on the uniqueness of euclidean distance matrix completions the case of points in general position the euclidean distance matrix completion problem edmcp is the problem of determining whether or not a given partial matrix can be completed into a euclidean distance matrix edm in this paper we investigate the necessary and sufficient conditions for the uniqueness of a given edm completion in the case where this edm completion is generated by points in general position we also show that the problem of checking the validity of these conditions can be formulated as a semidefinite programming problem

further relaxations of the semidefinite programming approach to sensor network localization recently a semidefinite programming sdp relaxation approach has been proposed to solve the sensor network localization problem although it achieves high accuracy in estimating the sensor locations the speed of the sdp approach is not satisfactory for practical applications in this paper we propose methods to further relax the sdp relaxation more precisely to relax the single semidefinite matrix cone into a set of small size semidefinite submatrix cones which we call a sub sdp ssdp approach we present two such relaxations although they are weaker than the original sdp relaxation they retain the key theoretical property and numerical experiments show that they are both efficient and accurate the speed of the ssdp is even faster than that of other approaches based on weaker relaxations the ssdp approach may also pave a way to efficiently solving general sdp problems without sacrificing the solution quality

an efficient algorithm for testing the generic rigidity of graphs in the plane given a structure made up of n sites connected by b bars the problem of recognizing which subsets of sites form rigid units is not a trivial one because of the non local character of rigidity in central force systems even though this is a very old problem of statics no simple algorithms are available for it so the most usual approach has been to solve the elastic equations which is very time consuming for large systems recently an integer algorithm was proposed for this problem in two dimensions using matching methods from graph theory and laman s theorem for two dimensional graphs the method is relatively simple but its time complexity grows as in the worst case and almost as fast on practical cases so that an improvement is highly desirable in this paper we describe a further elaboration of that procedure which relies upon the description of the system as a collection of rigid bodies connected by bars instead of sites connected by bars sets of rigidly connected objects are replaced by a unique body and this is done recursively as more rigid connections between bodies are discovered at larger scales as a consequence of this rescaling transformation our algorithm has much improved average behaviour even when its worst case complexity remains the time complexity of the body bar algorithm is found to scale as for the randomly diluted triangular lattice while the original site bar version scales as for the same problem

large scale molecular optimization from distance matrices by a d c optimization approach a so called dca method based on a d c difference of convex functions optimization approach algorithm for solving large scale distance geometry problems is developed different formulations of equivalent d c programs in the l 1 approach are stated via the lagrangian duality without gap relative to d c programming and new nonstandard nonsmooth reformulations in the l infty approach resp the l 1 l infty approach are introduced substantial subdifferential calculations permit us to compute sequences of iterations in the dca quite simply the computations actually require matrix vector products and only one cholesky factorization resp with an additional solution of a convex program in the l 1 approach resp the l 1 l infty approach and allow the exploitation of sparsity in the large scale setting two techniques respectively using shortest paths between all pairs of atoms to generate the complete dissimilarity matrix and the spanning trees procedure are investigated in order to compute a good starting point for the dca finally many numerical simulations of the molecular optimization problems with up to 12567 variables are reported which prove the practical usefulness of the nonstandard nonsmooth reformulations the globality of found solutions and the robustness and efficiency of our algorithms

detect overlapping and hierarchical community structure in networks abstract clustering and community structure is crucial for many network systems and the related dynamic processes it has been shown that communities are usually overlapping and hierarchical however previous methods investigate these two properties of community structure separately this paper proposes an algorithm eagle to detect both the overlapping and hierarchical properties of complex community structure together this algorithm deals with the set of maximal cliques and adopts an agglomerative framework the quality function of modularity is extended to evaluate the goodness of a cover the examples of application to real world networks give excellent results

effects of high order correlations on personalized recommendations for bipartite networks in this paper we introduce a modified collaborative filtering mcf algorithm which has remarkably higher accuracy than the standard collaborative filtering in the mcf instead of the cosine similarity index the user user correlations are obtained by a diffusion process furthermore by considering the second order correlations we design an effective algorithm that depresses the influence of mainstream preferences simulation results show that the algorithmic accuracy measured by the average ranking score is further improved by 20 45 and 33 25 in the optimal cases of movielens and netflix data more importantly the optimal value depends approximately monotonously on the sparsity of the training set given a real system we could estimate the optimal parameter according to the data sparsity which makes this algorithm easy to be applied in addition two significant criteria of algorithmic performance diversity and popularity are also taken into account numerical results show that as the sparsity increases the algorithm considering the second order correlation can outperform the mcf simultaneously in all three criteria

characterizing generic global rigidity a d dimensional framework is a graph and a map from its vertices to ed such a framework is globally rigid if it is the only framework in ed with the same graph and edge lengths up to rigid motions for which underlying graphs is a generic framework globally rigid we answer this question by proving a conjecture by connelly that his sucient condition is also necessary the condition comes from considering the geometry of the length squared mapping essentially the graph is generically locally rigid i the rank of is maximal and it is generically globally rigid i the rank of the gauss map on the image of is maximal we also show that this condition is eciently checkable with a randomized algorithm

rigidity transition in two dimensional random fiber networks rigidity percolation is analyzed in two dimensional random fibrous networks the model consists of central forces between the adjacent crossing points of the fibers two strategies are used to incorporate rigidity adding extra constraints between second nearest crossing points with a probability p sn and welding individual crossing points by adding there four additional constraints with a probability p weld and thus fixing the angles between the fibers these additional constraints will make the model rigid at a critical probability p sn p sn c and p weld p weld c respectively accurate estimates are given for the transition thresholds and for some of the associated critical exponents the transition is found in both cases to be in the same universality class as that of the two dimensional central force rigidity percolation in diluted lattices

