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

on multiscale entropy analysis for physiological data abstract we perform an analysis of cardiac data using multiscale entropy as proposed in costa et al multiscale entropy analysis of complex physiological time series phys rev lett 89 2002 068102 we reproduce the signatures of the multiscale entropy for the three cases of young healthy hearts atrial fibrillation and congestive heart failure we show that one has to be cautious how to interpret these signatures in terms of the underlying dynamics in particular we show that different dynamical systems can exhibit the same signatures depending on the sampling time and that similar systems may have different signatures depending on the time scales involved besides the total amount of data we identify the sampling time the correlation time and the period of possible nonlinear oscillations as important time scales which have to be involved in a detailed analysis of the signatures of multiscale entropies we illustrate our ideas with the lorenz equation as a simple deterministic chaotic system

measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

entropy solutions to a strongly degenerate anisotropic convection diffusion equation with application to utility theory we study the deterministic counterpart of a backward forward stochastic differential utility which has recently been characterized as the solution to the cauchy problem related to a pde of degenerate parabolic type with a conservative first order term we first establish a local existence result for strong solutions and a continuation principle and we produce a counterexample showing that in general strong solutions fail to be globally smooth afterward we deal with discontinuous entropy solutions and obtain the global well posedness of the cauchy problem in this class eventually we select a sufficient condition of geometric type which guarantees the continuity of entropy solutions for special initial data as a byproduct we establish the existence of an utility process which is a solution to a backward forward stochastic differential equation for a given class of final utilities which is relevant for financial applications

graph optimisation problems and the potts glass the np complete problems of partitioning and colouring of random graphs with p partitions and colours respectively are mapped onto the statistical mechanical problem of p state potts glasses an estimate of the cost functions of these optimisation problems has been derived using the potts glass mean field theory this estimate applies to dense graphs in the thermodynamic limit an exact expression for the cost function in the large p limit is given

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

kernel spectral clustering with memory effect evolving graphs describe many natural phenomena changing over time such as social relationships trade markets methabolic networks etc in this framework performing community detection and analyzing the cluster evolution represents a criti cal task here we propose a new model for this purpose where the smoothness of the clustering results over time can be considered as a valid prior knowledge it is based on a constrained optimization formulation typical of least squares support vector machines ls svm where the objective function is designed to explicitly incorporate temporal smoothness the latter allo ws the model to cluster the current data well and to be consistent with the recent history we also propose new model selection criteria in order to carefully choose the hyper parameters of our model which is a crucial issue to achieve good performances we successfully test the model on four toy problems and on a real world network we also compare our model with evolutionary spectral clustering which is a stateof the art algorithm for community detection of evolving networks illustrating that the kernel spectral clustering with memory effect can achieve better or equal performances

statistical significance of communities in networks nodes in real world networks are usually organized in local modules these groups called communities are intuitively defined as subgraphs with a larger density of internal connections than of external links in this work we define a measure aimed at quantifying the statistical significance of single communities extreme and order statistics are used to predict the statistics associated with individual clusters in random graphs these distributions allows us to define one community significance as the probability that a generic clustering algorithm finds such a group in a random graph the method is successfully applied in the case of real world networks for the evaluation of the significance of their communities

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

multiresolution community detection for megascale networks by information based replica correlations we use a potts model community detection algorithm to accurately and quantitatively evaluate the hierarchical or multiresolution structure of a graph our multiresolution algorithm calculates correlations among multiple copies replicas of the same graph over a range of resolutions significant multiresolution structures are identified by strongly correlated replicas the average normalized mutual information the variation in information and other measures in principle give a quantitative estimate of the best resolutions and indicate the relative strength of the structures in the graph because the method is based on information comparisons it can in principle be used with any community detection model that can examine multiple resolutions our approach may be extended to other optimization problems as a local measure our potts model avoids the resolution limit that affects other popular models with this model our community detection algorithm has an accuracy that ranks among the best of currently available methods using it we can examine graphs over 40 ifmmode times else texttimes fi 10 6 nodes and more than 1 ifmmode times else texttimes fi 10 9 edges we further report that the multiresolution variant of our algorithm can solve systems of at least 200 text 000 nodes and 10 ifmmode times else texttimes fi 10 6 edges on a single processor with exceptionally high accuracy for typical cases we find a superlinear scaling o l 1 3 for community detection and o l 1 3 text text log text n for the multiresolution algorithm where l is the number of edges and n is the number of nodes in the system

