computationally efficient measure of topological redundancy of biological and social networks it is well known that biological and social interaction networks have a varying degree of redundancy though a consensus of the precise cause of this is so far lacking in this paper we introduce a topological redundancy measure for labeled directed networks that is formal computationally efficient and applicable to a variety of directed networks such as cellular signaling and metabolic and social interaction networks we demonstrate the computational efficiency of our measure by computing its value and statistical significance on a number of biological and social networks with up to several thousands of nodes and edges our results suggest a number of interesting observations 1 social networks are more redundant that their biological counterparts 2 transcriptional networks are less redundant than signaling networks 3 the topological redundancy of the c elegans metabolic network is largely due to its inclusion of currency metabolites and 4 the redundancy of signaling networks is highly negatively correlated with the monotonicity of their dynamics

generalized multiple scale reproducing kernel particle methods abstract an approach to unify reproducing kernel methods under one large umbrella and an extension to include time and spatial shifting are proposed the study is divided into three major topics the groundwork is set by revisiting the fourier analysis of discrete systems the multiresolution concept and its significance in devising the reproducing kernel methods and its discrete counterpart reproducing kernel particle methods are explained an edge detection technique based on multiresolution analysis is developed this wavelet approach together with particle methods gives rise to a straightforward h adaptivity algorithm by using this framework a hermite reproducing kernel method is also proposed and its relation to wavelet methods is presented it is also shown that the new approach generalizes existing kernel methods and it can easily be degenerated into other widely used methods such as partition of unity moving least square interpolants smooth particle hydrodynamics scaling functions and wavelets and multiple scale analysis furthermore the hermite reproducing kernel particle method a particle based discrete version of the hermite reproducing kernel method is developed finally multiple scale methods based on frequency and wave number shifting techniques are presented a stability analysis is also presented for newmark time integration schemes for the low frequency equation numerical examples are presented throughout the paper to illustrate the flexibility and accuracy of this class of multiple scale methods

evaluation of the integral terms in reproducing kernel methods reproducing kernel method rkm has its origins in wavelets and it is based on convolution theory being their continuous version rkm is often referred as the general framework for meshless methods in fact since in real computation discretization is inevitable these integrals need to be evaluated numerically leading to the creation of reproducing kernel particle method rkpm and moving least squares mls approximation nevertheless in this paper the integrals in rkm are explicitly evaluated for polynomials basis function and simple geometries in one two and three dimensions even with conforming holes moreover a general formula is provided for complicated shapes also for multiple connected domains this is possible through a boundary formulation where domain integrals involved in rkm are transformed by gauss theorem in circular or flux integrals parallelization is readily enabled since no preliminary arrangements of nodes is needed for the moments matrix furthermore using symbolic inversion computation of shape functions in rkm is considerably speeded up

a parallelized meshfree method with boundary enrichment for large scale cfd a parallel computational implementation of a meshfree method the reproducing kernel particle method rkpm is used for 3 d implicit cfd analysis a novel procedure for implementing the essential boundary condition using the hierarchical enrichment method is presented using this enrichment along the essential boundaries produces results that more closely match experimental and analytical results for a flow past a cylinder problem than does either the finite element method or other meshfree methods that require matrix inversion for the application of essential boundary conditions this technique also allows the efficient parallelization of the algorithm and leads to higher parallel speedups than do other boundary condition implementations many of which involve inherently serial steps this is important because the expense of meshfree computations makes parallelization crucial for large size problems the performance of the parallelization technique and the accuracy of the implicit cfd algorithm are demonstrated in two example problems

convergence analysis of a hierarchical enrichment of dirichlet boundary conditions in a mesh free method implementation of dirichlet boundary conditions in mesh free methods is problematic in wagner and liu international journal for numerical methods in engineering 2001 50 507 a hierarchical enrichment technique is introduced that allows a simple implementation of the dirichlet boundary conditions in this paper we provide some error analysis for the hierarchical enrichment mesh free technique we derive optimal order error estimates for the hierarchical enrichment mesh free interpolants for one dimensional elliptic boundary value problems we can directly apply the interpolation error estimates to obtain error estimates for the mesh free solutions for higher dimensional problems derivation of error estimates for the mesh free solutions depends on the availability of an inverse inequality numerical examples in 1d and 2d are included showing the convergence behaviour of mesh free interpolants and mesh free solutions when the hierarchical enrichment mesh free technique is employed copyright 2001 john wiley sons ltd