rigidity of random networks of stiff fibers in the low density limit rigidity percolation is analyzed in two dimensional random networks of stiff fibers as fibers are randomly added to the system there exists a density threshold q q min above which a rigid stress bearing percolation cluster appears this threshold is found to be above the connectivity percolation threshold q q c such that q min 1 1698 0 0004 q c the transition is found to be continuous and in the universality class of the two dimensional central force rigidity percolation on lattices at percolation threshold the rigid backbone of the percolating cluster was found to break into rigid clusters whose number diverges in the limit of infinite system size when a critical bond is removed the scaling with system size of the average size of these clusters was found to give a new scaling exponent delta 1 61 0 04

reconstructing a three dimensional model with arbitrary errors a number of current technologies allow for the determination of interatomic distance information in structures such as proteins and rna thus the reconstruction of a three dimensional set of points using information about its interpoint distances has become a task of basic importance in determining molecular structure the distance measurements one obtains from techniques such as nmr are typically sparse and error prone greatly complicating the reconstruction task many of these errors result in distance measurements that can be safely assumed to lie within certain fixed tolerances but a number of sources of systematic error in these experiments lead to inaccuracies in the data that are very hard to quantify in effect one must treat certain entries of the measured distance matrix as being arbitrarily corrupted the existence of arbitrary errors leads to an interesting sort of error correction problem how many corrupted entries in a distance matrix can be efficiently corrected to produce a consistent three dimensional structure for the case of an n n matrix in which every entry is specified we provide a randomized algorithm running in time o n log n that enumerates all structures consistent with at most 1 2 e n errors per row with high probability in the case of randomly located errors we can correct errors of the same density in a sparse matrix one in which only a fraction of the entries in each row are given for any constant gt 0

adaptive fuzzy control via observer design for uncertain nonlinear systems with unmodeled dynamics in this paper the problems of stability and tracking control for a class of large scale nonlinear systems with unmodeled dynamics are addressed by designing the decentralized adaptive fuzzy output feedback approach because the dynamic surface control technique is introduced the designed controllers can avoid the issue of explosion of complexity which comes from the traditional backstepping design procedure that deals with large scale nonlinear systems with unmodeled dynamics in addition a reduced order observer is designed to estimate those immeasurable states based on the lyapunov stability method it is proven that all the signals in the closed loop system are bounded and the system outputs track the reference signals to a small neighborhood of the origin by choosing the design parameters appropriately the simulation examples are given to verify the effectiveness of the proposed techniques

graph realizations associated with minimizing the maximum eigenvalue of the laplacian in analogy to the absolute algebraic connectivity of fiedler we study the problem of minimizing the maximum eigenvalue of the laplacian of a graph by redistributing the edge weights via semidefinite duality this leads to a graph realization problem in which nodes should be placed as close as possible to the origin while adjacent nodes must keep a distance of at least one we prove three main results for a slightly generalized form of this embedding problem first given a set of vertices partitioning the graph into several or just one part the barycenter of each part is embedded on the same side of the affine hull of the set as the origin second there is an optimal realization of dimension at most the tree width of the graph plus one and this bound is best possible in general finally bipartite graphs possess a one dimensional optimal embedding

adaptive nn controller design for a class of nonlinear mimo discrete time systems an adaptive neural network tracking control is studied for a class of multiple input multiple output mimo nonlinear systems the studied systems are in discrete time form and the discretized dead zone inputs are considered in addition the studied mimo systems are composed of n subsystems and each subsystem contains unknown functions and external disturbance due to the complicated framework of the discrete time systems the existence of the dead zone and the noncausal problem in discrete time it brings about difficulties for controlling such a class of systems to overcome the noncausal problem by defining the coordinate transformations the studied systems are transformed into a special form which is suitable for the backstepping design the radial basis functions nns are utilized to approximate the unknown functions of the systems the adaptation laws and the controllers are designed based on the transformed systems by using the lyapunov method it is proved that the closed loop system is stable in the sense that the semiglobally uniformly ultimately bounded of all the signals and the tracking errors converge to a bounded compact set the simulation examples and the comparisons with previous approaches are provided to illustrate the effectiveness of the proposed control algorithm

interrogating witnesses for geometric constraint solving classically geometric constraint solvers use graph based methods to decompose systems of geometric constraints these methods have intrinsic limitations which the witness method overcomes a witness is a solution of a variant of the system this paper details the computation of a basis of the vector space of free infinitesimal motions of a typical witness and explains how to use this basis to interrogate the witness for dependence detection the paper shows that the witness method detects all kinds of dependences structural dependences already detectable by graph based methods but also non structural dependences due to known or unknown geometric theorems which are undetectable by graph based methods it also discusses how to decide about the rigidity of a witness and how to decompose it

bearing rigidity and almost global bearing only formation stabilization a fundamental problem that the bearing rigidity theory studies is to determine when a framework can be uniquely determined up to a translation and a scaling factor by its inter neighbor bearings while many previous works focused on the bearing rigidity of two dimensional frameworks a first contribution of this paper is to extend these results to arbitrary dimensions it is shown that a framework in an arbitrary dimension can be uniquely determined up to a translation and a scaling factor by the bearings if and only if the framework is infinitesimally bearing rigid in this paper the proposed bearing rigidity theory is further applied to the bearing only formation stabilization problem where the target formation is defined by inter neighbor bearings and the feedback control uses only bearing measurements nonlinear distributed bearing only formation control laws are proposed for the cases with and without a global orientation it is proved that the control laws can almost globally stabilize infinitesimally bearing rigid formations numerical simulations are provided to support the analysis

