asymptotic constraint qualifications and global error bounds for convex inequalities in this paper we study various asymptotic constraint qualifications for the existence of global error bounds for approximate solutions of convex inequalities many known conditions that ensure the existence of such a global error bound are shown to be equivalent to one of the following three conditions i the bounded excess condition ii slater condition together with the asymptotic constraint qualification defined by auslender and crouzeix 1 and iii positivity of normal directional derivatives of the maximum of the constraint functions introduced by lewis and pang 12

analysis of infeasible interior point paths arising with semidefinite linear complementarity problems we consider semidefinite monotone linear complementarity problems sdlcp in the space equation here n of real symmetric n n matrices equipped with the cone equation here n of all symmetric positive semidefinite matrices one may define weighted using any m equation here n as weight infeasible interior point paths by replacing the standard condition xy ri r 0 that defines the usual central path by xy yx 2 rm under some mild assumptions the most stringent is the existence of some strictly complementary solution of sdlcp these paths have a limit as r 0 and they depend analytically on all path parameters such as r and m even at the limit point r 0

link prediction based on path entropy abstract information theory has been taken as a prospective tool for quantifying the complexity of complex networks in this paper first we study the information entropy or uncertainty of a path using the information theory after that we apply the path entropy to the link prediction problem in real world networks specifically we propose a new similarity index namely path entropy pe index which considers the information entropies of shortest paths between node pairs with penalization to long paths empirical experiments demonstrate that pe index outperforms the mainstream of link predictors

superlinear convergence of a symmetric primal dual path following algorithm for semidefinite programming this paper establishes the superlinear convergence of a symmetric primal dual path following algorithm for semidefinite programming sdp under the assumptions that the semidefinite program has a strictly complementary primal dual optimal solution and that the size of the central path neighborhood tends to zero the interior point algorithm considered here closely resembles the mizuno todd ye predictor corrector method for linear programming which is known to be quadratically convergent it is shown that when the iterates are well centered the duality gap is reduced superlinearly after each predictor step indeed if each predictor step is succeeded by r consecutive corrector steps then the predictor reduces the duality gap superlinearly with order 2 1 2 r the proof relies on a careful analysis of the central path for sdp it is shown that under the strict complementarity assumption the primal dual central path converges to the analytic center of the primal dual optimal solution set and the distance from any point on the central path to this analytic center is bounded by the duality gap

solving conic optimization problems via self dual embedding and facial reduction a unified approach we establish connections between the facial reduction algorithm of borwein and wolkowicz and the self dual homogeneous model of goldman and tucker when applied to conic optimization problems specifically we show that the self dual homogeneous model returns facial reduction certificates when it fails to return a primal dual optimal solution or a certificate of infeasibility using this observation we give an algorithm based on facial reduction for solving the primal problem that in principle always succeeds an analogous algorithm is easily stated for the dual problem this algorithm has the appealing property that it only performs facial reduction when it is required not when it is possible e g if a primal dual optimal solution exists it will be found in lieu of a facial reduction certificate even if slater s condition fails for the case of linear second order and semidefinite optimization we show that the algorithm can be implemented by assuming oracle access to the central path limit poi

facial reduction and partial polyhedrality we present fra poly a facial reduction algorithm fra for conic linear programs that is sensitive to the presence of polyhedral faces in the cone the main goals of fra and fra poly are the same

centrality measures and thermodynamic formalism for complex networks in the study of small and large networks it is customary to perform a simple random walk where the random walker jumps from one node to one of its neighbors with uniform probability the properties of this random walk are intimately related to the combinatorial properties of the network in this paper we propose to use the ruelle bowens random walk instead whose probability transitions are chosen in order to maximize the entropy rate of the walk on an unweighted graph if the graph is weighted then a free energy is optimized instead of the entropy rate specifically we introduce a centrality measure for large networks which is the stationary distribution attained by the ruelle bowens random walk we name it entropy rank we introduce a more general version which is able to deal with disconnected networks under the name of free energy rank we compare the properties of those centrality measures with the classic pagerank and hyperlink induced topic search hits on both toy and real life examples in particular their robustness to small modifications of the network we show that our centrality measures are more discriminating than pagerank since they are able to distinguish clearly pages that pagerank regards as almost equally interesting and are more sensitive to the medium scale details of the graph

fast computation of von neumann entropy for large scale graphs via quadratic approximations abstract the von neumann graph entropy vnge can be used as a measure of graph complexity which can be the measure of information divergence and distance between graphs however computing vnge is extensively demanding for a large scale graph we propose novel quadratic approximations for fast computing vnge various inequalities for error between the quadratic approximations and the exact vnge are found our methods reduce the cubic complexity of vnge to linear complexity computational simulations on random graph models and various real network datasets demonstrate superior performance

levy walk in complex networks an efficient way of mobility abstract to obtain an efficient diffusion process is an intriguing and important issue in the study of dynamical behaviors on real networks most previous studies are mainly focused on the analysis based on the random walk strategy whose entropy rate is bounded by the logarithm of the largest node degree of a given graph in this paper we take into account a novel strategy named levy walk and derive the general expression of entropy rate of levy walk on networks we present numerical evidences for how the levy walk strategy delivers an efficient diffusion process on networks and significantly increases the entropy rate compared with the random walk strategy it is further demonstrated that the capability of levy walk heavily relies on the network topology as well as the amount of information available regarding the network structure specifically the behavior of levy walk is highly sensitive to the distribution of shortest distances of the network and its variation to address this finding we thereby give a theoretical explanation of the relationship between the variation of shortest distances and the entropy rate of levy walk this work may help to enrich our understanding of the behavior of levy walk and further guide us to find an efficient diffusion process on networks