a reproducing kernel method with nodal interpolation property a general formulation for developing reproducing kernel rk interpolation is presented this is based on the coupling of a primitive function and an enrichment function the primitive function introduces discrete kronecker delta properties while the enrichment function constitutes reproducing conditions a necessary condition for obtaining a rk interpolation function is an orthogonality condition between the vector of enrichment functions and the vector of shifted monomial functions at the discrete points a normalized kernel function with relative small support is employed as the primitive function this approach does not employ a finite element shape function and therefore the interpolation function can be arbitrarily smooth to maintain the convergence properties of the original rk approximation a mixed interpolation is introduced a rigorous error analysis is provided for the proposed method optimal order error estimates are shown for the meshfree interpolation in any sobolev norms optimal order convergence is maintained when the proposed method is employed to solve one dimensional boundary value problems numerical experiments are done demonstrating the theoretical error estimates the performance of the method is illustrated in several sample problems copyright 2003 john wiley sons ltd

applying clique decomposition for computing gromov hyperbolicity given a graph its hyperbolicity is a measure of how close its distance distribution is to the one of a tree this parameter has gained recent attention in the analysis of some graph algorithms and the classification of complex networks we study on practical improvements for the computation of hyperbolicity in large graphs precisely we investigate on relations between the hyperbolicity of a graph g and the hyperbolicity of its atoms that are the subgraphs output by the clique decomposition invented by tarjan 51 65 we prove that the maximum hyperbolicity taken over the atoms is at most one unit off from the hyperbol icity of g and the bound is sharp we also give an algorithm to slightly modify the atoms called the substitution method which is at no extra cost than computing the clique decomposition and so that the maximum hyperbolicity taken over the resulting graphs is exactly the hyperbolicity of the input graph g an experimental evaluation of our method for computing the hyperbolicity of a given graph from its atoms is provided for collaboration networks and biological networks finally on a more theoretical side we deduce from our results the first linear time algorithm for computing the hyperbolicity of an outerplanar graph

reproducing kernel particle methods for large deformation analysis of non linear structures abstract large deformation analysis of non linear elastic and inelastic structures based on reproducing kernel particle methods rkpm is presented the method requires no explicit mesh in computation and therefore avoids mesh distortion difficulties in large deformation analysis the current formulation considers hyperelastic and elasto plastic materials since they represent path independent and path dependent material behaviors respectively in this paper a material kernel function and an rkpm material shape function are introduced for large deformation analysis the support of the rkpm material shape function covers the same set of particles during material deformation and hence no tension instability is encountered in the large deformation computation the essential boundary conditions are introduced by the use of a transformation method the transformation matrix is formed only once at the initial stage if the rkpm material shape functions are employed the appropriate integration procedures for the moment matrix and its derivative are studied from the standpoint of reproducing conditions in transient problems with an explicit time integration method the lumped mass matrices are constructed at nodal coordinate so that masses are lumped at the particles several hyperelasticity and elasto plasticity problems are studied to demonstrate the effectiveness of the method the numerical results indicated that rkpm handles large material distortion more effectively than finite elements due to its smoother shape functions and consequently provides a higher solution accuracy under large deformation unlike the conventional finite element approach the nodal spacing irregularity in rkpm does not lead to irregular mesh shape that significantly deteriorates solution accuracy no volumetric locking is observed when applying non linear rkpm to nearly incompressible hyperelasticity and perfect plasticity problems further model adaptivity in rkpm can be accomplished simply by adding more points in the highly deformed areas without remeshing

