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

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

on lower bounds for cohomology growth in p adic analytic towers let p and ell be two distinct prime numbers and let gamma be a group we study the asymptotic behaviour of the mod ell betti numbers in p adic analytic towers of finite index subgroups if theta is a finite ell group of automorphisms of gamma our main theorem allows to lift lower bounds for the mod ell cohomology growth in the fixed point group gamma theta to lower bounds for the growth in gamma we give applications to s arithmetic groups and we also obtain a similar result for cohomology with rational coefficients

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

community detection by signaling on complex networks based on a signaling process of complex networks a method for identification of community structure is proposed for a network with n nodes every node is assumed to be a system which can send receive and record signals each node is taken as the initial signal source to excite the whole network one time then the source node is associated with an n dimensional vector which records the effects of the signaling process by this process the topological relationship of nodes on the network could be transferred into a geometrical structure of vectors in n dimensional euclidean space then the best partition of groups is determined by f statistics and the final community structure is given by the k means clustering method this method can detect community structure both in unweighted and weighted networks it has been applied to ad hoc networks and some real networks such as the zachary karate club network and football team network the results indicate that the algorithm based on the signaling process works well

the monotone iterative method and zeros of bessel functions for nonlinear singular derivative dependent bvp in the presence of upper and lower solutions abstract in this paper we consider a class of nonlinear singular boundary value problems x y x x f x y x x y x 0 0 x 1 y 0 y 1 0 for 1 we assume that the source function f x y x y is lipschitz in x y and one sided lipschitz in y the initial approximations are an upper solution u 0 x and a lower solution v 0 x which can be ordered in one way v 0 x u 0 x or the other u 0 x v 0 x we propose an iterative scheme and establish the existence of solutions bounded by v 0 and u 0 and allow f y to take both positive and negative values the method is constructive in nature and can be used to generate solutions of the nonlinear singular boundary value problems

detecting the optimal number of communities in complex networks abstract to obtain the optimal number of communities is an important problem in detecting community structures in this paper we use the extended measurement of community detecting algorithms to find the optimal community number based on the normalized mutual information index which has been used as a measure for similarity of communities a statistic c is proposed to detect the optimal number of communities in general when c reaches its local maximum especially the first one the corresponding number of communities c is likely to be optimal in community detection moreover the statistic c can also measure the significance of community structures in complex networks which has been paid more attention recently numerical and empirical results show that the index c is effective in both artificial and real world networks

combinatorial approach to modularity communities are clusters of nodes with a higher than average density of internal connections their detection is of great relevance to better understand the structure and hierarchies present in a network modularity has become a standard tool in the area of community detection providing at the same time a way to evaluate partitions and by maximizing it a method to find communities in this work we study the modularity from a combinatorial point of view our analysis as the modularity definition relies on the use of the configurational model a technique that given a graph produces a series of randomized copies keeping the degree sequence invariant we develop an approach that enumerates the null model partitions and can be used to calculate the probability distribution function of the modularity our theory allows for a deep inquiry of several interesting features characterizing modularity such as its resolution limit and the statistics of the partitions that maximize it additionally the study of the probability of extremes of the modularity in the random graph partitions opens the way for a definition of the statistical significance of network partitions

community detection in complex networks using extremal optimization we propose a method to find the community structure in complex networks based on an extremal optimization of the value of modularity the method outperforms the optimal modularity found by the existing algorithms in the literature giving a better understanding of the community structure we present the results of the algorithm for computer simulated and real networks and compare them with other approaches the efficiency and accuracy of the method make it feasible to be used for the accurate identification of community structure in large complex networks

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

a note on existence uniqueness results for a class of doubly singular boundary value problems abstract in this paper we establish the existence and uniqueness of solution for nonlinear singular boundary value problem p x y x q x f x y for 0 x b and lim x 0 p x y x 0 1 y b 1 p b y b 1 here p 0 0 and also q is not assumed to be continuous at 0 so the problem may be doubly singular in this note we discuss the cases when y 0 0 at the singular end is not possible but lim x 0 p x y x 0 regions for multiple solutions have also been determined

a fourth order spline method for singular two point boundary value problems this paper describes a new fourth order spline approximation method for the class of singular two point boundary value problems y frac alpha x y f x y 0 0 x 1 y 0 0 y 1 a for all alpha geqslant 1

accuracy and precision of methods for community identification in weighted networks based on brief review of approaches for community identification and measurement for sensitivity characterization the accuracy and precision of several approaches for detecting communities in weighted networks are investigated in weighted networks the community structure should take both links and link weights into account and the partition of networks should be evaluated by weighted modularity q w the results reveal that link weight has important effects on communities especially in dense networks potts model and weighted extremal optimization weo algorithm work well on weighted networks then potts model and weo algorithms are used to detect communities in rhesus monkey network the results gives nice understanding for real community structure

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

existence and uniqueness of solutions of a class of two point singular nonlinear boundary value problems abstract el gebeily m a a boumenir and m b m elgindi existence and uniqueness of solutions of a class of two point singular nonlinear boundary value problems journal of computational and applied mathematics 46 1993 345 355 this paper is concerned with the existence and uniqueness of solution of a class of two point singular nonlinear boundary value problems it is shown that the problem has a unique solution only for certain boundary conditions under the assumption that the range of f y has empty intersection with the closure of the spectrum of the singular differential operator where f denotes the nonlinearity