spectral properties of the alignment matrices in manifold learning local methods for manifold learning generate a collection of local parameterizations which is then aligned to produce a global parameterization of the underlying manifold the alignment procedure is carried out through the computation of a partial eigendecomposition of a so called alignment matrix in this paper we present an analysis of the eigenstructure of the alignment matrix giving both necessary and sufficient conditions under which the null space of the alignment matrix recovers the global parameterization we show that the gap in the spectrum of the alignment matrix is proportional to the square of the size the precise definition of the size is given in section sec sp of the overlap of the local parameterizations thus deriving a quantitative measure of how stably the null space can be computed numerically we also give a perturbation analysis of the null space of the alignment matrix when the computation of the local parameterizations is subject to error our analysis provides insights into the behaviors and performance of local manifold learning algorithms

generic rigidity in three dimensional bond bending networks much progress can be made in studying the mechanical stability of frameworks when they are treated as generic which lack any special symmetries this is because testing for rigidity becomes topological in nature rather than geometrical generic rigidity synonymous with graph rigidity depends only on the connectivity of the network making it a simpler problem to deal with in principle a complete combinatorial constraint counting characterization of graph rigidity is given by laman s theorem for two dimensions unfortunately there is no known corresponding theorem for three dimensions herein it is proposed that the theorem of laman generalizes to three dimensions for bar joint networks that have no implied hinge joints particular attention is given to bond bending networks having a truss structure with constraints between nearest and next nearest neighbours that are suitable for modelling many covalent network glasses and macromolecules it is shown that implied hinge joints do not exist in bond bending networks based on the proposition that an all subgraph constraint counting characterization of generic rigidity is recovered in three dimensional bar joint networks having no implied hinge joints an efficient combinatorial algorithm is constructed for bond bending networks complete agreement is found with exact calculations involving diagonalization of dynamical matrices for systems up to degrees of freedom

global registration of multiple point clouds using semidefinite programming consider n points in r d and m local coordinate systems that are related through unknown rigid transforms for each point we are given possibly noisy measurements of its local coordinates in some of the coordinate systems alternatively for each coordinate system we observe the coordinates of a subset of the points the problem of estimating the global coordinates of the n points up to a rigid transform from such measurements comes up in distributed approaches to molecular conformation and sensor network localization and also in computer vision and graphics the least squares formulation of this problem although nonconvex has a well known closed form solution when m 2 based on the singular value decomposition svd however no closed form solution is known for m geq 3 in this paper we demonstrate how the least squares formulation can be relaxed into a convex program namely a semidefinite program sdp by setting up connections between the uniqueness of this sdp and results fr

a simpler approach to matrix completion this paper provides the best bounds to date on the number of randomly sampled entries required to reconstruct an unknown low rank matrix these results improve on prior work by candes and recht 2009 candes and tao 2009 and keshavan et al 2009 the reconstruction is accomplished by minimizing the nuclear norm or sum of the singular values of the hidden matrix subject to agreement with the provided entries if the underlying matrix satisfies a certain incoherence condition then the number of entries required is equal to a quadratic logarithmic factor times the number of parameters in the singular value decomposition the proof of this assertion is short self contained and uses very elementary analysis the novel techniques herein are based on recent work in quantum information theory

an sdp based divide and conquer algorithm for large scale noisy anchor free graph realization we propose the disco algorithm for graph realization in mathbb r d given sparse and noisy short range intervertex distances as inputs our divide and conquer algorithm works as follows when a group has a sufficiently small number of vertices the basis step is to form a graph realization by solving a semidefinite program the recursive step is to break a large group of vertices into two smaller groups with overlapping vertices these two groups are solved recursively and the subconfigurations are stitched together using the overlapping atoms to form a configuration for the larger group at intermediate stages the configurations are improved by gradient descent refinement the algorithm is applied to the problem of determining protein moleculer structure tests are performed on molecules taken from the protein data bank database for each molecule given 20 30 of the inter atom distances less than 6a that are corrupted by a high level of noise disco is able to reliably and efficiently reconstruct the conformation of large molecules in particular given 30 of distances with 20 multiplicative noise a 13000 atom conformation problem is solved within an hour with a root mean square deviation of 1 6a

generalized weak rigidity theory and local and global convergence of formations abstract this paper proposes a generalized weak rigidity theory and aims to apply the theory to formation control problems with a gradient descent flow law the generalized weak rigidity theory is utilized in order to characterize desired rigid formations by a general set of pure inter agent distances and subtended angles where the rigid formation shape with distances and subtended angles is determined up to translations and rotations if the formation shape is composed of only subtended angles then it is determined up to translations rotations and additionally scaling factors as the first result of its applications this paper provides analysis of local exponential stability for a formation control system with pure distance angle or only angle constraints in 2 and 3 dimensional spaces then as the second result it is shown that if there are three agents in 2 dimensional space then almost global exponential stability is ensured for a formation control system with pure distance angle or only angle constraints

global continuation for distance geometry problems distance geometry problems arise in the determination of protein structure we consider the case where only a subset of the distances between atoms is given and formulate this distance geometry problem as a global minimization problem with special structure we show that global smoothing techniques and a continuation approach for global optimization can be used to determine global solutions of this problem reliably and efficiently the global continuation approach determines a global solution with less computational effort than is required by a standard multistart algorithm moreover the continuation approach usually finds the global solution from any given starting point while the multistart algorithm tends to fail

