deim has been combined with the lrbf dq technique for checking the ability of the new procedure the groundwater equation is solved this equation has been classified in category of system of advection diffusion equations the solutions of advection equations have some shock thus special numerical methods should be applied for example discontinuous galerkin and finite volume methods moreover several test problems are given that show the acceptable accuracy and efficiency of the proposed schemes

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

tight wavelet frames on multislice graphs we present a framework for the design of wavelet transforms tailored to data defined on multislice graphs i e multiplex or dynamic graphs graphs with multiple types of interactions are ubiquitous in real life motivating the extension of wavelets to these complex domains our framework generalizes the recently proposed spectral graph wavelet transform sgwt d hammond p vandergheynst and r gribonval wavelets on graphs via spectral graph theory appl comput harmon anal vol 30 pp 129 150 mar 2011 which is designed in the spectral frequency domain of an arbitrary finite weighted graph we extend the sgwt to form a tight frame which conserves energy in the wavelet domain and define the relationship between conventional and spectral graph wavelets we then propose a design for multislice graphs that is based on the higher order singular value decomposition hosvd a powerful tool from multilinear algebra in particular the multiple adjacency matrices are stacked to form a tensor and the hosvd decomposition provides information about its third order structure analogous to that provided by matrix factorizations we obtain a set of eigennetworks and from these graph wavelets which exploit the variability across the graphs we demonstrate the feasibility of our method 1 by capturing different orientations of a gray scale image and 2 by decomposing brain signals from functional magnetic resonance imaging we show its effectiveness to identify variability across graph edges and provide meaningful decompositions

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

unitary representations branching rules and matrix elements for the non compact symplectic groups the complementarity of the symplectic and orthogonal groups is used to infer properties of the infinite dimensional unirreps of the former from the character theory of the latter the complete set of d series metaplectic unirreps of sp n r is identified and branching rules are given for their restrictions to the maximal compact subgroup u n developed in terms of the properties of schur functions a known algorithm for the evaluation of matrix elements of the sp 3 r lie algebra is extended to any sp n r and analytic expressions are given for important classes of unirreps and multiplicity free states

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

numerical models for 2d free boundary analysis of groundwater in slopes stabilized by drain trenches a numerical model for 2d free boundary analysis of groundwater in slopes stabilized by drain trenches has been developed it consists of a front tracking method based on an original way of adapting the space derivatives very effective in saving calculation time respect to classical fix grid methods the method analyses the trenches effect inside slopes in which the soils above the water table are partially saturated for which a boundary can be recognized between the saturated domain water table and the unsaturated one above the water table in this case pore pressure lowering due to trenches can be analyzed considering the progressively reduction of the saturated domain this approach efficiently solves the problem of fixing hydraulic boundary conditions on the sides of the trenches results have been compared with those obtained by a fix grid method observing difference less than 0 14 applying the method the capability of drain trenches to control the effect of heavy rainfalls has been investigated calculating during the transient process of water table lowering limit values of water recharge for which water table keeps on constant

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

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

subgraph based filterbanks for graph signals we design a critically sampled compact support biorthogonal transform for graph signals via graph filterbanks instead of partitioning the nodes in two sets so as to remove one every two nodes in the filterbank downsampling operations the design is based on a partition of the graph in connected subgraphs coarsening is achieved by defining one supernode for each subgraph and the edges for this coarsened graph derives from the connectivity between the subgraphs unlike the one every two nodes downsampling on bipartite graphs this coarsening operation does not have an exact formulation in the graph fourier domain instead we rely on the local fourier bases of each subgraph to define filtering operations we apply successfully this method to decompose graph signals and show promising performance on compression and denoising

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

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

solution of nonlinear problems in applied sciences by generalized collocation methods and mathematica abstract this paper deals with the developments of mathematical methods for the discretization of continuous models and the solution of nonlinear problems of interest in applied sciences the contents refer to developments of the differential quadrature method which leads to the so called generalized collocation methods the method is developed and applied to the solution of initial boundary value problems the computational problems are technically solved with mathematica

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

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

simple unified proofs of four duality theorems duality relationships between the irreps irreducible representations of pairs of distinct commuting groups g1 and g2 on hilbert spaces of interest have long played important roles in the atomic and nuclear shell models in addition to the well known schur weyl duality the most widely used duality relationships are the so called unitary unitary orthogonal symplectic i e noncompact symplectic symplectic symplectic compact symplectics and orthogonal orthogonal dualities proofs of these dualities exist in the literature but most of them are not readily accessible to physicists or give little insight into how they might be used in practice this paper presents unified proofs of the above mentioned dualities based on the explicit construction of states which are simultaneously of extreme weight for the actions of both g1 and g2 the proofs expressed in language familiar to physicists are simple systematic and provide useful insights

sp 3 r coadjoint orbit theory the algebraic mean field method is applied to the symplectic lie algebra sp 3 r that describes geometrical collective states in atomic nuclei the expectations of sp 3 r generators define a symplectic density matrix the mean field approximation restricts the densities to a manifold that is a coadjoint orbit of the transformation group sp 3 r and a level surface of the symplectic casimir functions compared to representation theory mean field theory is technically simpler but yields similar predictions for physical properties of collective states the critical points of the energy functional on a coadjoint orbit surface define rotational bands the deformation kinetic energy and kelvin circulation of principal axis symplectic rotors are determined as a function of the angular momentum illustrative applications of coadjoint orbit theory are reported for the yrast rotational bands of a light 20ne and a heavy 166er deformed isotope

vector coherent state theory and its application to the orthogonal groups vector coherent state theory is developed and presented in a form that explicitly exhibits its general applicability to the ladder representations of all semisimple lie groups and their lie algebras it is shown that in a suitable basis the vector coherent state inner product can be inferred algebraically by k matrix theory and changed to a simpler bargmann inner product thereby facilitating the explicit calculation of the matrix representaions of lie algebras applications are made to the even and odd orthogonal lie algebras

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

analysis and application of the interpolating element free galerkin iefg method to simulate the prevention of groundwater contamination with application in fluid flow abstract we develop a meshless numerical procedure to simulate the groundwater equation gwe the used technique is based on the interpolating element free galerkin iefg method the interpolating moving least squares imls approximation produces a set of functions such that they are well known as shape functions the iefg technique employs the shape functions of imls approximation the shape functions of imls approximation vanish on the boundary and also they satisfy the property of the kronecker delta function thus dirichlet boundary conditions can be exactly imposed in this paper we check the unconditional stability and convergence of the proposed numerical scheme based on the energy method the numerical results confirm the theoretical analysis

estimating the number of clusters in a data set via the gap statistic we propose a method the gap statistic for estimating the number of clusters groups in a set of data the technique uses the output of any clustering algorithm e g k means or hierarchical comparing the change in within cluster dispersion with that expected under an appropriate reference null distribution some theory is developed for the proposal and a simulation study shows that the gap statistic usually outperforms other methods that have been proposed in the literature

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