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

intersection numbers of special cycles let g q be a connected semisimple algebraic group defined over q and let a r be two q rational automorphisms of finite order of g which commute with each other we assume that the group g r of real points of g is noncompact let i be a torsionfree arithmetic subgroup of g q which is stable under the group a r generated by a and r then a and r act on the space x of maximal compact subgroups of g r and on the locally symmetric space x i in a natural way if g f l denotes the reductive algebraic q group of fixed points of f l a r acting on g then the corresponding symmetric space x f l is acted upon by the arithmetic group 1 f l of fixed points of f l acting on i these give rise to nonempty closed immersed submanifolds c a x a 1 a and c r x r 1 r of x i they are called special cycles we assume that x i c a and c r are oriented and that the intersection c a n c r is compact then if dim c a dim c r dim x the intersection number

properties of highly clustered networks we propose and solve exactly a model of a network that has both a tunable degree distribution and a tunable clustering coefficient among other things our results indicate that increased clustering leads to a decrease in the size of the giant component of the network we also study susceptible infective recovered type epidemic processes within the model and find that clustering decreases the size of epidemics but also decreases the epidemic threshold making it easier for diseases to spread in addition clustering causes epidemics to saturate sooner meaning that they infect a near maximal fraction of the network for quite low transmission rates

on the existence of solutions of a class of singular nonlinear two point boundary value problems we examine the existence of solutions of the class of singular nonlinear two point boundary value problems y a x y ffi f x y 0 l we show that for every a l a unique solution of the singular two point boundary value problem exists provided u kl where u sup 3f ay and k i is the first positive zero of jta l 2 fk j z is bessel s function of the first kind of order s interestingly k i is a monotonically increasing function of a values of k i for some values of a are tabulated

automorphisms and cohomology of discrete groups let g denote a discrete group of finite cohomological dimension calculating the cohomology of such groups is notoriously difficult involving complicated geometric information attached to the group for example the cohomology of torsion free arithmetic groups involves delicate questions about symmetric spaces and number theory perhaps the key difficulty lies in that there is no practical method for building up the cohomology of g from that of its subgroups such as one can do for finite groups in this paper we outline a method for constructing non trivial classes in u h g f where f denotes a field with p elements based on the use of p p finite automorphism groups of g given an explicit presentation for g aut g can often be approached and in particular its finite subgroups are sometimes accessible furthermore it is an elementary observation that many interesting classes of groups admit numerous finite symmetries an obvious but very important example is given by a normal torsion free subgroup g in an arithmetic group u any finite subgroup k u will act on it via conjugation given g aut g we can form the semidirect product g s g g w let h g denote a finite subgroup mapping onto g under the natural projection map then c h s c h l g g and in the particular case g g 4 when h s 1 x x g g we have that

nonresonant nonlinear singular problems in the limit circle case existence results are presented for the resonant singular boundary value problem tfrac 1 p py prime prime lambda m qy f t y a e on 0 1 with limt 0 py y 1 0 here we donot assume int 0 1 tfrac ds p s infty but only that int 0 1 tfrac 1 p s left int 0 s p x q x dx right tfrac 1 2 ds infty

wavelet galerkin method for solving nonlinear singular boundary value problems arising in physiology abstract in this study wavelet galerkin method has been developed for solving nonlinear singular boundary value problems associated with physiology science the quasilinearization technique is applied to reduce the given nonlinear problem to a sequence of linear problems we modify the resulting set of differential equations at the singular point then treat this set of boundary value problems by using compactly supported cubic b spline wavelets and galerkin method these wavelets are used as testing and weighting functions the method is computationally attractive and applications are demonstrated through illustrative examples

smith theory and hecke operators abstract of the connections between the cohomology of arithmetic groups and representations of galois groups some are known and more are conjectured this paper proves one of these conjectures for certain monomial galois representations let p be a prime we show that the representation of the absolute galois group of q induced from an f p valued ray class character of the cyclotomic field q p is attached to a hecke eigenclass in the mod p cohomology of a torsion free congruence subgroup m of gl p 1 z the method involves constructing an action of the hecke algebra on the fundamental exact sequence of smith theory arising from the action by conjugation of an element of order p in gl p 1 z on m

on a class of regular singular two point boundary value problems abstract existence of a unique solution for a class of regular singular two point boundary value problems p x y p x f x y 0 x b lim x 0 y x 0 y b b has been established wherep x satisfies i p x 0 on 0 b ii p x c1 0 r and for somer b iii xp x p x is analytic in z z xp x p x b 0 b 1 x b 0 1 and with quite general conditions onf x y these conditions onf x y are sharp which is seen through one example regions for multiple solutions have also been determined

rational computations of the topological k theory of classifying spaces of discrete groups we compute rationally the topological complex k theory of the classifying space bg of a discrete group provided that g has a cocompact g cw model for its classifying space for proper g actions for instance word hyperbolic groups and cocompact discrete subgroups of connected lie groups satisfy this assumption the answer is given in terms of the group cohomology of g and of the centralizers of finite cyclic subgroups of prime power order we also analyze the multiplicative structure

representation theory and homological stability abstract we introduce the idea of representation stability and several variations for a sequence of representations v n of groups g n a central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability this makes it possible to extend classical theorems of homological stability to a much broader variety of examples representation stability also provides a framework in which to find and to predict patterns from classical representation theory littlewood richardson and murnaghan rules stability of schur functors to cohomology of groups pure braid torelli and congruence groups to lie algebras and their homology to the equivariant cohomology of flag and schubert varieties to combinatorics the n 1 n 1 conjecture the majority of this paper is devoted to exposing this phenomenon through examples in doing this we obtain applications theorems and conjectures beyond the discovery of new phenomena the viewpoint of representation stability can be useful in solving problems outside the theory in addition to the applications given in this paper it is applied by church ellenberg farb in preparation 20 to counting problems in number theory and finite group theory representation stability is also used by church 2012 19 to give broad generalizations and new proofs of classical homological stability theorems for configuration spaces on oriented manifolds