new boundary condition treatments in meshfree computation of contact problems due to the loss of kronecker delta properties in the meshfree shape functions the imposition of essential boundary conditions consumes significant cpu time in meshfree computation in this work two boundary condition treatments are proposed to enhance the computational efficiency of meshfree methods for contact problems the mixed transformation method is modified from the previous full transformation method by introducing a node partitioning and a mixed coordinate so that the matrix inversion and multiplication of coordinate transformation involves operations only on the sub degrees of freedom associated with the boundary group the boundary singular kernel method introduces singularities to the kernel functions of the essential and contact boundary nodes so that the corresponding coefficients of the singular kernel shape functions recover nodal values and consequently kinematic constraints can be imposed directly the effectiveness of the proposed methods is demonstrated in several numerical examples

moving least square reproducing kernel method part ii fourier analysis in part i of this work the moving least square reproducing kernel mlsrk method is formulated and implemented based on its generic construction an m consistency structure is discovered and the convergence theorems are established in this part of the work a systematic fourier analysis is employed to evaluate and further establish the method the preliminary fourier analysis reveals that the mlsrk method is stable for sufficiently dense non degenerated particle distribution in the sense that the kernel function family satisfies the riesz bound one of the novelties of the current approach is to treat the mlsrk method as a variant of the standard finite element method and depart from there to make a connection with the multiresolution approximation in the spirits of multiresolution analysis we propose the following mlsrk transformation fm k hu i 1npu k ik h x xi x w1 the highlight of this paper is to embrace the mlsrk formulation with the notion of the controlled fp approximation based on its characterization the strang fix condition for example a systematic procedure is proposed to design new window functions so they can enhance the computational performance of the mlsrk algorithm the main effort here is to obtain a constant correction function in the interior region of a general domain i e c h 1 this can create a leap in the approximation order of the mlsrk algorithm significantly if a highly smooth window function is embedded within the kernel one consequence of this development is the synchronized convergence phenomenon a unique convergence mechanism for the mlsrk method i e by properly tuning the dilation parameter the convergence rate of higher order error norms will approach the same order convergence rate of the l2 error norm they are synchronized

arbitrarily smooth generalized finite element approximations this paper presents a procedure to build ck k arbitrarily large generalized finite element fe shape functions defined on non structured finite element meshes the functions have the same support as corresponding global fe lagrangian shape functions meshes with both convex and non convex clouds set of elements sharing a vertex node can be used the so called r functions are used to build ck fe based partition of unity functions with non convex support a technique to combine c0 lagrangian fe shape functions with the proposed ck partition of unity is presented the technique allows the use of ck generalized fe shape functions in parts of the computational domain where their high smoothness is required as in the case of problems with distributional boundary conditions and the less computationally demanding c0 generalized fe shape functions elsewhere in the domain a linear elasticity problem with a concentrated moment is solved using the proposed ck generalized fe method higher order distributional boundary conditions can also be handled by the method a detailed convergence analysis is presented for this class of problems as well as for problems in energy space the integrability of the functions using standard gauss legendre rules is also investigated

a meshfree enriched finite element method for compressible and near incompressible elasticity summary in this paper a two dimensional displacement based meshfree enriched fem me fem is presented for the linear analysis of compressible and near incompressible planar elasticity the me fem element is established by injecting a first order convex meshfree approximation into a low order finite element with an additional node the convex meshfree approximation is constructed using the generalized meshfree approximation method and it possesses the kronecker delta property on the element boundaries the gradient matrix of me fem element satisfies the integration constraint for nodal integration and the resultant me fem formulation is shown to pass the constant stress test for the compressible media the me fem interpolation is an element wise meshfree interpolation and is proven to be discrete divergence free in the incompressible limit to prevent possible pressure oscillation in the near incompressible problems an area weighted strain smoothing scheme incorporated with the divergence free me fem interpolation is introduced to provide the smoothing on strains and pressure with this smoothed strain field the discrete equations are derived based on a modified hu washizu variational principle several numerical examples are presented to demonstrate the effectiveness of the proposed method for the compressible and near incompressible problems copyright 2012 john wiley sons ltd