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

multivariate multiscale fractional order weighted permutation entropy of nonlinear time series abstract in this letter multivariate multiscale fractional permutation entropy mmfpe and multivariate weighted multiscale fractional permutation entropy mwmfpe have been proposed to provide insights for the study of time series when measuring the dynamics of complex systems the mmfpe and mwmfpe methods are sensitive to the signal evolution meanwhile they can provide some analysis of complexity over multiple time series as well as multiple channel signals we perform these methods on synthetic tri variate time series to explore some of the interesting properties especially for negative information and information deception it can be seen that more complex system is more likely to be deceptive the amplitude information of time series which is taken into account in the mwmfpe can weaken this deception the methods are also employed to the closing prices and trade volume of financial stock markets from different areas according to the mwmfpe results the indices can be divided into three groups 1 cac40 hsi nasdaq s p500 2 n225 and 3 shencheng implying that it has a capacity to distinguish these financial stock market

on the regularity of solutions to a nonlinear ultraparabolic equation arising in mathematical finance we consider the following nonlinear degenerate parabolic equation which arises in some recent problems of mathematical finance partial xx u u partial y u partial t u f using a harmonic analysis technique on lie groups we prove that if the solution u satisfies condition partial x u neq 0 in an open set omega subset mathbb r 3 and f in c infty omega then u in c infty omega

application of statistical mechanics to np complete problems in combinatorial optimisation recently developed techniques of the statistical mechanics of random systems are applied to the graph partitioning problem the averaged cost function is calculated and agrees well with numerical results the problem bears close resemblance to that of spin glasses the authors find a spin glass transition in the system and the low temperature phase space has an ultrametric structure this sheds light on the nature of hard computation problems

weighted multiscale renyi permutation entropy of nonlinear time series abstract in this paper based on renyi permutation entropy rpe which has been recently suggested as a relative measure of complexity in nonlinear systems we propose multiscale renyi permutation entropy mrpe and weighted multiscale renyi permutation entropy wmrpe to quantify the complexity of nonlinear time series over multiple time scales first we apply mpre and wmpre to the synthetic data and make a comparison of modified methods and rpe meanwhile the influence of the change of parameters is discussed besides we interpret the necessity of considering not only multiscale but also weight by taking the amplitude into account then mrpe and wmrpe methods are employed to the closing prices of financial stock markets from different areas by observing the curves of wmrpe and analyzing the common statistics stock markets are divided into 4 groups 1 dji s p500 and hsi 2 nasdaq and ftse100 3 dax40 and cac40 and 4 shangzheng and shencheng results show that the standard deviations of weighted methods are smaller showing wmrpe is able to ensure the results more robust besides wmpre can provide abundant dynamical properties of complex systems and demonstrate the intrinsic mechanism

subgraphs and network motifs in geometric networks many real world networks describe systems in which interactions decay with the distance between nodes examples include systems constrained in real space such as transportation and communication networks as well as systems constrained in abstract spaces such as multivariate biological or economic data sets and models of social networks these networks often display network motifs subgraphs that recur in the network much more often than in randomized networks to understand the origin of the network motifs in these networks it is important to study the subgraphs and network motifs that arise solely from geometric constraints to address this we analyze geometric network models in which nodes are arranged on a lattice and edges are formed with a probability that decays with the distance between nodes we present analytical solutions for the numbers of all three and four node subgraphs in both directed and nondirected geometric networks we also analyze geometric networks with arbitrary degree sequences and models with a bias for directed edges in one direction scaling rules for scaling of subgraph numbers with system size lattice dimension and interaction range are given several invariant measures are found such as the ratio of feedback and feed forward loops which do not depend on system size dimension or connectivity function we find that network motifs in many real world networks including social networks and neuronal networks are not captured solely by these geometric models this is in line with recent evidence that biological network motifs were selected as basic circuit elements with defined information processing functions

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

identifying communities within energy landscapes potential energy landscapes can be represented as a network of minima linked by transition states the community structure of such networks has been obtained for a series of small lennard jones lj clusters this community structure is compared to the concept of funnels in the potential energy landscape two existing algorithms have been used to find community structure one involving removing edges with high betweenness the other involving optimization of the modularity the definition of the modularity has been refined making it more appropriate for networks such as these where multiple edges and self connections are not included the optimization algorithm has also been improved using monte carlo methods with simulated annealing and basin hopping both often used successfully in other optimization problems in addition to the small clusters two examples with known heterogeneous landscapes the 13 atom cluster lj13 with one labeled atom and the 38 atom cluster lj38 were studied with this approach the network methods found communities that are comparable to those expected from landscape analyses this is particularly interesting since the network model does not take any barrier heights or energies of minima into account for comparison the network associated with a two dimensional hexagonal lattice is also studied and is found to have high modularity thus raising some questions about the interpretation of the community structure associated with such partitions

