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

models of social networks based on social distance attachment we propose a class of models of social network formation based on a mathematical abstraction of the concept of social distance social distance attachment is represented by the tendency of peers to establish acquaintances via a decreasing function of the relative distance in a representative social space we derive analytical results corroborated by extensive numerical simulations showing that the model reproduces the main statistical characteristics of real social networks large clustering coefficient positive degree correlations and the emergence of a hierarchy of communities the model is confronted with the social network formed by people that shares confidential information using the pretty good privacy pgp encryption algorithm the so called web of trust of pgp

finite element approximation of a sixth order nonlinear degenerate parabolic equation we consider a finite element approximation of the sixth order nonlinear degenerate parabolic equation equation here where generically equation here for any given equation here in addition to showing well posedness of our approximation we prove convergence in space dimensions d leq 3 furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed finally some numerical experiments in one and two space dimensions are presented

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

entropy stable and entropy dissipative approximations of a fourth order quantum diffusion equation structure preserving numerical schemes for a nonlinear parabolic fourth order equation modeling the electron transport in quantum semiconductors with periodic boundary conditions are analyzed first a two step backward differentiation formula bdf semi discretization in time is investigated the scheme preserves the nonnegativity of the solution is entropy stable and dissipates a modified entropy functional the existence of a weak semi discrete solution and in a particular case its temporal second order convergence to the continuous solution is proved the proofs employ an algebraic relation which implies the g stability of the two step bdf second an implicit euler and q q step bdf discrete variational derivative method are considered this scheme which exploits the variational structure of the equation dissipates the discrete fisher information or energy numerical experiments show that the discrete relative entropies and fisher information decay even exponentially fast to zero

analysis and numerical solution of coupled volume surface reaction diffusion systems with application to cell biology abstract we consider the numerical solution of coupled volume surface reaction diffusion systems having a detailed balance equilibrium based on the conservation of mass an appropriate quadratic entropy functional is identified and an entropy entropy dissipation inequality is proven this allows us to show exponential convergence to equilibrium by the entropy method we then investigate the discretization of the system by a finite element method and an implicit time stepping scheme including the domain approximation by polyhedral meshes mass conservation and exponential convergence to equilibrium are established on the discrete level by arguments similar to those on the continuous level and we obtain estimates of optimal order for the discretization error that hold uniformly in time some numerical tests are presented to illustrate these theoretical results the analysis and the numerical approximation are discussed in detail for a simple model problem the basic arguments however apply also in a more general context this is demonstrated by investigation of a particular volume surface reaction diffusion system arising as a mathematical model for asymmetric stem cell division

a posteriori error estimation with finite element semi and fully discrete methods for nonlinear parabolic equations in one space dimension a posteriors error estimates for semi and fully discrete finite element methods using a pth degree polynomial basis are considered for nonlinear parabolic equations the error estimates are obtained by solving local parabolic or elliptic equations for corrections to the solution on each element using a p 1 st degree polynomial which is zero at the nodes singly implicit runge kutta sirk and backward difference formula bdf methods are considered in the fully discrete case local errors are defined in an analogous manner to the respective methods for ordinary differential equations the a posteriors error estimates are shown to converge to the local errors even in case the local correction is missing from the nonlinear term computational examples compare and verify the theoretical results

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

existence and positivity of solutions of a fourth order nonlinear pde describing interface fluctuations we study the partial differential equation which arose originally as a scaling limit in the study of interface fluctuations in a certain spin system in that application x lies in r but here we study primarily the periodic case x e s we establish existence uniqueness and regularity of solutions locally in time for positive initial data in h s and prove the existence of several families of lyapunov functions for the evolution from the latter we establish a sharp connection between existence globally in time and positivity preservation if o t is a maximal half open interval of existence for a positive solution of the equation with t 1 16r2 then t 03 and lim m w r exists and is constant we discuss also some explicit solutions and propose a generalization to higher dimensions 0 1994 john wiley

traveling wave solutions and the coupled korteweg de vries equation some coupled nonlinear equations are considered for studying traveling wave solutions by introducing a stream function it is shown that if one of the solutions is of the form v v x ct the other also must be of the form u u x ct in addition the possibility of including cubic nonlinear terms has been considered and such a system assuming that the solutions are of the traveling wave type has been solved

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

energy estimates for continuous and discretized electro reaction diffusion systems abstract we consider electro reaction diffusion systems consisting of continuity equations for a finite number of species coupled with a poisson equation we take into account heterostructures anisotropic materials and rather general statistic relations we investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the free energy to its equilibrium value here the essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly the same properties are shown for an implicit time discretized version of the problem moreover we provide a space discretized scheme for the electro reaction diffusion system which is dissipative the free energy decays monotonously on a fixed grid we use for each species different voronoi boxes which are defined with respect to the anisotropy matrix occurring in the flux term of this species

