type cotype and twisted sums induced by complex interpolation abstract this paper deals with extensions or twisted sums of banach spaces that come induced by complex interpolation and the relation between the type and cotype of the spaces in the interpolation scale and the nontriviality and singularity of the induced extension the results are presented in the context of interpolation of families of banach spaces and are applied to the study of submodules of schatten classes we also obtain nontrivial extensions of spaces without the cap which also fail the cap

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

efficient modularity density heuristics for large graphs abstract modularity density maximization is a community detection optimization problem which improves the resolution limit degeneracy of modularity maximization this paper presents seven scalable heuristics for modularity density and compares them with literature results from exact mixed integer linear programming and gaod imeme net hain and bmd heuristics the results are also compared with cnm and louvain which are scalable heuristics for modularity maximization the results suggest that our seven heuristics are faster than gaod imeme net hain and bmd modularity density heuristics our experiments also show that some of our heuristics surpassed the objective function value reported by imeme net hain and bmd for some real graphs our seven heuristics were tested with real graphs from the stanford large network dataset collection and the experiments show that they are scalable this feature was confirmed by an amortized complexity analysis which reveals average linear time for three of our heuristics hypothesis tests suggest that four proposed heuristics are state of the art since they are scalable for hundreds of thousands of nodes for the modularity density problem and they find the high objective value partitions for the largest instances ground truth experiments in artificial random graphs were performed and suggest that our heuristics lead to better cluster detection than both cnm and louvain

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

detection of topological patterns in complex networks correlation profile of the internet abstract a general scheme for detecting and analyzing topological patterns in large complex networks is presented in this scheme the network in question is compared with its properly randomized version that preserves some of its low level topological properties statistically significant deviation of any topological property of a network from this null model likely reflects its design principles and or evolutionary history we illustrate this basic scheme using the example of the correlation profile of the internet quantifying correlations between degrees of its neighboring nodes this profile distinguishes the internet from previously studied molecular networks with a similar scale free degree distribution we finally demonstrate that the clustering in a network is very sensitive to both the degree distribution and its correlation profile and compare the clustering in the internet to the appropriate null model

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

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

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

real and complex interpolation methods for finite and infinite families of banach spaces abstract a comparative study is made of the various interpolation spaces generated with respect to n tuples or infinite families of compatible banach spaces by real and complex interpolation methods due to sparr favini lions coifman cwikel rochberg sagher weiss and fernandez certain inclusions are established between these spaces and examples are given showing that in general they do not coincide it is also shown that in contrast to the case of couples of spaces the spaces generated by the above methods may depend on the structure of the containing space in which the banach spaces of the n tuple n 3 or infinite family are embedded finally a construction is given which enables the spaces of sparr and favini lions hitherto defined only with respect to n tuples to also be defined with respect to infinite families of banach spaces

a theory of complex interpolation for families of banach spaces abstract a detailed development is given of a theory of complex interpolation for families of banach spaces which extends the well known theory for pairs of spaces

column generation algorithms for exact modularity maximization in networks finding modules or clusters in networks currently attracts much attention in several domains the most studied criterion for doing so due to newman and girvan phys rev e 69 026113 2004 is modularity maximization many heuristics have been proposed for maximizing modularity and yield rapidly near optimal solution or sometimes optimal ones but without a guarantee of optimality there are few exact algorithms prominent among which is a paper by xu et al eur phys j b 60 231 2007 modularity maximization can also be expressed as a clique partitioning problem and the row generation algorithm of grotschel and wakabayashi math program 45 59 1989 applied we propose to extend both of these algorithms using the powerful column generation methods for linear and non linear integer programming performance of the four resulting algorithms is compared on problems from the literature instances with up to 512 entities are solved exactly moreover the computing time of previously solved problems are reduced substantially

partition signed social networks via clustering dynamics abstract inspired by the dynamics phenomenon occurred in social networks the wjjlgs model is modified to imitate the clustering dynamics of signed social networks analyses show that the clustering dynamics of the model can be applied to partition signed social networks traditionally blockmodel is applied to partition signed networks in this paper a detailed dynamics based algorithm for signed social networks dbas is presented simulations on several typical real world and illustrative networks that have been analyzed by the blockmodel verify the correctness of the proposed algorithm the efficiency of the algorithm is verified on large scale synthetic networks

the domain spaces of quasilogarithmic operators the construction of intermediate banach spaces in interpolation theory and the study of commutator inequalities in this context are closely related to certain nonlinear operators q here an explicit characterization of the domain spaces of these operators q is obtained and the characterization is related to logarithmic sobolev inequalities

l p bounds for the commutators of singular integrals and maximal singular integrals with rough kernels the commutator of convolution type calderon zygmund singular integral operators with rough kernels p v x n are studied the authors established the l p 1 p boundedness of the commutators of singular integrals and maximal singular integrals with the kernel condition which is different from the condition h1 sn 1 mr 2000 subject classification 42b20 42b25 42b99

hypercontractivity for the heat semigroup for ultraspherical polynomials and on the n sphere a logarithmic sobolev inequality with respect to the probability measure a 1 x2 12 dx on 1 1 is proved for all 12 from this inequality sharp hypercontractive estimates are derived for the heat semigroups for ultraspherical polynomials and on the n sphere