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

explicit sensor network localization using semidefinite representations and facial reductions the sensor network localization snl problem in embedding dimension r consists of locating the positions of wireless sensors given only the distances between sensors that are within radio range and the positions of a subset of the sensors called anchors current solution techniques relax this problem to a weighted nearest positive semidefinite programming sdp completion problem by using the linear mapping between euclidean distance matrices edm and semidefinite matrices the resulting sdp is solved using primal dual interior point solvers yielding an expensive and inexact solution this relaxation is highly degenerate in the sense that the feasible set is restricted to a low dimensional face of the sdp cone implying that the slater constraint qualification fails cliques in the graph of the snl problem give rise to this degeneracy in the sdp relaxation in this paper we take advantage of the absence of the slater constraint qualification and derive a technique for the snl problem with exact data that explicitly solves the corresponding rank restricted sdp problem no sdp solvers are used for randomly generated instances we are able to efficiently solve many huge instances of this np hard problem to high accuracy by finding a representation of the minimal face of the sdp cone that contains the sdp matrix representation of the edm the main work of our algorithm consists in repeatedly finding the intersection of subspaces that represent the faces of the sdp cone that correspond to cliques of the snl problem

graph rigidity via euclidean distance matrices abstract let g v e be an incomplete graph with node set v edge set e and nonnegative weights ij s on the edges let each edge v i v j be viewed as a rigid bar of length ij which can rotate freely around its end nodes a realization of a graph g is an assignment of coordinates in some euclidean space to each node of g in this paper we consider the problem of determining whether or not a given realization of a graph g is rigid we show that each realization of g can be epresented as a point in a compact convex set r m and that a generic realization of g is rigid if and only if its corresponding point is a vertex of i e an extreme point with full dimensional normal cone

the molecule problem exploiting structure in global optimization the molecule problem is that of determining the relative locations of a set of objects in euclidean space relying only upon a sparse set of pairwise distance measurements this np hard problem has applications in the determination of molecular conformation the molecule problem can be naturally expressed as a continuous global optimization problem but it also has a rich combinatorial structure this paper investigates how that structure can be exploited to simplify the optimization problem in particular we present a novel divide and conquer algorithm in which a large global optimization problem is replaced by a sequence of smaller ones since the cost of the optimization can grow exponentially with problem size this approach holds the promise of a substantial improvement in performance our algorithmic development relies upon some recently published results in graph theory we describe an implementation of this algorithm and report some results of its performance on a sample molecule

combinatorial models of rigidity and renormalization we first introduce the percolation problems associated with the graph theoretical concepts of k l sparsity and make contact with the physical concepts of ordinary and rigidity percolation we then devise a renormalization transformation for k l percolation problems and investigate its domain of validity in particular we show that it allows an exact solution of k l percolation problems on hierarchical graphs for k l 2k we introduce and solve by renormalization such a model which has the interesting feature of showing both ordinary percolation and rigidity percolation phase transitions depending on the values of the parameters

solving partial differential equations on manifolds from incomplete interpoint distance solutions of partial differential equations pdes on manifolds have provided important applications in different fields in science and engineering existing methods are mainly based on discretization of manifolds as implicit functions triangle meshes or point clouds where the manifold structure is approximated by either zero level set of an implicit function or a set of points in many applications manifolds might be only provided as an interpoint distance matrix with possible missing values this paper discusses a framework to discretize pdes on manifolds represented as incomplete interpoint distance information without conducting a time consuming global coordinates reconstruction we propose a more efficient strategy by discretizing differential operators only based on pointwisely local reconstruction our local reconstruction model is based on the recent advances of low rank matrix completion theory where only a very small random portion of distance information is required this method enables u

a novel cluster validity index for fuzzy clustering based on bipartite modularity a novel cluster validity index whose implementation is based on the membership degrees and improved bipartite modularity of bipartite network is proposed for the validation of partitions produced by the fuzzy c means fcm algorithm fcm algorithm is employed to group the dataset in order to obtain the membership degree of samples then a weighted bipartite network is constructed by samples and centroids of each cluster this allows the introduction of a new measurement for optimizing the numbers of clusters for fuzzy partitions the proposed index utilizes the optimum membership as its global property and the modularity of bipartite network as its local independent property the proposed index is compared with a number of popular validation indices on fifteen datasets the experimental results show that the effectiveness and reliability of the proposal is superior to other indices

rigidity percolation in a field rigidity percolation with g degrees of freedom per site is analyzed on randomly diluted erdos renyi graphs with average connectivity gamma in the presence of a field h in the gamma h plane the rigid and flexible phases are separated by a line of first order transitions whose location is determined exactly this line ends at a critical point with classical critical exponents analytic expressions are given for the densities n f of uncanceled degrees of freedom and gamma r of redundant bonds upon crossing the coexistence line gamma r and n f are continuous although their first derivatives are discontinuous we extend for the case of nonzero field a recently proposed hypothesis namely that the density of uncanceled degrees of freedom is a free energy for rigidity percolation analytic expressions are obtained for the energy entropy and specific heat some analogies with a liquid vapor transition are discussed particularizing to zero field we find that the existence of a g 1 core is a necessary condition for rigidity percolation with g degrees of freedom at the transition point gamma c maxwell counting of degrees of freedom is exact on the rigid cluster and on the g 1 rigid core i e the average coordination of these subgraphs is exactly 2g although gamma c the average coordination of the whole system is smaller than 2g gamma c is found to converge to 2g for large g i e in this limit maxwell counting is exact globally as well