solution of forward backward stochastic differential equations in this paper we study the existence and uniqueness of the solution to forward backward stochastic differential equations without the nondegeneracy condition for the forward equation under a certain monotonicity condition we prove the existence and uniqueness of the solution to forward backward stochastic differential equations

on uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic pde s we prove several comparison and existence theorems for viscosity solutions of fully nonlinear degenerate elliptic equations one of them extends some recent uniqueness results by jensen some establish the uniqueness of solutions for second order isaacs equations and hence include the uniqueness results for bellman equations by p l lions our comparison results apply even for discontinuous solutions and so perron s method readily yields the existence of continuous solutions

graph bipartitioning problem replica symmetric solutions of the graph bipartitioning problem with finite connectivity are presented with the constraint sum mathrm i 1 mathrm n mathrm s mathrm i 0 strictly enforced another solution can be obtained which gives a lower cost function than that given by the spin glass solution it is believed that this new solution is currently the best candidate for the true solution of the graph bipartitioning problem

density shrinking algorithm for community detection with path based similarity abstract community structure is ubiquitous in real world complex networks finding the communities is the key to understand the functions of those networks a lot of works have been done in designing algorithms for community detection but it remains a challenge in the field traditional modularity optimization suffers from the resolution limit problem recent researches show that combining the density based technique with the modularity optimization can overcome the resolution limit and an efficient algorithm named denshrink was provided the main procedure of denshrink is repeatedly finding and merging micro communities broad sense into super nodes until they cannot merge analyses in this paper show that if the procedure is replaced by finding and merging only dense pairs both of the detection accuracy and runtime can be obviously improved thus an improved density based algorithm imds is provided since the time complexity path based similarity indexes are difficult to be applied in community detection for high performance in this paper the path based katz index is simplified and used in the imds algorithm

mixing patterns in networks we study assortative mixing in networks the tendency for vertices in networks to be connected to other vertices that are like or unlike them in some way we consider mixing according to discrete characteristics such as language or race in social networks and scalar characteristics such as age as a special example of the latter we consider mixing according to vertex degree i e according to the number of connections vertices have to other vertices do gregarious people tend to associate with other gregarious people we propose a number of measures of assortative mixing appropriate to the various mixing types and apply them to a variety of real world networks showing that assortative mixing is a pervasive phenomenon found in many networks we also propose several models of assortatively mixed networks both analytic ones based on generating function methods and numerical ones based on monte carlo graph generation techniques we use these models to probe the properties of networks as their level of assortativity is varied in the particular case of mixing by degree we find strong variation with assortativity in the connectivity of the network and in the resilience of the network to the removal of vertices

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

optimization problems and replica symmetry breaking in finite connectivity spin glasses a formalism capable of handling the first step of hierarchical replica symmetry breaking rsb in finite connectivity models is introduced the emerging order parameter is claimed to be a probability distribution over the space of field distributions or equivalently magnetization distributions inside the cluster of states the approach is shown to coincide with previous works in the replica symmetric case and in the two limiting cases m 0 and 1 where m is parisi s break point as an application to the study of optimization problems the gs properties of the random 3 satisfiability problem are investigated and we present a first rsb solution improving replica symmetric results

minimum spanning trees for community detection abstract a simple deterministic algorithm for community detection is provided by using two rounds of minimum spanning trees by comparing the first round minimum spanning tree 1st mst with the second round spanning tree 2nd mst of the network communities are detected and their overlapping nodes are also identified to generate the two msts a distance matrix is defined and computed from the adjacent matrix of the network compared with the resistance matrix or the communicability matrix used in community detection in the literature the proposed distance matrix is very simple in computation the proposed algorithm is tested on real world social networks graphs which are failed by the modularity maximization and the lfr benchmark graphs for community detection

pointwise estimates for a class of non homogeneous kolmogorov equations we consider a class of ultraparabolic differential equations that satisfy the hormander s hypoellipticity condition and we prove that the weak solutions to the equation with measurable coefficients are locally bounded functions the method extends the moser s iteration procedure and has previously been employed in the case of operators verifying a further homogeneity assumption here we remove that assumption by proving some potential estimates and some ad hoc sobolev type inequalities for solutions

holder gradient estimates for fully nonlinear elliptic equations in this paper we prove interior and global holder estimates for lipschitz viscosity solutions of second order nonlinear uniformly elliptic equations the smoothness hypotheses on the operators are more general than previously considered for classical solutions so that our estimates are also new in this case and readily extend to embrace obstacle problems in particular isaac s equations of stochastic differential game theory constitute a special case of our results and moreover our techniques in combination with recent existence theorems of ishii lead to existence theorems for continuously differentiable viscosity solutions of the uniformly elliptic isaac s equation