edge ratio and community structure in networks a hierarchical divisive algorithm is proposed for identifying communities in complex networks to that effect the definition of community in the weak sense of radicchi et al proc natl acad sci u s a 101 2658 2004 is extended into a criterion for a bipartition to be optimal one seeks to maximize the minimum for both classes of the bipartition of the ratio of inner edges to cut edges a mathematical program is used within a dichotomous search to do this in an optimal way for each bipartition this includes an exact solution of the problem of detecting indivisible communities the resulting hierarchical divisive algorithm is compared with exact modularity maximization on both artificial and real world data sets for two problems of the former kind optimal solutions are found for five problems of the latter kind the edge ratio algorithm always appears to be competitive moreover it provides additional information in several cases notably through the use of the dendrogram summarizing the resolution finally both algorithms are compared on reduced versions of the data sets of girvan and newman proc natl acad sci u s a 99 7821 2002 and of lancichinetti et al phys rev e 78 046110 2008 results for these instances appear to be comparable

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

nonlinear centralizers with values in l0 abstract it is shown that every centralizer from any metric function space x to l 0 is continuous at the origin of x as a consequence every short exact sequence of l modules 0 l 0 z x 0 splits if x is a minimal function space and in particular if x l 0 there are pairs of orlicz function spaces u v such that hom u v 0 but ext u v 0

differentials of complex interpolation processes for kothe function spaces we continue the study of centralizers on kothe function spaces and the commutator estimates they generate see 29 our main result is that if x is a super reflexive kothe function space then for every real centralizer on x there is a complex interpolation scale of kothe function spaces through x inducing as a derivative up to equivalence and a scalar multiple thus in a loose sense all real centralizers can be identified with derivatives of complex interpolation processes we apply our ideas in an appendix to show for example that there is a twisted sum of two hilbert spaces which fails to be a umd space

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

function theoretic results for complex interpolation families of banach spaces the theory of complex interpolation of banach spaces is viewed as a branch of the theory of vector valued holomorphic functions versions of the schwarz lemma liouville s theorem the identity theorem and the reflection principle are proved and are interpreted from the point of view of interpolation theory

loops and multiple edges in modularity maximization of networks the modularity maximization model proposed by newman and girvan for the identification of communities in networks works for general graphs possibly with loops and multiple edges however the applications usually correspond to simple graphs these graphs are compared to a null model where the degree distribution is maintained but edges are placed at random therefore in this null model there will be loops and possibly multiple edges sharp bounds on the expected number of loops and their impact on the modularity are derived then building upon the work of massen and doye but using algebra rather than simulation we propose modified null models associated with graphs without loops but with multiple edges graphs with loops but without multiple edges and graphs without loops nor multiple edges we validate our models by using the exact algorithm for clique partitioning of grotschel and wakabayashi

detecting the community structure in complex networks based on quantum mechanics abstract in this paper we develop a novel method to detect the community structure in complex networks this approach is based on the combination of kernel based clustering using quantum mechanics the spectral clustering technique and the concept of the bayesian information criterion we test the proposed algorithm on zachary s karate club network and the world of american college football experimental results indicate that our algorithm is efficient and effective at finding both the optimal number of clusters and the best clustering of community structures

a characterization of certain shimura curves in the moduli stack of abelian varieties let f x y be a semistable family of complex abelian varieties over a curve y of genus g y and smooth over the com

a distance between orbits that controls commutator estimates and invertibility of operators abstract a distance between orbit spaces generated by a single element is introduced and it is shown that if an operator is invertible in one orbit it is also invertible in nearby orbits thus proving a version of shneiberg s theorem for orbital methods the same machinery is used to extend the celebrated rochberg weiss commutator theorem to the setting of orbital methods it is shown that these results apply to the real and complex methods of interpolation by proving that these methods can be suitably obtained as orbits of a single element

efficient modularity optimization by multistep greedy algorithm and vertex mover refinement identifying strongly connected substructures in large networks provides insight into their coarse grained organization several approaches based on the optimization of a quality function e g the modularity have been proposed we present here a multistep extension of the greedy algorithm msg that allows the merging of more than one pair of communities at each iteration step the essential idea is to prevent the premature condensation into few large communities upon convergence of the msg a simple refinement procedure called vertex mover vm is used for reassigning vertices to neighboring communities to improve the final modularity value with an appropriate choice of the step width the combined msg vm algorithm is able to find solutions of higher modularity than those reported previously the multistep extension does not alter the scaling of computational cost of the greedy algorithm

isogeny classes of abelian varieties over function fields we study finiteness problems for isogeny classes of abelian vari eties over an algebraic function field k in one variable over the field of complex numbers in particular we construct explicitly a non isotrivial absolutely sim ple abelian fourfold x over a certain k such that the isogeny class of x x contains infinitely many mutually non isomorphic principally polarized abelian varieties such examples do not exist when the ground field is finitely gener ated over its prime subfield 2000 mathematics subject classification 14k02 primary 14d07 sec ondary