conditions for unique graph realizations the graph realization problem is that of computing the relative locations of a set of vertices placed in euclidean space relying only upon some set of inter vertex distance measurements this paper is concerned with the closely related problem of determining whether or not a graph has a unique realization both these problems are np hard but the proofs rely upon special combinations of edge lengths if one assumes the vertex locations are unrelated then the uniqueness question can be approached from a purely graph theoretic angle that ignores edge lengths this paper identifies three necessary graph theoretic conditions for a graph to have a unique realization in any dimension efficient sequential and nc algorithms are presented for each condition although these algorithms have very different flavors in different dimensions

designing optimal multiplex networks for certain laplacian spectral properties we discuss the design of interlayer edges in a multiplex network under a limited budget with the goal of improving its overall performance we analyze the following three problems separately first we maximize the smallest nonzero eigenvalue also known as the algebraic connectivity second we minimize the largest eigenvalue also known as the spectral radius and finally we minimize the spectral width maximizing the algebraic connectivity requires identical weights on the interlayer edges for budgets less than a threshold value however for larger budgets the optimal weights are generally nonuniform the dual formulation transforms the problem into a graph realization embedding problem that allows us to give a fuller picture namely before the threshold budget the optimal realization is one dimensional with nodes in the same layer embedded to a single point while beyond the threshold the optimal embeddings generally unfold into spaces with dimension bounded by the multiplicity of the algebraic connectivity finally for extremely large budgets the embeddings again revert to lower dimensions minimizing the largest eigenvalue is driven by the spectral radius of the individual networks and its corresponding eigenvector before a threshold the total budget is distributed among interlayer edges corresponding to the nodal lines of this eigenvector and the optimal largest eigenvalue of the laplacian remains constant for larger budgets the weight distribution tends to be almost uniform in the dual picture the optimal graph embedding is one dimensional and nonhomogeneous at first with the nodes corresponding to the layer with the largest spectral radius distributed on a line according to its eigenvector while the other layer is embedded at the origin beyond this threshold the optimal embedding expands to be multidimensional and for larger values of the budget the two layers fill the embedding space finally we show how these two problems are connected to minimizing the spectral width

modularity and community detection in bipartite networks the modularity of a network quantifies the extent relative to a null model network to which vertices cluster into community groups we define a null model appropriate for bipartite networks and use it to define a bipartite modularity the bipartite modularity is presented in terms of a modularity matrix b some key properties of the eigenspectrum of b are identified and used to describe an algorithm for identifying modules in bipartite networks the algorithm is based on the idea that the modules in the two parts of the network are dependent with each part mutually being used to induce the vertices for the other part into the modules we apply the algorithm to real world network data showing that the algorithm successfully identifies the modular structure of bipartite networks

a connection between positive semidefinite and euclidean distance matrix completion problems abstract the positive semidefinite and euclidean distance matrix completion problems have received a lot of attention in the literature results have been obtained for these two problems that are very similar in their formulations although there is a strong relationship between positive semidefinite matrices and euclidean distance matrices it was not clear as noted by johnson et al how to link the two completion problems the purpose of this note is twofold first we show how the results for the euclidean distance matrix completion problem can be derived from the corresponding results for the positive semidefinite completion problem using a functional transform introduced by schoenberg second we introduce a new set of necessary conditions that are stronger than some previously known ones and we identify the graphs for which these conditions suffice for ensuring completability

degree distributions of evolving alphabetic bipartite networks and their projections many real world systems are modeled as evolving bipartite networks and their one mode projections in particular discrete combinatorial systems dcss which consist of a finite set of elementary units and different combinations of these units can be modeled by a subclass of bipartite networks known as alphabetic bipartite networks or a bins where the bottom partite set contains a fixed number of nodes the elementary units and the top partite set grows unboundedly with time through the addition of nodes the combinations the principal questions in the study of a bin evolution are to predict the degree distribution of the bottom set and that of the projection onto the bottom set the bottom projection from a knowledge of the bipartite growth dynamics in this paper we propose a realistic growth model for a bins where the degree distribution of the top set i e the distribution of the number of elementary units in the composite entities in the dcs can be any arbitrary distribution with finite first and second moments utilizing an exact correspondence between the preferential growth of a bins and the polya urn scheme we analytically solve the model to compute exact degree distributions of the bottom fixed set and the bottom projection to the best of our knowledge this is the first work which proposes and solves such a generalized growth model for a bins we also derive that the degree distributions of both the bottom set and the bottom projection suitably normalized with time converge to distributions that are invariant over time we also verify that this improved model can accurately explain the degree distributions of several real world dcss and their projections

local similarity solutions for boundary layer flow of a fene p fluid we study the boundary layer flow past a semi infinite plate of a viscoelastic fluid governed by the fene p model in the limit as the reynolds number re and the wessenberg number we is such that were 1 2 o 1 the stream function is governed by a generalized blasuis equation a local similarity solution is obtained by a shooting method the effect of extensibility on the skin friction and first normal stress coefficients are discussed we show that the skin friction increases with the extensibility parameter b as b approaches infinity the result reduces to blasius solution compared to blasius solution a reduction of up to 20 in the skin friction coefficient is possible for finite values of b

distance geometry algorithms in molecular modelling of polymer and composite systems abstract in this paper microscopic atomistically detailed models of polymer and composite systems are studied in the context of distance geometry algorithms the development of such algorithms in the general framework of molecular modelling is closely associated with problems in computational optimisation and is pursued here as a key to a better understanding of properties of polymer and composite systems at the macroscopic level applications of two distance geometry algorithms to topological optimisation of alkane chains and bulk decane structures are discussed with results of numerical simulations