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

on the convergence of entropy consistent schemes for lubrication type equations in multiple space dimensions we present nonnegativity preserving finite element schemes for a general class of thin film equations in multiple space dimensions the equations are fourth order degenerate parabolic and may contain singular terms of second order which are to model van der waals interactions a subtle discretization of the arising nonlinearities allows us to prove discrete counterparts of the essential estimates found in the continuous setting by use of the entropy estimate strong convergence results for discrete solutions are obtained in particular the limit of discrete fluxes mh uh ph will be identified with the flux m u w u u in the continuous setting as a by product first results on existence and positivity almost everywhere of solutions to equations with singular lower order terms can be established in the continuous setting

positivity preserving numerical schemes for lubrication type equations lubrication equations are fourth order degenerate diffusion equations of the form h t nabla cdot f h nabla delta h 0 describing thin films or liquid layers driven by surface tension recent studies of singularities in which h to 0 at a point describing rupture of the fluid layer show that such equations exhibit complex dynamics which can be difficult to simulate accurately in particular one must ensure that the numerical approximation of the interface does not show a false premature rupture generic finite difference schemes have the potential to manifest such instabilities especially when underresolved we present new numerical methods in one and two space dimensions that preserve positivity of the solution regardless of the spatial resolution whenever the pde has such a property we also show that the schemes can preserve positivity even when the pde itself is only known to be nonnegativity preserving we prove that positivity preserving finite difference schemes have unique positive solutions at all times we prove stability and convergence of both positivity preserving and generic methods in one and two space dimensions to positive solutions of the pde showing that the generic methods also preserve positivity and have global solutions for sufficiently fine meshes we generalize the positivity preserving property to a finite element framework and show via concrete examples how this leads to the design of other positivity preserving 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

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

solutions of an equation that lead to solutions of a modified emden equation and a coupled korteweg devries equation in a recent paper moreira res rep if ufrj 83 25 universidade federal do rio de janeiro inst de fisica cidade univ ilha do fundao rio de janeiro brazil obtained a nonlinear second order differential equation that leads to the first integral of a modified emden equation he also obtained two particular solutions of his equation this paper completely integrates moreira s equation and uses it to get a class of solutions of a coupled korteweg devries kdv equation recently studied by guha ray bagchi and sinha j math phys 27 2558 1986

the wasserstein gradient flow of the fisher information and the quantum drift diffusion equation we prove the global existence of non negative variational solutions to the drift diffusion evolution equation partial t u div left u mathrm d left 2 frac delta sqrt u sqrt u f right right 0 under variational boundary condition despite the lack of a maximum principle for fourth order equations non negative solutions can be obtained as a limit of a variational approximation scheme by exploiting the particular structure of this equation which is the gradient flow of the perturbed fisher information functional fancyscript f f u frac 12 int left mathrm d log u right 2 u dx int fu dx with respect to the kantorovich rubinstein wasserstein distance between probability measures we also study long time behavior of the solutions proving their exponential decay to the equilibrium state g e v characterized by delta v frac12 left mathrm d v right 2 f quad int rm e v dx int u 0 dx when the potential v is uniformly convex in this case the functional fancyscript f f coincides with the relative fisher information fancyscript f f u frac12 fancyscript i u g int left mathrm d log u g right 2u dx

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

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

nonnegativity preserving convergent schemes for the thin film equation we present numerical schemes for fourth order degenerate parabolic equations that arise e g in lubrication theory for time evolution of thin films of viscous fluids we prove convergence and nonnegativity results in arbitrary space dimensions a proper choice of the discrete mobility enables us to establish discrete counterparts of the essential integral estimates known from the continuous setting hence the numerical cost in each time step reduces to the solution of a linear system involving a sparse matrix furthermore by introducing a time step control that makes use of an explicit formula for the normal velocity of the free boundary we keep the numerical cost for tracing the free boundary low

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

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

a positivity preserving numerical scheme for a nonlinear fourth order parabolic system noindent a positivity preserving numerical scheme for a fourth order nonlinear parabolic system arising in quantum semiconductor modeling is studied the system is numerically treated by introducing an additional nonlinear potential and a subsequent semidiscretization in time the resulting sequence of nonlinear second order elliptic systems admits at each time level strictly positive solutions which is proved by an exponential transformation of variables the stability of the scheme is shown and convergence is proved in one space dimension numerical results concerning the switching behavior of a resonant tunneling diode are presented

first integrals for the modified emden equation q α t q qn 0 it is shown that the modified emden equation q t q qn 0 possesses first integrals for functions t other than kt 1 the function t is obtained explicitly in the case n 3 and parametrically for other n 2 the case n 2 is seen to be particularly difficult to solve