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

an investigation of oil water two phase flow instability using multivariate multi scale weighted permutation entropy abstract the present study is devoted to the investigation of oil water two phase flow instability using multivariate multi scale weighted permutation entropy mwmpe four typical multivariate multi scale entropies namely mwmpe multivariate multi scale permutation entropy mmpe multivariate multi scale sample entropy mmse and multivariate multi scale approximate entropy mmae are applied in lorenz system with the addition of different kinds of noises the comparison results indicate that mwmpe presents the superiorities of being sensitive to the variation in scale showing a monotonous increasing trend as well as the best anti noise ability accordingly with the fluctuating signals from an eight electrode rotating electrical field conductance sensor used for the measurement of oil water flows we extract mwmpe for the whole experimental flow conditions in terms of which the effects of mixture velocity and water cut on oil water two phase flow instability are illuminated our research provides an effective method for uncovering the underlying evolution instability of the flow structures in oil water flows

moser s estimates for degenerate kolmogorov equations with non negative divergence lower order coefficients abstract we prove l loc estimates for positive solutions to the following degenerate second order partial differential equation of kolmogorov type with measurable coefficients of the form i j 1 m 0 x i a i j x t x j u x t i j 1 n b i j x j x i u x t t u x t i 1 m 0 b i x t i u x t i 1 m 0 x i a i x t u x t c x t u x t 0 where x t x 1 x n t z is a point of r n 1 and 1 m 0 n a i j is a uniformly positive symmetric matrix with bounded measurable coefficients b i j is a constant matrix we apply the moser s iteration method to prove the local boundedness of the solution u under minimal integrability assumption on the coefficients

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

when are networks truly modular abstract the study of cluster or community structure of complex networks contributes to the understanding of networks at a functional level in many networks latent classes of nodes are suspected which manifest themselves as communities i e groups of nodes with a high link density among the nodes of the same class and low link density between nodes of different classes community detection algorithms are used to infer these classes e g by finding a partition of the network which maximizes a quality function such as the network modularity q m newman m girvan finding and evaluating community structure in networks phys rev e 69 2004 026113 however it is known from numerical experiments that even purely random networks display intrinsic modularity and may be partitioned yielding high values of q extending on our earlier results j reichardt s bornholdt statistical mechanics of community detection phys rev e 74 2006 016110 the mapping of the community detection problem onto finding the ground state of a spin glass is exploited in order to derive analytical expressions for the expected modularity in random graphs and assess the theoretical limits to community detection the results are independent of any specific community detection algorithm and allow for differentiation between modularity arising purely due to the search process in the large configuration space of possible partitionings on the one hand or due to the actual presence of different classes of nodes on the other hand

maximizing modularity intensity for community partition and evolution most previous studies of community partition have often focused on the network topology the topology and link weights are in fact closely associated with each other for community formation in complex networks this paper proposes a function modularity intensity a variation of modularity density d value for evaluating the cohesiveness of a community which considers links between vertices as well as link weights the results showed that maximizing the modularity intensity not only can resolve the resolution limits problem but also achieve better performance for community partition to further evaluate the function and clarify the weight topology correlation with communities we give a simple model to simulate the topology and link weights development for community evolution and use the modularity intensity to capture communities of networks in each step of this process in this model a network is treated as a fuzzy relation and two operations of the fuzzy relation are used to make the link weights stronger and weaker respectively with the growth of the network in each evolutionary step by simulation experiments we found that in the model the modularity intensity catches communities of networks that undergo a gradual transition from faintness to clearness our model also reproduces the finding of granovetter that strong links are confined mainly in tight communities and the links between communities are predominantly weak from the results above we believe that the modularity intensity gives a more comprehensive evaluation for communities and by studying the weight topology correlation this model provides a new view for community evolution

generalized modularity matrices various modularity matrices appeared in the recent literature on network analysis and algebraic graph theory their purpose is to allow writing as quadratic forms certain combinatorial functions appearing in the framework of graph clustering problems in this paper we put in evidence certain common traits of various modularity matrices and shed light on their spectral properties that are at the basis of various theoretical results and practical spectral type algorithms for community detection

dispersive graded entropy on computing dynamical complexity abstract we propose a phase space based statistical disorder to investigate the dynamical complexity of chaotic models the statistical disorder is defined by introducing a grade function inversely maps the mean dispersion of the trajectories in the phase space we denote the associate entropy by the dispersive graded entropy dge numerical investigation shows that dge can quantify the dynamical complexity of discrete as well as continuous chaotic systems a comparative study is also made with the other phase space based entropy measures finally the proposed measure has been applied on three types of heart rate variability hrv signals the results support the clinical observations related to the dynamics of healthy and congestive hearts