euclidean distance geometry and applications euclidean distance geometry is the study of euclidean geometry based on the concept of distance this is useful in several applications where the input data consist of an incomplete set of distances and the output is a set of points in euclidean space realizing those given distances we survey the theory of euclidean distance geometry and its most important applications with special emphasis on molecular conformation problems

exponential stability for formation control systems with generalized controllers a unified approach this paper discusses generalized controllers for distance based rigid formation shape stabilization and aims to provide a unified approach for the convergence analysis we consider two types of formation control systems according to different characterizations of target formations minimally rigid target formation and non minimally rigid target formation for the former case we firstly prove the local exponential stability for rigid formation systems when using a general form of shape controllers with certain properties from this viewpoint different formation controllers proposed in previous literature can be included in a unified framework we then extend the result to the case that the target formation is non minimally rigid and show that exponential stability of the formation system is still guaranteed with generalized controllers

cycles and clustering in bipartite networks we investigate the clustering coefficient in bipartite networks where cycles of size three are absent and therefore the standard definition of clustering coefficient cannot be used instead we use another coefficient given by the fraction of cycles with size four showing that both coefficients yield the same clustering properties the new coefficient is computed for two networks of sexual contacts one bipartite and another where no distinction between the nodes is made monopartite in both cases the clustering coefficient is similar furthermore combining both clustering coefficients we deduce an expression for estimating cycles of larger size which improves previous estimations and is suitable for either monopartite and multipartite networks and discuss the applicability of such analytical estimations

an algorithm for two dimensional rigidity percolation many important macroscopic properties of materials depend upon the number of microscopic degrees of freedom the task of counting the number of such degrees of freedom can be computationally very expensive we describe a new approach for this calculation which is appropriate for two dimensional glass like networks building upon recent work in graph rigidity this purely combinatorial algorithm is formulated in terms of a simplepebble game it has allowed for the first studies of the rigidity transition ingenericnetworks which are models of amorphous materials and glasses in the context of generic rigidity percolation we show how to calculate the number of internal degrees of freedom identify all rigid clusters and locate the overconstrained regions for a network ofnsites the pebble game has a worst case performance ofo n2 in our applications its performance scaled asn1 15at the rigidity transition while away from the transition region it grew linearly

lra local rigid averaging of stretchable non rigid shapes we present a novel algorithm for generating the mean structure of non rigid stretchable shapes following an alignment process which supports local affine deformations we translate the search of the mean shape into a diagonalization problem where the structure is hidden within the kernel of a matrix this is the first step required in many practical applications where one needs to model bendable and stretchable shapes from multiple observations

rigidity percolation on aperiodic lattices we studied the rigidity percolation rp model for two dimensional aperiodic quasi crystal lattices the rp thresholds for bond dilution were obtained for several aperiodic lattices via computer simulation using the pebble game algorithm it was found that the two rhombi penrose lattice is always floppy in view of the rp model the same was found for the ammann s octagonal lattice and the socolar s dodecagonal lattice in order to impose the percolation transition we used the so called ferro modification of these aperiodic tilings we studied as well the pinwheel tiling which has infinitely many orientations of edges the obtained estimates for the modified penrose ammann and socolar lattices are respectively pcp 0 836 0 002 pca 0 769 0 002 pcs 0 938 0 001 the bond rp threshold of the pinwheel tiling was estimated to pc 0 69 0 01 it was found that these results are very close to the maxwell the mean field like approximation for them

rigid formation control of double integrator systems in this chapter we move our focus from formation control systems modelled by single integrators e g those considered in previous chapters to formation systems modelled by double integrators two kinds of double integrator formation systems are considered namely formation stabilization systems and flocking control systems novel observations on the measurement requirement the null space and eigenvalues of the system jacobian matrix will be provided which reveal important properties of system dynamics and the associated convergence results we also establish some new links between single integrator formation systems and double integrator formation systems via a parameterized hamiltonian system which in addition provide novel stability criteria for different equilibria in double integrator formation systems by using available results from single integrator formation systems

detecting one mode communities in bipartite networks by bipartite clustering triangular abstract in this paper an algorithm is proposed to detect one mode community structures in bipartite networks and to deduce which one mode community structures are weighted after analyzing the topological properties in bipartite networks bipartite clustering triangular is introduced first bipartite networks are projected into two weighted one mode networks by bipartite clustering triangular then all the maximal sub graphs from two one mode weighted networks are extracted and the maximal sub graphs are merged together using a weighted clustering threshold in addition the proposed algorithm successfully finds overlapping vertices between one mode communities experimental results using some real world network data shows that the performance of the proposed algorithm is satisfactory

modeling and simulation of three dimensional planar contraction flow of viscoelastic fluids with ptt giesekus and fene p constitutive models abstract the planar contraction flow is a representative benchmark problem that is often used as a stringent test for the robustness of the numerical method to predict real flow patterns of viscoelastic fluids a numerical methodology based on the penalty finite element method with a decoupled algorithm is presented in the study to simulate three dimensional flow of viscoelastic fluids through planar contraction the viscoelastic rheological responses are described by using three kind differential constitutive models including the phan thien tanner ptt model the giesekus model and the finite extensible nonlinear elastic dumbbell with a peterlin closure approximation fene p model the discrete elastic viscous split stress devss formulation in cooperating with the inconsistent streamline upwind su scheme is employed to improve the computational stability the simulation results of flow velocity and stress on different sections of the flow field are compared with quinzani s experimental results that detected by laser doppler velocimetry and flow induced birefringence technologies it is found that the simulation results predicted with three kind differential constitutive models agree well with the experimental results the numerical methodology proposed in the study can be used successfully to predict complex flow patterns of viscoelastic fluids