a cutting plane algorithm for a clustering problem in this paper we consider a clustering problem that arises in qualitative data analysis this problem can be transformed to a combinatorial optimization problem the clique partitioning problem we have studied the latter problem from a polyhedral point of view and determined large classes of facets of the associated polytope these theoretical results are utilized in this paper we describe a cutting plane algorithm that is based on the simplex method and uses exact and heuristic separation routines for some of the classes of facets mentioned before we discuss some details of the implementation of our code and present our computational results we mention applications from e g zoology economics and the political sciences

twisting non commutative lp spaces abstract the paper makes the first steps into the study of extensions twisted sums of noncommutative l p spaces regarded as banach modules over the underlying von neumann algebra m our approach combines kalton s description of extensions by centralizers these are certain maps which are in general neither linear nor bounded with a general principle due to rochberg and weiss saying that whenever one finds a given banach space y as an intermediate space in a complex interpolation scale one automatically gets a self extension 0 y x y 0 for semifinite algebras considering l p l p m as an interpolation space between m and its predual m one arrives at a certain self extension of l p that is a kind of noncommutative kalton peck space and carries a natural bimodule structure some interesting properties of these spaces are presented for general algebras including those of type iii the interpolation mechanism produces two rather than one extensions of one sided modules one of left modules and the other of right modules whether or not one may find nontrivial self extensions of bimodules in all cases is left open

locally optimal heuristic for modularity maximization of networks community detection in networks based on modularity maximization is currently done with hierarchical divisive or agglomerative as well as partitioning heuristics hybrids and in a few papers exact algorithms we consider here the case of hierarchical networks in which communities should be detected and propose a divisive heuristic which is locally optimal in the sense that each of the successive bipartitions is done in a provably optimal way this heuristic is compared with the spectral based hierarchical divisive heuristic of newman proc natl acad sci usa 103 8577 2006 and with the hierarchical agglomerative heuristic of clauset newman and moore phys rev e 70 066111 2004 computational results are given for a series of problems of the literature with up to 4941 vertices and 6594 edges they show that the proposed divisive heuristic gives better results than the divisive heuristic of newman and than the agglomerative heuristic of clauset et al

a unified theory of commutator estimates for a class of interpolation methods abstract a general family of interpolation methods is introduced which includes as special cases the real and complex methods and also the so called or g1 and g2 methods defined by peetre and gustavsson peetre derivation operators and translation operators r are introduced for all methods of this family a theorem is proved about the boundedness of the commutators t and t r for operators t which are bounded on the spaces of the pair to which the interpolation method is applied this extends and unifies results previously known for derivation and translation operators in the contexts of the real and complex methods other results deal with higher order commutators and also include an equivalence theorem i e it is shown that as previously known only for real interpolation spaces all these interpolation spaces have two different equivalent definitions in the style of the j method and k method auxiliary results which may also be of independent interest include the equivalence of lions schechter complex interpolation spaces defined using an annulus with the same spaces defined in the usual way using a strip

a new inequality on the hodge number h 1 1 of algebraic surfaces we get a new inequality on the hodge number h 1 1 s of fibred algebraic complex surfaces s which is a generalization of an inequality of beauville our inequality implies the arakelov type inequalities due to arakelov faltings viehweg and zuo respectively

surveying network community structure in the hidden metric space abstract most real world networks from various fields share a universal topological property as community structure in this paper we propose a node similarity based mechanism to explore the formation of modular networks by applying the concept of hidden metric spaces of complex networks it is demonstrated that network community structure could be formed according to node similarity in the underlying hidden metric space to clarify this we generate a set of observed networks using a typical kind of hidden metric space model by detecting and analyzing corresponding communities both in the observed network and the hidden space we show that the values of the fitness are rather close and the assignments of nodes for these two kinds of community structures detected based on the fitness parameter are extremely matching ones furthermore our research also shows that networks with strong clustering tend to display prominent community structures with large values of network modularity and fitness

detecting complex network modularity by dynamical clustering based on cluster desynchronization properties of phase oscillators we introduce an efficient method for the detection and identification of modules in complex networks the performance of the algorithm is tested on computer generated and real world networks whose modular structure is already known or has been studied by means of other methods the algorithm attains a high level of precision especially when the modular units are very mixed and hardly detectable by the other methods with a computational effort o kn on a generic graph with n nodes and k links

derivatives of analytic families of banach spaces the theory of complex interpolation of banach spaces and operators which was developed by calderon lions and s g krein and extended by us and others centers on the use of the maximum principle for analytic functions in this paper we investigate the significance in interpolation theory of estimates for derivatives of analytic functions we obtain norm estimates for certain linear and non linear commutators and obtain new classical interpolation theorems the paper has four sections in the first section we show how thorin s proof of the riesz thorin theorem can be extended to give an estimate for a non linear commutator and how analogous computations can be done in families of interpolation spaces the second section introduces an extension of the complex interpolation theory which is designed to incorporate in a systematic way the type of estimates obtained in the first section in the third section the details of the abstract theory of the second section are filled in for several standard examples including lp spaces and weighted lp spaces a brief final section contains some general comments and observations