adaptive multi resolution modularity for detecting communities in networks abstract community structure is a common topological property of complex networks which attracted much attention from various fields optimizing quality functions for community structures is a kind of popular strategy for community detection such as modularity optimization here we introduce a general definition of modularity by which several classical multi resolution modularity can be derived and then propose a kind of adaptive multi resolution modularity that can combine the advantages of different modularity by applying the modularity to various synthetic and real world networks we study the behaviors of the methods showing the validity and advantages of the multi resolution modularity in community detection the adaptive modularity as a kind of multi resolution method can naturally solve the first type limit of modularity and detect communities at different scales it can quicken the disconnecting of communities and delay the breakup of communities in heterogeneous networks and thus it is expected to generate the stable community structures in networks more effectively and have stronger tolerance against the second type limit of modularity

local resolution limit free potts model for community detection we report on an exceptionally accurate spin glass type potts model for community detection with a simple algorithm we find that our approach is at least as accurate as the best currently available algorithms and robust to the effects of noise it is also competitive with the best currently available algorithms in terms of speed and size of solvable systems we find that the computational demand often exhibits superlinear scaling o l 1 3 where l is the number of edges in the system and we have applied the algorithm to synthetic systems as large as 40 ifmmode times else texttimes fi 10 6 nodes and over 1 ifmmode times else texttimes fi 10 9 edges a previous stumbling block encountered by popular community detection methods is the so called resolution limit being a local measure of community structure our potts model is free from this resolution limit effect and it further remains a local measure on weighted and directed graphs we also address the mitigation of resolution limit effects for two other popular potts models

the maximum principle for viscosity solutions of fully nonlinear second order partial differential equations we prove that viscosity solutions in w1 of the second order fully nonlinear equation f d2u du u 0 are unique when i f is degenerate elliptic and decreasing in u or ii f is uniformly elliptic and nonincreasing in u we do not assume that f is convex the method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions respectively this method is completely different from that used in lions 8 9 for second order problems with f convex in d 2 u and from that used by crandall lions 3 and crandall evans lions 2 for fully nonlinear first order problems

forward backward stochastic differential equations and quasilinear parabolic pdes this paper studies under some natural monotonicity conditions the theory existence and uniqueness a priori estimate continuous dependence on a parameter of forward backward stochastic differential equations and their connection with quasilinear parabolic partial differential equations we use a purely probabilistic approach and allow the forward equation to be degenerate

detection of community structures in networks via global optimization abstract we present an analysis of communality structure in networks based on the application of simulated annealing techniques in this case we use as cost function the already introduced modularity q 1 which is based on the relative number of links within a commune against the number of links that would correspond in case the links were distributed randomly we compare the results of our approach against other methodologies based on betweenness analysis and show that in all cases a better community structure can be attained

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

modularity from fluctuations in random graphs and complex networks the mechanisms by which modularity emerges in complex networks are not well understood but recent reports have suggested that modularity may arise from evolutionary selection we show that finding the modularity of a network is analogous to finding the ground state energy of a spin system moreover we demonstrate that due to fluctuations stochastic network models give rise to modular networks specifically we show both numerically and analytically that random graphs and scale free networks have modularity we argue that this fact must be taken into consideration to define statistically significant modularity in complex networks

robustness of community structure in networks the discovery of community structure is a common challenge in the analysis of network data many methods have been proposed for finding community structure but few have been proposed for determining whether the structure found is statistically significant or whether conversely it could have arisen purely as a result of chance in this paper we show that the significance of community structure can be effectively quantified by measuring its robustness to small perturbations in network structure we propose a suitable method for perturbing networks and a measure of the resulting change in community structure and use them to assess the significance of community structure in a variety of networks both real and computer generated

axioms for graph clustering quality functions we investigate properties that intuitively ought to be satisfied by graph clustering quality functions that is functions that assign a score to a clustering of a graph graph clustering also known as network community detection is often performed by optimizing such a function two axioms tailored for graph clustering quality functions are introduced and the four axioms introduced in previous work on distance based clustering are reformulated and generalized for the graph setting we show that modularity a standard quality function for graph clustering does not satisfy all of these six properties this motivates the derivation of a new family of quality functions adaptive scale modularity which does satisfy the proposed axioms adaptive scale modularity has two parameters which give greater flexibility in the kinds of clusterings that can be found standard graph clustering quality functions such as normalized cut and unnormalized cut are obtained as special cases of adaptive scale modularity in general the results of our investigation indicate that the considered axiomatic framework covers existing good quality functions for graph clustering and can be used to derive an interesting new family of quality functions

solving forward backward stochastic differential equations explicitly a four step scheme in this paper we investigate the nature of the adapted solutions to a class of forward backward stochastic differential equations sdes for short in which the forward equation is non degenerate we prove that in this case the adapted solution can always be sought in an ordinary sense over an arbitrarily prescribed time duration via a direct four step scheme using this scheme we further prove that the backward components of the adapted solution are determined explicitly by the forward components via the solution of a certain quasilinear parabolic pde system moreover the uniqueness of the adapted solutions over an arbitrary time duration as well as the continuous dependence of the solutions on the parameters can all be proved within this unified framework some special cases are studied separately in particular we derive a new form of the integral representation of the clark haussmann ocone type for functionals or functions of diffusions in which the conditional expectation is no longer needed

