the bergman space the bloch space and commutators of multiplication operators soit d le disque unite ouvert dans le plan complexe c soit h d l ensemble des fonctions analytiques bornees sur d on caracterise les fonctions feh d telles que t f t f tft f est un operateur compact ou t f est l operateur multiplication t f l a 2 l a 2 defini par t f g fg

extremal approximately convex functions and estimating the size of convex hulls abstract a real valued function f defined on a convex set k is an approximately convex function iff it satisfies f x y 2 f x f y 2 1 a thorough study of approximately convex functions is made the principal results are a sharp universal upper bound for lower semi continuous approximately convex functions that vanish on the vertices of a simplex and an explicit description of the unique largest bounded approximately convex function e vanishing on the vertices of a simplex a set a in a normed space is an approximately convex set iff for all a b a the distance of the midpoint a b 2 to a is 1 the bounds on approximately convex functions are used to show that in rn with the euclidean norm for any approximately convex set a any point z of the convex hull of a is at a distance of at most log2 n 1 1 n 1 2 log2 n 1 from a examples are given to show this is the sharp bound bounds for general norms on rn are also given

approximate convexity and submonotonicity abstract it is shown that a locally lipschitz function is approximately convex if and only if its clarke subdifferential is a submonotone operator consequently in finite dimensions the class of locally lipschitz approximately convex functions coincides with the class of lower c 1 functions directional approximate convexity is introduced and shown to be a natural extension of the class of lower c 1 functions in infinite dimensions the following characterization is established a multivalued operator is maximal cyclically submonotone if and only if it coincides with the clarke subdifferential of a locally lipschitz directionally approximately convex function which is unique up to a constant furthermore it is shown that in asplund spaces every regular function is generically approximately convex

advanced modularity specialized label propagation algorithm for detecting communities in networks abstract a modularity specialized label propagation algorithm lpam for detecting network communities was recently proposed this promising algorithm offers some desirable qualities however lpam favors community divisions where all communities are similar in total degree and thus it is prone to get stuck in poor local maxima in the modularity space to escape local maxima we employ a multistep greedy agglomerative algorithm msg that can merge multiple pairs of communities at a time combining lpam and msg we propose an advanced modularity specialized label propagation algorithm lpam experiments show that lpam successfully detects communities with higher modularity values than ever reported in two commonly used real world networks moreover lpam offers a fair compromise between accuracy and speed

analysis of weighted networks the connections in many networks are not merely binary entities either present or not but have associated weights that record their strengths relative to one another recent studies of networks have by and large steered clear of such weighted networks which are often perceived as being harder to analyze than their unweighted counterparts here we point out that weighted networks can in many cases be analyzed using a simple mapping from a weighted network to an unweighted multigraph allowing us to apply standard techniques for unweighted graphs to weighted ones as well we give a number of examples of the method including an algorithm for detecting community structure in weighted networks and a simple proof of the maximum flow minimum cut theorem

coauthorship networks and patterns of scientific collaboration by using data from three bibliographic databases in biology physics and mathematics respectively networks are constructed in which the nodes are scientists and two scientists are connected if they have coauthored a paper we use these networks to answer a broad variety of questions about collaboration patterns such as the numbers of papers authors write how many people they write them with what the typical distance between scientists is through the network and how patterns of collaboration vary between subjects and over time we also summarize a number of recent results by other authors on coauthorship patterns

bmo in the bergman metric on bounded symmetric domains for bounded symmetric domains in cn a notion of bounded mean oscillation in terms of the bergman metric is introduced it is shown that for in l2 dv is in bmo if and only if the densely defined operator m p m p pm on l2 dv is bounded here m is multiplication by and p is the bergman projection with range the bergman subspace h2 dv la2 dv of holomorphic functions in l2 dv an analogous characterization of compactness for m p is provided by functions of vanishing mean oscillation at the boundary of

embedding theorems for bergman spaces via harmonic analysis let a p omega denote the bergman space in the unit disc induced by a radial weight omega with the doubling property int r 1 omega s ds le c int frac 1 r 2 1 omega s ds the positive borel measures such that the differentiation operator of order n in mathbb n cup 0 is bounded from a p omega into l q mu are characterized in terms of geometric conditions when 0 p q infty en route to the proof a theory of tent spaces for weighted bergman spaces is built

function theory on cartan domains and the berezin toeplitz symbol calculus cn with dv z normalized lebesgue measure on 9 we let l l2 q dv be the usual space of lebesgue square integrable complex valued functions on 9 by h2 h2 q dv we denote the bergman subspace of l2 consisting of holomorphic functions the orthogonal projection operator from l2 onto h2 is denoted by p forf in l q the space of essentially bounded measurable functions on 9 we will consider the multiplication operator

