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

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

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

growth and integrability in the dynamics of mappings the growth of some numerical characteristics of the mappings under their iterations in the context of the general problem of integrability is discussed in the general case such characteristics as complexity by arnold or the number of the different images for the multiple valued mappings are growing exponentially it is shown that the integrability is closely related with thepolynomial growth the analogies with quantum integrable systems are discussed

singular nonlinear boundary value problems for second order ordinary differential equations abstract the boundary value problem y f x y 0 y 0 y 0 0 y 1 y 1 0 is studied under conditions which are modeled on the problem y a x y p 0y 0 y 1 0 or y 0 y 1 0 which occurs in a number of applications the technical arguments involve properties of solutions i e concavity and the use of iterative techniques a new fixed point theorem for cones for decreasing mappings is proved and used it is of interest in its own right since most such theorems are for monotone increasing mappings

some existence principles and some general results for singular nonlinear two point boundary value problems abstract existence results for singular second order nonlinear two point boundary value problems of the form 1 p t p t y q t f t y 0 0 where limy 0 f t y are presented here we may allow singularities at y 0 t 0 and or t 1 also existence principles are presented for the general two point boundary value problem 1 p py qf t y py 0 0 where f 0 1 r 2 r is continuous the proof uses the nonlinear alternative of leray and schauder

existence theorems for certain classes of singular boundary value problems abstract existence results for nonlinear singular second order two point boundary value problems of the form 1 p py f t y 0 satisfies appropriate boundary conditions are presented the class of problems considered includes both the thomas fermi and singular emden fowler equations

singular boundary value problems for superlinear second order ordinary and delay differential equations abstract existence criteria are established for singular boundary value problems for nonlinear second order ordinary and delay differential equations the theorems obtained are very general and complement previous known results

an online expectation maximization algorithm for exploring general structure in massive networks abstract mixture model and stochastic block model sbm for structure discovery employ a broad and flexible definition of vertex classes such that they are able to explore a wide variety of structure compared to the existing algorithms based on the sbm their time complexities are o m c 2 where m and c are the number of edges and clusters the algorithms of mixture model are capable of dealing with networks with a large number of communities more efficiently due to their o m c time complexity however the algorithms of mixture model using expectation maximization em technique are still too slow to deal with real million node networks since they compute hidden variables on the entire network in each iteration in this paper an online variational em algorithm is designed to improve the efficiency of the em algorithms in each iteration our online algorithm samples a node and estimates its cluster memberships only by its adjacency links and model parameters are then estimated by the memberships of the sampled node and old model parameters obtained in the previous iteration the provided online algorithm updates model parameters subsequently by the links of a new sampled node and explores the general structure of massive and growing networks with millions of nodes and hundreds of clusters in hours compared to the relevant algorithms on synthetic and real networks the proposed online algorithm costs less with little or no degradation of accuracy results illustrate that the presented algorithm offers a good trade off between precision and efficiency

integrable mappings and soliton equations ii abstract some simple solutions of discrete soliton equations are shown to satisfy 2d mappings we show that these belong to a recently introduced 18 parameter family of integrable reversible mappings of the plane thus lending weight to a previous conjecture we also give an example of an integrable mapping occuring in an exactly solvable model in statistical mechanics finally we discuss the notion of generalized reversibility

the discrete chazy iii system of labrunie conte is not integrable we analyse the discrete form of the chazy iii equation proposed by labrunie and conte with the help of two different integrability criteria singularity confinement and algebraic entropy we show that for all values of the free parameter this third order mapping fails both criteria and thus cannot be integrable

singularity confinement and algebraic integrability two important notions of integrability for discrete mappings algebraic integrability and singularity confinement have been used for discrete mappings algebraic integrability is related to the existence of sufficiently many conserved quantities and singularity confinement is associated with the local analysis of singularities in this article the relationship between these two notions is explored for birational autonomous mappings the main result of this article is that algebraically integrable mappings are shown to have the singularity confinement property using this result the proof of the nonexistence of algebraic conserved quantities for a class of discrete systems is given

positive solutions to singular and non singular second order boundary value problems abstract the existence of positive solutions to second order singular boundary value problems is established our analysis is based on the topological transversality theorem of andrzej granas which relies on the notions of an essential map and a priori estimates on solutions

existence of positive solutions to some singular and nonsingular second order boundary value problems abstract the topological transversality theorem is employed to study existence of solutions to a variety of singular and nonsingular second order boundary value problems of the form y f t y y 0 0 t t 0 t 1 y 0 and or y 0 the principal tools are the arzela ascoli theorem essential maps and a priori bounds on solutions

integrability criteria for differential difference systems a comparison of singularity confinement and low growth requirements a new approach for the study of the integrability of differential difference systems is introduced given a differential difference system of such a form that it can be iterated without necessitating the integration of a differential equation at any step a two stage strategy is used first the singularity confinement necessary integrability criterion is used in order to limit the possible choices once the system is sufficiently reduced the much stronger requirement of nonexponential growth of the degree of the iterates of some initial condition is implemented this method turns out to be powerful and practical the investigation of a given class of differential difference systems has resulted in some well known integrable systems but also to two promising integrability candidates one of which is reduced to a known integrable case

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

solvability of some nonlinear boundary value problems on etablit l existence des solutions positives de problemes a deux points limites singuliers pour des equations differentielles de la forme y f t y y o o t 1

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

the existence of solutions to second order singular boundary value problems abstract this article analyzes qualitative properties of solutions to two point boundary value problems for singular ordinary differential equations in particular we form new approaches that ensure that all possible solutions satisfy certain a priori bounds the methods involve differential inequalities the ideas are then applied to generate new existence results for solutions to the boundary value problems under consideration many of the results are novel for both the singular and the nonsingular cases the main concepts are further illustrated via applications to various equations arising in modeling such as the poisson boltzmann equation and the thomas fermi equation with an ionized atom

how to detect the integrability of discrete systems several integrability tests for discrete equations will be reviewed all tests considered can be applied directly to a given discrete equation and do not rely on the a priori knowledge of the existence of related structures such as lax pairs specifically singularity confinement algebraic entropy nevanlinna theory diophantine integrability and discrete systems over finite fields will be described

exploring the random genesis of co occurrence networks using the network random generation models from gustedt 2009 we simulate and analyze several characteristics such as the number of components the degree sequence and the clustering coefficient of the generated networks this is done for a variety of distributions fixed value bernoulli poisson binomial that are used to control the parameters of generation process these parameters are in particular the size of newly appearing sets of objects the number of contexts in which new elements appear initially the number of objects that are shared with parent contexts and the time period inside which a context may serve as a parent context aging the results show that these models allow to fine tune the generation process such that the graphs adopt properties as they can be found in real world graphs

singular problems an upper and lower solution approach this paper presents an upper and lower solution theory for singular boundary value problems where the nonlinearity may be singular in the dependent variable in addition our nonlinearity is allowed to change sign

nonlinear superlinear singular and nonsingular second order boundary value problems abstract many new existence theorems are presented for singular and nonsingular second order boundary value problems singularities at y 0 t 0 and t 1 will be discussed

non proliferation of pre images in integrable mappings we present an integrability criterion for rational mappings based on two requirements first that a given point should have a unique pre image under the mapping and second that the spontaneously appearing singularities be confined to a few iteration steps we present several examples of known integrable mappings that meet these requirements and also use our algorithm in order to derive new examples of integrable mappings