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

limits of modularity maximization in community detection modularity maximization is the most popular technique for the detection of community structure in graphs the resolution limit of the method is supposedly solvable with the introduction of modified versions of the measure with tunable resolution parameters we show that multiresolution modularity suffers from two opposite coexisting problems the tendency to merge small subgraphs which dominates when the resolution is low the tendency to split large subgraphs which dominates when the resolution is high in benchmark networks with heterogeneous distributions of cluster sizes the simultaneous elimination of both biases is not possible and multiresolution modularity is not capable to recover the planted community structure not even when it is pronounced and easily detectable by other methods for any value of the resolution parameter this holds for other multiresolution techniques and it is likely to be a general problem of methods based on global optimization

weighted multiscale cumulative residual renyi permutation entropy of financial time series abstract in this paper based on cumulative residual entropy cre and renyi permutation entropy rpe multiscale cumulative residual renyi permutation entropy mcrrrpe and weighted multiscale cumulative residual renyi permutation entropy wmcrrpe are proposed as novelty measures to quantify the uncertainty of the nonlinear time series first the mcrrpe and wmcrrpe methods are performed on the synthetic data and the impact of the changes to parameters is discussed then the mcrrpe and wmcrrpe methods are applied to the closing prices of the us european and chinese stock markets we analyze the statistics and draw some conclusions from the comparison the stock markets can be divided into four categories 1 nasdaq and ftse100 2 s p500 dji and hsi 3 dax30 and cac40 and 4 sse and szse the standard deviation has a certain relationship with wmcrrpe compared with the mcrrpe method the wmcrrpe method can distinguish these financial stock markets effectively and better estimate the complexity of the time series containing amplitude information

generalized statistical complexity measures geometrical and analytical properties abstract we discuss bounds on the values adopted by the generalized statistical complexity measures m t martin et al phys lett a 311 2003 126 p w lamberti et al physica a 334 2004 119 introduced by lopez ruiz et al phys lett a 209 1995 321 and shiner et al phys rev e 59 1999 1459 several new theorems are proved and illustrated with reference to the celebrated logistic map

a practical method for calculating largest lyapunov exponents from small data sets detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest lyapunov exponent lyapunov exponents quantify the exponential divergence of initially close state space trajectories and estimate the amount of chaos in a system we present a new method for calculating the largest lyapunov exponent from an experimental time series the method follows directly from the definition of the largest lyapunov exponent and is accurate because it takes advantage of all the available data we show that the algorithm is fast easy to implement and robust to changes in the following quantities embedding dimension size of data set reconstruction delay and noise level furthermore one may use the algorithm to calculate simultaneously the correlation dimension thus one sequence of computations will yield an estimate of both the level of chaos and the system complexity

holder regularity for a kolmogorov equation we study the interior regularity properties of the solutions to the degenerate parabolic equation x u b y u t u f x y t r n r r which arises in mathematical finance and in the theory of diffusion processes

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

graph bipartitioning and spin glasses on a random network of fixed finite valence the authors study the problem of bipartitioning a random graph of fixed finite valence using a mean field replica symmetric theory of an ising ferromagnet with zero magnetisation constraint the thermodynamics is determined by the probability distribution of an auxiliary field the expression for the ground state energy agrees with that proposed by mezard and parisi 1986 using a cavity field method but their expression for the fraction of crazy spins is reinterpreted

multiscale entropy analysis of complex physiologic time series there has been considerable interest in quantifying the complexity of physiologic time series such as heart rate however traditional algorithms indicate higher complexity for certain pathologic processes associated with random outputs than for healthy dynamics exhibiting long range correlations this paradox may be due to the fact that conventional algorithms fail to account for the multiple time scales inherent in healthy physiologic dynamics we introduce a method to calculate multiscale entropy mse for complex time series we find that mse robustly separates healthy and pathologic groups and consistently yields higher values for simulated long range correlated noise compared to uncorrelated noise