hierarchical enrichment for bridging scales and mesh free boundary conditions the finite element method when used with a basis made up of piecewise polynomials often requires the generation of a very fine computational mesh in order to capture localized solution phenomena such as boundary layers or near singularities enrichment of the basis with additional functions obtained through analytical or experimental means can allow for a coarser mesh and more accurate solution we introduce an enrichment scheme in which an interaction or bridging scale term is used to separate the basis formed by the enrichment functions from the original set of basis functions in effect making the enrichment hierarchical this separation of scales allows the simple application of essential boundary conditions it also allows a quantification of the effects of the enrichment leading to strategies for error estimation and control of the stiffness matrix condition number we also find that this formulation allows for the simple application of essential boundary conditions for mesh free shape functions which are notoriously problematic we find that for multiple dimensions care must be taken to derive a weak form which is truly consistent with the strong form on the essential boundary copyright 2001 john wiley sons ltd

reproducing kernel element method part ii globally conforming im cn hierarchies in this part of the work a minimal degrees of freedom arbitrary smooth globally compatible i m c n interpolation hierarchy is constructed in the framework of reproducing kernel element method rkem for arbitrary multiple dimensional domains this is the first interpolation hierarchical structure that has been constructed with both minimal degrees of freedom and higher order smoothness or continuity over multi dimensional domain the proposed hierarchical structure possesses the generalized kronecker property i e o a w b i ox a x j dijdab jaj jbj 6 m

hyperbolicity measures democracy in real world networks in this work we analyze the hyperbolicity of real world networks a geometric quantity that measures if a space is negatively curved we provide two improvements in our understanding of this quantity first of all in our interpretation a hyperbolic network is aristocratic since few elements connect the system while a non hyperbolic network has a more democratic structure with a larger number of crucial elements the second contribution is the introduction of the average hyperbolicity of the neighbors of a given node through this definition we outline an influence area for the vertices in the graph we show that in real networks the influence area of the highest degree vertex is small in what we define local networks i e social or peer to peer networks and large in global networks i e power grid metabolic networks or autonomous system networks

reproducing kernel element method part i theoretical formulation in this paper and its sequels we introduce and analyze a new class of methods collectively called the reproducing kernel element method rkem the central idea in the development of the new method is to combine the strengths of both finite element methods fem and meshfree methods two distinguished features of rkem are the arbitrarily high order smoothness and the interpolation property of the shape functions these properties are desirable especially in solving galerkin weak forms of higher order partial differential equations and in treating dirichlet boundary conditions so unlike the fem there is no need for special treatment with the rkem in solving high order equations compared to meshfree methods dirichlet boundary conditions do not present any difficulty in using the rkem a rigorous error analysis and convergence study of the method are presented the performance of the method is illustrated and assessed through some numerical examples

topological implications of negative curvature for biological and social networks network measures that reflect the most salient properties of complex large scale networks are in high demand in the network research community in this paper we adapt a combinatorial measure of negative curvature also called hyperbolicity to parametrized finite networks and show that a variety of biological and social networks are hyperbolic this hyperbolicity property has strong implications on the higher order connectivity and other topological properties of these networks specifically we derive and prove bounds on the distance among shortest or approximately shortest paths in hyperbolic networks we describe two implications of these bounds to crosstalk in biological networks and to the existence of central influential neighborhoods in both biological and social networks

error analysis of the reproducing kernel particle method interest in meshfree or meshless methods has grown rapidly in recent years in solving boundary value problems bvps arising in mechanics especially in dealing with difficult problems involving large deformation moving discontinuities etc in this paper we provide a theoretical analysis of the reproducing kernel particle method rkpm which belongs to the family of meshfree methods one goal of the paper is to set up a framework for error estimates of rkpm we introduce the concept of a regular family of particle distributions and derive optimal order error estimates for rkp interpolants on a regular family of particle distributions the interpolation error estimates can be used to yield error estimates for rkp solutions of bvps

mls moving least square based finite elements for three dimensional nonmatching meshes and adaptive mesh refinement abstract with the aid of moving least square mls approximation a new class of three dimensional finite elements are proposed for treating nonmatching meshes and adaptive mesh refinement for which the existing finite elements are hardly efficient with a special choice of the weight function supports and the base functions the method results in useful elements with the polynomial shape function for which the c 1 continuity breaks down on the boundaries between the neighboring subdomains comprising one element the effectiveness of the new elements in handling the discontinuities due to nonmatching interfaces and automatic mesh refinement is demonstrated via three dimensional examples