a parameter free community detection method based on centrality and dispersion of nodes in complex networks abstract k means is a simple and efficient clustering algorithm to detect communities in networks however it may suffer from a bad choice of initial seeds also called centers that seriously affect the clustering accuracy and the convergence rate additionally in k means the number of communities should be specified in advance till now it is still an open problem on how to select initial seeds and how to determine the number of communities in this study a new parameter free community detection method named k rank d was proposed first based on the fact that good initial seeds usually have high importance and are dispersedly located in a network we proposed a modified pagerank centrality to evaluate the importance of a node and drew a decision graph to depict the importance and the dispersion of nodes then the initial seeds and the number of communities were selected from the decision graph actively and intuitively as the start parameter of k means experimental results on synthetic and real world networks demonstrate the superior performance of our approach over competing methods for community detection

normalized modularity optimization method for community identification with degree adjustment as a fundamental problem in network study community identification has attracted much attention from different fields representing a seminal work in this area the modularity optimization method has been widely applied and studied however this method has issues in resolution limit and extreme degeneracy and may not perform well for networks with unbalanced structures although several methods have been proposed to overcome these limitations they are all based on the original idea of defining modularity through comparing the total number of edges within the putative communities in the observed network with that in an equivalent randomly generated network in this paper we show that this modularity definition is not suitable to analyze some networks such as those with unbalanced structures instead we propose to define modularity through the average degree within the communities and formulate modularity as comparing the sum of average degree within communities of the observed network to that of an equivalent randomly generated network in addition we also propose a degree adjusted approach for further improvement when there are unbalanced structures we analyze the theoretical properties of our degree adjusted method numerical experiments for both artificial networks and real networks demonstrate that average degree plays an important role in network community identification and our proposed methods have better performance than existing ones

structural information and dynamical complexity of networks in 1953 shannon proposed the question of quantification of structural information to analyze communication systems the question has become one of the longest great challenges in information science and computer science here we propose the first metric for structural information given a graph g we define the k dimensional structural information of g or structure entropy of g denoted by mathcal h k g to be the minimum overall number of bits required to determine the k dimensional code of the node that is accessible from random walk in g the k dimensional structural information provides the principle for completely detecting the natural or true structure which consists of the rules regulations and orders of the graphs for fully distinguishing the order from disorder in structured noisy data and for analyzing communication systems solving the shannon s problem and opening up new directions the k dimensional structural information is also the first metric of dynamical complexity of networks measuring the complexity of interactions communications operations and even evolution of networks the metric satisfies a number of fundamental properties including additivity locality robustness local and incremental computability and so on we establish the fundamental theorems of the one and two dimensional structural information of networks including both lower and upper bounds of the metrics of classic data structures general graphs the networks of models and the networks of natural evolution we propose algorithms to approximate the k dimensional structural information of graphs by finding the k dimensional structure of the graphs that minimizes the k dimensional structure entropy we find that the k dimensional structure entropy minimization is the principle for detecting the natural or true structures in real world networks consequently our structural information provides the foundation for knowledge discovering from noisy data we establish a black hole principle by using the two dimensional structure information of graphs we propose the natural rank of locally listing algorithms by the structure entropy minimization principle providing the basis for a next generation search engine

weak factorization and hankel forms for weighted bergman spaces on the unit ball we establish weak factorizations for a weighted bergman space a p alpha with 1 p infty into two weighted bergman spaces on the unit ball of mathbb c n to obtain this result we characterize bounded hankel forms on weighted bergman spaces on the unit ball of mathbb c n

characterizing the functional connectivity diversity of the insula cortex subregions diversity curves and behavior the connectivity of the insula cortex is diverse we present new models to characterize the resting state connectional diversity of the human insula cortex and perform model selection using high quality fmri data from the human connectome project we first attempt to parcellate the insula into distinct subregions using traditional clustering methods but find that the resulting subregions are not homogeneous and that the optimal number of subregions is substantially influenced by data smoothness we then introduce the concept of a diversity curve which we use to continuously parameterize the insula s laplacian eigenmap with respect to streamlines propagated through the eigenmap s gradient field to perform model selection we compare the insula s diversity curve to benchmark diversity curves for i two distinct regions ii a continuum of gradual change and iii an absence of any connectional diversity i e homogenous region of the three benchmarks tested we find that the insula s connectional diversity is most parsimoniously modeled as continuum of gradual change from dorsal posterior to ventral anterior we find that individuals who score high on measures of positive affect self efficacy emotion recognition motor dexterity and gustation show greater diversity within the anterior insula our findings are replicated using data from a second fmri session we conclude that the functional connectivity diversity of the insula can be characterized parsimoniously as a continuum avoiding the vexed task of determining an optimal number of insula subregions and that inter individual variation in this continuum can explain significant variation in behavior