weighted networks of scientific communication the measurement and topological role of weight abstract in order to take the weight of connection into consideration and to find a natural measurement of weight we have collected papers in econophysics and constructed a network of scientific communication to integrate idea transportation among econophysicists by collaboration citation and personal discussion some basic statistics such as weight per degree are discussed in fan et al j mod phys b 17 19 2004 2505 in this paper by including the papers published recently further statistical results for the network are reported clustering coefficient of weighted networks is introduced and empirically studied in this network we also compare the typical statistics on this network under different weight assignments including random and inverse weight the conclusion from weight redistributed network is helpful to the investigation of the topological role of weight

on the uniqueness of euclidean distance matrix completions the euclidean distance matrix completion problem edmcp is the problem of determining whether or not a given partial matrix can be completed into a euclidean distance matrix in this paper we present necessary and sufficient conditions for a given solution of the edmcp to be unique

bi clique communities we present a novel method for detecting communities in bipartite networks based on an extension of the k clique community detection algorithm we demonstrate how modular structure in bipartite networks presents itself as overlapping bicliques if bipartite information is available the bi clique community detection algorithm retains all of the advantages of the k clique algorithm but avoids discarding important structural information when performing a one mode projection of the network further the bi clique community detection algorithm provides a new level of flexibility by incorporating independent clique thresholds for each of the non overlapping node sets in the bipartite network

a numerical method for the solution of the falkner skan equation we present a numerical method for the solution of the falkner skan equation the method uses a coordinate transformation to convert the physical problem on a semi infinite physical domain to a problem on a fixed computational domain then after using a suitable change of variables the third order boundary value problem is solved using a shooting technique the numerical solutions obtained by the present method are in agreement with those obtained by previous authors

on stress matrices of d 1 lateration frameworks in general position let g p be a bar framework of n vertices in general position in mathbb r d for d n 1 where g is a d 1 lateration graph in this paper we present a constructive proof that g p admits a positive semidefinite stress matrix with rank n d 1 we also prove a similar result for a sensor network where the graph consists of m d 1 anchors

properties of euclidean and non euclidean distance matrices abstract a distance matrix d of order n is symmetric with elements 1 2 d ij 2 where dii 0 d is euclidean when the 1 2 n n 1 quantities dij can be generated as the distances between a set of n points x n p in a euclidean space of dimension p the dimensionality of d is defined as the least value of p rank x of any generating x in general p 1 and p 2 are also acceptable but may include imaginary coordinates even when d is euclidean basic properties of euclidean distance matrices are established in particular when rank d it is shown that depending on whether etd e is not or is zero the generating points lie in either p 1 dimensions in which case they lie on a hypersphere or in p 2 dimensions in which case they do not the notation e is used for a vector all of whose values are one when d is non euclidean its dimensionality p r s will comprise r real and s imaginary columns of x and r s are invariant for all generating x of minimal rank higher ranking representations can arise only from p 1 r 1 s or p 1 r s 1 or p 2 r 1 s 1 so that not only are r s invariant but they are both minimal for all admissible representations x

generic rigidity percolation in two dimensions we study rigidity percolation for random central force networks on the bondand site diluted generic triangular lattice here each site location is randomly displaced from the perfect lattice removing any special symmetries using the pebble game algorithm the total number of floppy modes are counted exactly and exhibit a cusp singularity in the second derivative at the transition from a rigid to a floppy structure the critical thresholds for bond and site dilution are found to be 0 66020 ifmmode pm else textpm fi 0 0003 and 0 69755 ifmmode pm else textpm fi 0 0003 respectively the network is decomposed into unique rigid clusters and we apply the usual percolation scaling theory from finite size scaling we find that the generic rigidity percolation transition is second order but in a different universality class from connectivity percolation with the exponents ensuremath alpha 0 48 ifmmode pm else textpm fi 0 05 ensuremath beta 0 175 ifmmode pm else textpm fi 0 02 and ensuremath nu 1 21 ifmmode pm else textpm fi 0 06 the fractal dimension of the spanning rigid clusters and the spanning stressed regions at the critical threshold are found to be mathit d mathit f 1 86 ifmmode pm else textpm fi 0 02 and mathit d mathit bb 1 80 ifmmode pm else textpm fi 0 03 respectively textcopyright 1996 the american physical society

cuts matrix completions and graph rigidity this paper brings together several topics arising in distinct areas polyhedral combinatorics in particular cut and metric polyhedra matrix theory and semidefinite programming in particular completion problems for positive semidefinite matrices and euclidean distance matrices distance geometry and structural topology in particular graph realization and rigidity problems cuts and metrics provide the unifying theme indeed cuts can be encoded as positive semidefinite matrices this fact underlies the approximative algorithm for max cut of goemans and williamson and both positive semidefinite and euclidean distance matrices yield points of the cut polytope or cone after applying the functions 1 arccos or when fixing the dimension in the euclidean distance matrix completion problem we find the graph realization problem and the related question of unicity of realization which leads to the question of graph rigidity our main objective here is to present in a unified setting a number of results and questions concerning matrix completion graph realization and rigidity problems these problems contain indeed very interesting questions relevant to mathematical programming and we believe that research in this area could yield to cross fertilization between the various fields involved