encoding dynamics for multiscale community detection markov time sweeping for the map equation the detection of community structure in networks is intimately related to finding a concise description of the network in terms of its modules this notion has been recently exploited by the map equation formalism rosvall and bergstrom proc natl acad sci usa 105 1118 2008 through an information theoretic description of the process of coding inter and intracommunity transitions of a random walker in the network at stationarity however a thorough study of the relationship between the full markov dynamics and the coding mechanism is still lacking we show here that the original map coding scheme which is both block averaged and one step neglects the internal structure of the communities and introduces an upper scale the field of view limit in the communities it can detect as a consequence map is well tuned to detect clique like communities but can lead to undesirable overpartitioning when communities are far from clique like we show that a signature of this behavior is a large compression gap the map description length is far from its ideal limit to address this issue we propose a simple dynamic approach that introduces time explicitly into the map coding through the analysis of the weighted adjacency matrix of the time dependent multistep transition matrix of the markov process the resulting markov time sweeping induces a dynamical zooming across scales that can reveal potentially multiscale community structure above the field of view limit with the relevant partitions indicated by a small compression gap

application of statistical mechanics to combinatorial optimization problems the chromatic number problem and q partitioning of a graph methods of statistical mechanics are applied to two important np complete combinatorial optimization problems the first is the chromatic number problem which seeks the minimal number of colors necessary to color a graph such that no two sites connected by an edge have the same color the second is partitioning of a graph intoq equal subgraphs so as to minimize intersubgraph connections both models are mapped into a frustrated potts model which is related to theq state potts spin glass for the first problem we obtain very good agreement with numerical simulations and theoretical bounds using the annealed approximation the quenched model is also discussed for the second problem we obtain analytic and numerical results by evaluating the groundstate energy of theq 3 and 4 potts spin glass using parisi s replica symmetry breaking we also perform some numerical simulations to test the theoretical result and obtain very good agreement

on the regularity theory of fully nonlinear parabolic equations ii recently m crandall and p l lions 3 developed a very successful method for proving the existence of solutions of nonlinear second order partial differential equations their method called the theory of viscosity solutions also applies to fully nonlinear equations in which even the second order derivatives can enter in nonlinear fashion solutions produced by the viscosity method are guaranteed to be continuous but not necessarily smooth here we announce smoothness results for viscosity solutions our methods extend those of 1 we obtain krylov safonov i e c estimates 8 c 1 schauder c and w estimates for viscosity solutions of uniformly parabolic equations in general form the results can be viewed as a priori estimates on the classical c solutions our method produces in particular regularity results for a broad new array of nonlinear heat equations including the bellman equation 6

graph bipartitioning and the bethe spin glass the problem of the equipartitioning of a random graph of fixed finite valence is studied by comparison with a ferromagnetic bethe lattice with random boundary conditions the simplicity of recursion relations for effective fields due to descendents on bethe lattices provides simple approximations for the optimal cost in quite good agreement with simulations

penalized versions of the newman girvan modularity and their relation to normalized cuts and k means clustering two penalized balanced and normalized versions of the newman girvan modularity are introduced and estimated by the non negative eigenvalues of the modularity and normalized modularity matrix respectively in this way the partition of the vertices that maximizes the modularity can be obtained by applying the k means algorithm for the representatives of the vertices based on the eigenvectors belonging to the largest positive eigenvalues of the modularity or normalized modularity matrix the proper dimension depends on the number of the structural eigenvalues of positive sign while dominating negative eigenvalues indicate an anticommunity structure the balance between the negative and the positive eigenvalues determines whether the underlying graph has a community anticommunity or randomlike structure

graph bipartitioning and statistical mechanics the problem of bipartitioning a random graph of fixed finite local valence connectivity so as to minimise the number of cross edges is studied by the application of monte carlo simulation of thermodynamics and annealing this problem is np complete and is relevant as an idealisation of several practical organisation problems evidence is presented for the existence of a multiplicity of metastable states and for non self averaging and ultrametricity in the space of overlaps albeit with a possible critical valence an empirical formula is presented for the optimal cost in excellent accord with values obtained by simulation and those known exactly

on the viscosity solutions of a stochastic differential utility problem we prove existence uniqueness and gradient estimates of stochastic differential utility as a solution of the cauchy problem for the following equation in r 3 xxu u yu tu f u where f is lipschitz continuous we also characterize the solution in the vanishing viscosity sense r 2002 elsevier science usa all rights reserved

graph optimization problems on a bethe lattice thep partitioning andp coloring problems on a bethe lattice of coordination numberz are analyzed it is shown that these two np complete optimization problems turn out to be equivalent to finding the ground state energy ofp state potts models with frustration numerical calculation of the cost function of both problems are carried out for several values ofz andp in the case ofp 2 the results are identical to those obtained by mezard and parisi for the case of the bipartitioning problem a numerical upper bound to the chromatic number is found for several values ofz