discovering natural communities in networks abstract understanding and detecting natural communities in networks have been a fundamental challenge in networks and in science generally recently we proposed a hypothesis that homophyly kinship is the principle of natural communities based on real network experiments proposed a model of networks to explore the principle of natural selection in nature evolving and proposed the measure of structure entropy of networks here we proposed a community finding algorithm by our measure of structure entropy of networks we found that our community finding algorithm exactly identifies almost all natural communities of networks generated by natural selection if any and that the algorithm exactly identifies or precisely approximates almost all the communities planted in the networks of the existing models we verified that our algorithm identifies or very well approximates the ground truth communities of some real world networks if the ground truth communities are semantically well defined that our algorithm naturally finds the balanced communities and that the communities found by our algorithm may have larger modularity than that by the algorithms based on modularity for some networks our algorithm provides for the first time an approach to detecting and analyzing natural or true communities in real world networks our results demonstrate that structure entropy minimization is the principle of detecting the natural or true communities in large scale networks

personalized pagerank clustering a graph clustering algorithm based on random walks abstract graph clustering has been an essential part in many methods and thus its accuracy has a significant effect on many applications in addition exponential growth of real world graphs such as social networks biological networks and electrical circuits demands clustering algorithms with nearly linear time and space complexity in this paper we propose personalized pagerank clustering ppc that employs the inherent cluster exploratory property of random walks to reveal the clusters of a given graph we combine random walks and modularity to precisely and efficiently reveal the clusters of a graph ppc is a top down algorithm so it can reveal inherent clusters of a graph more accurately than other nearly linear approaches that are mainly bottom up it also gives a hierarchy of clusters that is useful in many applications ppc has a linear time and space complexity and has been superior to most of the available clustering algorithms on many datasets furthermore its top down approach makes it a flexible solution for clustering problems with different requirements

finding community structure in very large networks the discovery and analysis of community structure in networks is a topic of considerable recent interest within the physics community but most methods proposed so far are unsuitable for very large networks because of their computational cost here we present a hierarchical agglomeration algorithm for detecting community structure which is faster than many competing algorithms its running time on a network with n vertices and m edges is o md phantom rule 0 2em 0ex mathrm log phantom rule 0 2em 0ex n where d is the depth of the dendrogram describing the community structure many real world networks are sparse and hierarchical with m ensuremath sim n and d ensuremath sim mathrm log phantom rule 0 2em 0ex n in which case our algorithm runs in essentially linear time o n phantom rule 0 2em 0ex mathrm log 2 phantom rule 0 2em 0ex n as an example of the application of this algorithm we use it to analyze a network of items for sale on the web site of a large on line retailer items in the network being linked if they are frequently purchased by the same buyer the network has more than 400 000 vertices and 2 ifmmode times else texttimes fi 10 6 edges we show that our algorithm can extract meaningful communities from this network revealing large scale patterns present in the purchasing habits of customers

community mining with new node similarity by incorporating both global and local topological knowledge in a constrained random walk abstract detection of community is a crucial step to understand the structure and dynamics of complex networks most of conventional community detection methods focus on optimizing a certain objective function or on clustering nodes based on their similarities which leads to a phenomenon that they have preference for specific types of networks but are not general using constrained random walk we exploit global and local topology structures of network to propose a modified transition matrix and further to define a new similarity metric named isim between two nodes in contrast to the existing similarities isim does not work directly on the observed data but in a convergent stable space this feature makes isim robust to the observed noisy data in real world networks isim not only measures node s distance but also captures node s topology structure in network experiments on synthetic and real world networks demonstrate that isim can be successfully applied to community detection in broader types of networks and outperforms other community detection methods

a general theory of almost convex functions let m t 0 t m r m 1 t i 0 m i 0 t i 1 be the standard m dimensional simplex and let s c m 1 m then a function h c r with domain a convex set in a real vector space is s almost convex iff for all t 0 t m s and x 0 x m c the inequality h t 0 x 0 t m x m 1 t 0 h x 0 t m h x m holds a detailed study of the properties of s almost convex functions is made if s contains at least one point that is not a vertex then an extremal s almost convex function eg n r is constructed with the properties that it vanishes on the vertices of an and if h n r is any bounded s almost convex function with h e k 0 on the vertices of an then h x e s x for all x an in the special case s 1 m 1 1 m 1 the barycenter of m very explicit formulas are given for eg and s n sup x n e s x these are of interest as e s and s n are extremal in various geometric and analytic inequalities and theorems