moving least square reproducing kernel methods i methodology and convergence this paper formulates the moving least square interpolation scheme in a framework of the so called moving least square reproducing kernel mlsrk representation in this study the procedure of constructing moving least square interpolation function is facilitated by using the notion of reproducing kernel formulation which as a generalization of the early discrete approach establishes a continuous basis for a partition of unity this new formulation possesses the quality of simplicity and it is easy to implement moreover the reproducing kernel formula proposed is not only able to reproduce any mth order polynomial exactly on an irregular particle distribution but also serves as a projection operator that can approximate any smooth function globally with an optimal accuracy in this contribution a generic m consistency relation has been found which is the essential property of the mlsrk approximation an interpolation error estimate is given to assess the convergence rate of the approximation it is shown that for sufficiently smooth function the interpolant expansion in terms of sampled values will converge to the original function in the sobolev norms as a meshless method the convergence rate is measured by a new control variable dilation parameter of the window function instead of the mesh size h as usually done in the finite element analysis to illustrate the procedure convergence has been shown for the numerical solution of the second order elliptic differential equations in a galerkin procedure invoked with this interpolant in the numerical example a two point boundary problem is solved by using the method and an optimal convergence rate is observed with respect to various norms

implementation of boundary conditions for meshless methods abstract boundaries and boundary conditions are an aspect of the numerical solution of partial differential equations where meshless methods have had to surmount many initial difficulties due to the lack of a finite element like kronecker delta condition furthermore it is frequently desirable especially in fluid mechanics to impose general nonlinear boundary and interface constraints this paper describes a computationally efficient algorithm based on d alembert s principle that can be used for general constraints both in meshless methods and finite elements first a method of partitioning meshless shape functions suitable for imposing linear boundary conditions is developed subsequently an analogous method is developed for nonlinear constraints special attention is given to imposing general boundary and fluid structure interface conditions on the navier stokes equations in terms of conservative variables numerical results using d alembert s principle with the reproducing kernel particle method rkpm including viscous supersonic flow past a naca 7012 airfoil are shown

data center interconnection networks are not hyperbolic topologies for data center interconnection networks have been proposed in the literature through various graph classes and operations a common trait to most existing designs is that they enhance the symmetric properties of the underlying graphs indeed symmetry is a desirable property for interconnection networks because it minimizes congestion problems and it allows each entity to run the same routing protocol however despite sharing similarities these topologies all come with their own routing protocol recently generic routing schemes have been introduced which can be implemented for any interconnection network the performances of such universal routing schemes are intimately related to the hyperbolicity of the topology roughly graph hyperbolicity is a metric parameter which measures how close is the shortest path metric of a graph from a tree metric the smaller the gap the better motivated by the good performances in practice of these new routing schemes we propose the first general study of the hyperbolicity of data center interconnection networks our findings are disappointingly negative we prove that the hyperbolicity of most data center interconnection topologies scales linearly with their diameter that is the worst case possible for hyperbolicity to obtain these results we introduce original connection between hyperbolicity and the properties of the endomorphism monoid of a graph in particular our results extend to all vertex and edge transitive graphs additional results are obtained for de bruijn and kautz graphs grid like graphs and networks from the so called cayley model

euclidean skeletons of digital image and volume data in linear time by the integer medial axis transform a general algorithm for computing euclidean skeletons of 2d and 3d data sets in linear time is presented these skeletons are defined in terms of a new concept called the integer medial axis ima transform we prove a number of fundamental properties of the ima skeleton and compare these with properties of the cmd centers of maximal disks skeleton several pruning methods for ima skeletons are introduced constant linear and square root pruning and their properties studied the algorithm for computing the ima skeleton is based upon the feature transform using a modification of a linear time algorithm for euclidean distance transforms the skeletonization algorithm has a time complexity which is linear in the number of input points and can be easily parallelized we present experimental results for several data sets looking at skeleton quality memory usage and computation time both for 2d images and 3d volumes