randomness and degrees of irregularity the fundamental question are sequential data random arises in myriad contexts often with severe data length constraints furthermore there is frequently a critical need to delineate nonrandom sequences in terms of closeness to randomness e g to evaluate the efficacy of therapy in medicine we address both these issues from a computable framework via a quantification of regularity apen approximate entropy defining maximal randomness for sequences of arbitrary length indicating the applicability to sequences as short as n 5 points an infinite sequence formulation of randomness is introduced that retains the operational and computable features of the finite case in the infinite sequence setting we indicate how the foundational definition of independence in probability theory and the definition of normality in number theory reduce to limit theorems without rates of convergence from which we utilize apen to address rates of convergence of a deficit from maximal randomness refining the aforementioned concepts in a computationally essential manner representative applications among many are indicated to assess i random number generation output ii well shuffled arrangements and iii the quality of bootstrap replicates

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

partitioning and modularity of graphs with arbitrary degree distribution we solve the graph bipartitioning problem in dense graphs with arbitrary degree distribution using the replica method we find the cut size to scale universally with k in contrast earlier results studying the problem in graphs with a poissonian degree distribution had found a scaling with kfu and anderson j phys a 19 1605 1986 our results also generalize to the problem of q partitioning they can be used to find the expected modularity q newman and girvan phys rev e 69 026113 2004 of random graphs and allow for the assessment of the statistical significance of the output of community detection algorithms given a graph or network gn m of n nodes and m edges the problem of graph partitioning is finding a partition of the nodes into q equal sized parts such that the number of edges connecting different parts the cut size is minimal the solution of this problem has many important practical applications in multiprocessor scheduling for parallel computing very large scale integrated chip design vlsi and data mining 1 due to the problem being np complete 2 i e no algorithm is known which is guaranteed to find an optimal solution in a number of steps that grows only polynomially with the size of the graph only heuristics exist to approximate a solution furthermore one cannot determine in polynomial time whether a solution found is indeed optimal the only way to assess the quality of a heuristic algorithm or a particular result is to compare with rigorous bounds on the cut size

subgraphs in random networks understanding the subgraph distribution in random networks is important for modeling complex systems in classic erdos networks which exhibit a poissonian degree distribution the number of appearances of a subgraph g with n nodes and g edges scales with network size as approximately n n g however many natural networks have a non poissonian degree distribution here we present approximate equations for the average number of subgraphs in an ensemble of random sparse directed networks characterized by an arbitrary degree sequence we find scaling rules for the commonly occurring case of directed scale free networks in which the outgoing degree distribution scales as p k approximately k gamma considering the power exponent of the degree distribution gamma as a control parameter we show that random networks exhibit transitions between three regimes in each regime the subgraph number of appearances follows a different scaling law approximately nalpha where alpha n g s 1 for gamma gamma c where s is the maximal outdegree in the subgraph and gamma c s 1 we find that certain subgraphs appear much more frequently than in erdos networks these results are in very good agreement with numerical simulations this has implications for detecting network motifs subgraphs that occur in natural networks significantly more than in their randomized counterparts

entropy solutions for diffusion convection equations with partial diffusivity we consider the cauchy problem for the following scalar conservation law with partial viscosity ut a u 0 f u x y e r n t o the existence of solutions is proved by the vanishing viscosity method by introducing a suitable entropy condition we prove uniqueness of solutions this entropy condition is inspired by the entropy criterion introduced by kruzhkov for hyperbolic conservation laws but it takes into account the effect of diffusion introduction we present a theory of existence and uniqueness for suitable entropy solutions of a nonlinear equation which describes the combined effects of diffusion and convection of matter it has the form 0 1 ut axu ayf u posed in the space rn n 2 denoted by the variable z x y x e rn y e r the main characteristic of this equation is that it has mixed parabolichyperboolic type due to the directional separation of the diffusion and convection effects while matter is convected along the y axis it is simultaneously diffused along all orthogonal directions our interest in the model is not merely academic we have encountered it while studying in evz the asymptotic behavior of a standard nonlinear diffusion convection model of the form 0 2 ut azu ayf u in the case f u ululq l for 1 q n 1 n notice that azu axu o2u in fact certain self similar solutions of equation 0 1 will give the large time behaviour of the nonnegative solutions of 0 2 having finite mass the unique characterization of solutions of 0 1 is essential in establishing the asymptotic behaviour of 0 2 the comparison of both equations suggests the name reduced equation for 0 1 the existence of solutions of equation 0 1 can be obtained by the classical method of adding a vanishing artificial viscosity in other terms a diffusion in received by the editors december 21 1992 1991 mathematics subject classification primary 35l65 35k65 35a05