structure of n clique networks embedded in a complex network we propose the n clique network as a powerful tool for understanding global structures of combined highly interconnected subgraphs and provide theoretical predictions for statistical properties of the n clique networks embedded in a complex network using the degree distribution and the clustering spectrum furthermore using our theoretical predictions we find that the statistical properties are invariant between 3 clique networks and original networks for several observable real world networks with the scale free connectivity and the hierarchical modularity the result implies that structural properties are identical between the 3 clique networks and the original networks

partition of networks into basins of attraction we study partition of networks into basins of attraction based on a steepest ascent search for the node of highest degree each node is associated with or attracted to its neighbor of maximal degree as long as the degree is increasing a node that has no neighbors of higher degree is a peak attracting all the nodes in its basin maximally random scale free networks exhibit different behavior based on their degree distribution exponent gamma for small gamma broad distribution networks are dominated by a giant basin whereas for large gamma narrow distribution there are numerous basins with peaks attracting mainly their nearest neighbors we derive expressions for the first two moments of the number of basins we also obtain the complete distribution of basin sizes for a class of hierarchical deterministic scale free networks that resemble random nets finally we generalize the problem to regular networks and lattices where all degrees are equal and thus the attractiveness of a node must be determined by an assigned weight rather than the degree we derive the complete distribution of basins of attraction resulting from randomly assigned weights in one dimensional chains

extremal approximately convex functions and the best constants in a theorem of hyers and ulam abstract let n 1 and b 2 a real valued function f defined on the n simplex n is approximately convex with respect to b 1 if f i 1 b t i x i i 1 b t i f x i 1 for all x 1 x b n and all t 1 t b b 1 we determine the extremal function of this type which vanishes on the vertices of n we also prove a stability theorem of hyers ulam type which yields as a special case the best constants in the hyers ulam stability theorem for e convex functions

detecting network modules in fmri time series a weighted network analysis approach many network analyses of fmri data begin by defining a set of regions extracting the mean signal from each region and then analyzing the correlations between regions one essential question that has not been addressed in the literature is how to best define the network neighborhoods over which a signal is combined for network analyses here we present a novel unsupervised method for the identification of tightly interconnected voxels or modules from fmri data this approach weighted voxel coactivation network analysis wvcna is based on a method that was originally developed to find modules of genes in gene networks this approach differs from many of the standard network approaches in fmri in that connections between voxels are described by a continuous measure whereas typically voxels are considered to be either connected or not connected depending on whether the correlation between the two voxels survives a hard threshold value additionally instead of simply using pairwise correlations to describe the connection between two voxels wvcna relies on a measure of topological overlap which not only compares how correlated two voxels are but also the degree to which the pair of voxels is highly correlated with the same other voxels we demonstrate the use of wvcna to parcellate the brain into a set of modules that are reliably detected across data within the same subject and across subjects in addition we compare wvcna to ica and show that the wvcna modules have some of the same structure as the ica components but tend to be more spatially focused we also demonstrate the use of some of the wvcna network metrics for assessing a voxel s membership to a module and also how that voxel relates to other modules last we illustrate how wvcna modules can be used in a network analysis to find connections between regions of the brain and show that it produces reasonable results

generalized hilbert operators on weighted bergman spaces abstract the main purpose of this paper is to study the generalized hilbert operator h g f z 0 1 f t g t z d t acting on the weighted bergman space a p where the weight function belongs to the class r of regular radial weights and satisfies the muckenhoupt type condition sup 0 r 1 r 1 t 1 s d s p p d t p p 0 r 1 t p t 1 s d s d t if q p the condition on g that characterizes the boundedness or the compactness of h g a p a q depends on p only but the situation is completely different in the case q p in which the inducing weight plays a crucial role the results obtained also reveal a natural connection to the muckenhoupt type condition indeed it is shown that the classical hilbert operator the case g z log 1 1 z of h g is bounded from l t 1 s d s p 0 1 the natural restriction of a p to functions defined on 0 1 to a p if and only if satisfies the condition on the way to these results decomposition norms for the weighted bergman space a p are established

hankel operators induced by radial bekolle bonami weights on bergman spaces we study big hankel operators h f nu a p omega rightarrow l q nu generated by radial bekolle bonami weights nu when 1 p le q infty here the radial weight omega is assumed to satisfy a two sided doubling condition and a p omega denotes the corresponding weighted bergman space a characterization for simultaneous boundedness of h f nu and h overline f nu is provided in terms of a general weighted mean oscillation compared to the case of standard weights that was recently obtained by pau et al indiana univ math j 65 5 1639 1673 2016 the respective spaces depend on the weights omega and nu in an essentially stronger sense this makes our analysis deviate from the blueprint of this more classical setting as a consequence of our main result we also study the case of anti analytic symbols