universal rigidity and edge sparsification for sensor network localization owing to their high accuracy and ease of formulation there has been great interest in applying convex optimization techniques particularly that of semidefinite programming sdp relaxation to tackle the sensor network localization problem in recent years however a drawback of such techniques is that the resulting convex program is often expensive to solve in order to speed up computation various edge sparsification heuristics have been proposed whose aim is to reduce the number of edges in the input graph although these heuristics do reduce the size of the convex program and hence make it faster to solve they are often ad hoc in nature and do not preserve the localization properties of the input as such one often has to face a tradeoff between solution accuracy and computational effort in this paper we propose a novel edge sparsification heuristic that can provably preserve the localization properties of the original input at the heart of our heuristic is a graph decomposition procedure which allows us to identify certain sparse generically universally rigid subgraphs of the input graph our computational results show that the proposed approach can significantly reduce the computational and memory complexities of sdp based algorithms for solving the sensor network localization problem moreover it compares favorably with existing speedup approaches both in terms of accuracy and solution time

exact reconstruction of euclidean distance geometry problem using low rank matrix completion the euclidean distance geometry problem arises in a wide variety of applications from determining molecular conformations in computational chemistry to localization in sensor networks when the distance information is incomplete the problem can be formulated as a nuclear norm minimization problem in this paper this minimization program is recast as a matrix completion problem of a low rank r gram matrix with respect to a suitable basis the well known restricted isometry property cannot be satisfied in this scenario instead a dual basis approach is introduced to theoretically analyze the reconstruction problem if the gram matrix satisfies certain coherence conditions with parameter nu the main result shows that the underlying configuration of n points can be recovered with very high probability from o nr nu log 2 n uniformly random samples computationally simple and fast algorithms are designed to solve the euclidean distance geometry problem numerical tests on different 3 d data and protein molecules validate the effectiveness and efficiency of the proposed algorithms

sequential localization of sensor networks the sensor network localization problem with distance information is to determine the positions of all sensors in a network given the positions of some of the sensors and the distances between some pairs of sensors a definition is given for a sensor network in the plane to be sequentially localizable it is shown that the graph of a sequentially localizable network must have a bilateration ordering and a polynomial time algorithm is given for deciding whether or not a network s graph has such an ordering a provably correct algorithm is given which consists of solving a sequence of quadratic equations and it is shown that the algorithm can localize any localizable network in the plane whose graph has a bilateration ordering

adaptive neural output feedback tracking control for a class of uncertain discrete time nonlinear systems this brief studies an adaptive neural output feedback tracking control of uncertain nonlinear multi input multi output mimo systems in the discrete time form the considered mimo systems are composed of n subsystems with the couplings of inputs and states among subsystems in order to solve the noncausal problem and decouple the couplings it needs to transform the systems into a predictor form the higher order neural networks are utilized to approximate the desired controllers by using lyapunov analysis it is proven that all the signals in the closed loop system is the semi globally uniformly ultimately bounded and the output errors converge to a compact set in contrast to the existing results the advantage of the scheme is that the number of the adjustable parameters is highly reduced the effectiveness of the scheme is verified by a simulation example

clustering coefficient and community structure of bipartite networks abstract many real world networks display natural bipartite structure where the basic cycle is a square in this paper with the similar consideration of standard clustering coefficient in binary networks a definition of the clustering coefficient for bipartite networks based on the fraction of squares is proposed in order to detect community structures in bipartite networks two different edge clustering coefficients l c 4 and l c 3 of bipartite networks are defined which are based on squares and triples respectively with the algorithm of cutting the edge with the least clustering coefficient communities in artificial and real world networks are identified the results reveal that investigating bipartite networks based on the original structure can show the detailed properties that is helpful to get deep understanding about the networks

strong converse for identification via quantum channels we present a simple proof of the strong converse for identification via discrete memoryless quantum channels based on a novel covering lemma the new method is a generalization to quantum communication channels of ahlswede s 1979 1992 approach to classical channels it involves a development of explicit large deviation estimates to the case of random variables taking values in self adjoint operators on a hilbert space this theory is presented separately in an appendix and we illustrate it by showing its application to quantum generalizations of classical hypergraph covering problems

distributed maximum likelihood sensor network localization we propose a class of convex relaxations to solve the sensor network localization problem based on a maximum likelihood ml formulation this class as well as the tightness of the relaxations depends on the noise probability density function pdf of the collected measurements we derive a computational efficient edge based version of this ml convex relaxation class and we design a distributed algorithm that enables the sensor nodes to solve these edge based convex programs locally by communicating only with their close neighbors this algorithm relies on the alternating direction method of multipliers admm it converges to the centralized solution it can run asynchronously and it is computation error resilient finally we compare our proposed distributed scheme with other available methods both analytically and numerically and we argue the added value of admm especially for large scale networks

minimal and redundant bearing rigidity conditions and applications this article studies the notions of minimal and 1 redundant bearing rigidity a necessary and sufficient condition for the numbers of edges in a graph of n n geq 3 vertices to be minimally bearing rigid mbr in mathbb r d d geq 2 is proposed if 3 leq n leq d 1 a graph is mbr if and only if it is the cycle graph in case n d 1 a generically bearing rigid graph is minimal if it has precisely 1 lfloor frac n 2 d 1 rfloor times d text mod n 2 d 1 text sgn text mod n 2 d 1 edges then several conditions for 1 redundant bearing rigidity are derived based on the mathematical conditions some algorithms for generating generically minimally and 1 redundantly bearing rigid graphs are given furthermore two applications of the new notions to optimal network design and formation merging are also reported