clustering of random scale free networks we derive the finite size dependence of the clustering coefficient of scale free random graphs generated by the configuration model with degree distribution exponent 2l ensuremath gamma l3 degree heterogeneity increases the presence of triangles in the network up to levels that compare to those found in many real networks even for extremely large nets we also find that for values of ensuremath gamma ensuremath approx 2 clustering is virtually size independent and at the same time becomes a de facto non self averaging topological property this implies that a single instance network is not representative of the ensemble even for very large network sizes

statistical mechanics of multiedge networks statistical properties of binary complex networks are well understood and recently many attempts have been made to extend this knowledge to weighted ones there are however subtle yet important considerations to be made regarding the nature of the weights used in this generalization weights can be either continuous or discrete magnitudes and in the latter case they can additionally have undistinguishable or distinguishable nature this fact has not been addressed in the literature insofar and has deep implications on the network statistics in this work we face this problem introducing multiedge networks as graphs where multiple distinguishable connections between nodes are considered we develop a statistical mechanics framework where it is possible to get information about the most relevant observables given a large spectrum of linear and nonlinear constraints including those depending both on the number of multiedges per link and their binary projection the latter case is particularly interesting as we show that binary projections can be understood from multiedge processes the implications of these results are important as many real agent based problems mapped onto graphs require this treatment for a proper characterization of their collective behavior

depth based complexity traces of graphs in this paper we aim to characterize graphs in terms of a structural measure of complexity our idea is to decompose a graph into layered substructures of increasing size and then to measure the information content of these substructures to locate dominant substructures within a graph we commence by identifying a centroid vertex which has the minimum shortest path length variance to the remaining vertices for each graph a family of centroid expansion subgraphs is derived from the centroid vertex in order to capture dominant structural characteristics of the graph since the centroid vertex is identified through a global analysis of the shortest path length distribution the expansion subgraphs provide a fine representation of a graph structure we then show how to characterize graphs using depth based complexity traces here we explore two different strategies the first strategy is to measure how the entropies on the centroid expansion subgraphs vary with the increasing size of the subgraphs the second strategy is to measure how the entropy differences vary with the increasing size of the subgraphs we perform graph classification in the principal component space of the complexity trace vectors experiments on graph datasets abstracted from some bioinformatics and computer vision databases demonstrate the effectiveness and efficiency of the proposed graph complexity traces our methods are competitive to state of the art methods highlightswe identify a centroid vertex of a graph through its shortest path matrix we establish a family of expansion subgraphs of a graph from its centroid vertex we construct depth based complexity traces of a graph on its expansion subgraphs we evaluate the performance of our complexity traces on standard graph datasets we demonstrate the effectiveness and efficiency of the graph complexity traces

large deviations for a random walk model with state dependent noise in this paper we prove the large deviation principle for a class of random walks with state dependent noise this type of model has important applications in queueing and communication theory and in the area of stochastic approximation

on the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues we derive some basic results on the geometry of semidefinite programming sdp and eigenvalue optimization i e the minimization of the sum of the k largest eigenvalues of a smooth matrix valued function we provide upper bounds on the rank of extreme matrices in sdps and the first theoretically solid explanation of a phenomenon of intrinsic interest in eigenvalue optimization in the spectrum of an optimal matrix the kth and k 1 st largest eigenvalues tend to be equal and frequently have multiplicity greater than two this clustering is intuitively plausible and has been observed as early as 1975 when the matrix valued function is affine we prove that clustering must occur at extreme points of the set of optimal solutions if the number of variables is sufficiently large we also give a lower bound on the multiplicity of the critical eigenvalue these results generalize to the case of a general matrix valued function under appropriate conditions

partial facial reduction simplified equivalent sdps via approximations of the psd cone we develop a practical semidefinite programming sdp facial reduction procedure that utilizes computationally efficient approximations of the positive semidefinite cone the proposed method simplifies sdps with no strictly feasible solution a frequent output of parsers by solving a sequence of easier optimization problems and could be a useful pre processing technique for sdp solvers we demonstrate effectiveness of the method on sdps arising in practice and describe our publicly available software implementation we also show how to find maximum rank matrices in our psd cone approximations which helps us find maximal simplifications and we give a post processing procedure for dual solution recovery that generally applies to facial reduction based pre processing techniques finally we show how approximations can be chosen to preserve problem sparsity

first and second order analysis of nonlinear semidefinite programs in this paper we study nonlinear semidefinite programming problems convexity duality and first order optimality conditions for such problems are presented a second order analysis is also given second order necessary and sufficient optimality conditions are derived finally sensitivity analysis of such programs is discussed

quantum statistics in complex networks in this work we discuss the symmetric construction of bosonic and fermionic networks and we present a case of a network showing a mixed quantum statistics this model takes into account the different nature of nodes described by a random parameter that we call energy and includes rewiring of the links the system described by the mixed statistics is an inhomogeneous system formed by two class of nodes in fact there is a threshold energy epsilon s such that nodes with lower energy epsilon epsilon s decrease their connectivity in time

entropy of spatial network ensembles we analyze complexity in spatial network ensembles through the lens of graph entropy mathematically we model a spatial network as a soft random geometric graph i e a graph with two sources of randomness namely nodes located randomly in space and links formed independently between pairs of nodes with probability given by a specified function the pair connection function of their mutual distance we consider the general case where randomness arises in node positions as well as pairwise connections i e for a given pair distance the corresponding edge state is a random variable classical random geometric graph and exponential graph models can be recovered in certain limits we derive a simple bound for the entropy of a spatial network ensemble and calculate the conditional entropy of an ensemble given the node location distribution for hard and soft probabilistic pair connection functions under this formalism we derive the connection function that yields maximum entropy under general constraints finally we apply our analytical framework to study two practical examples ad hoc wireless networks and the us flight network through the study of these examples we illustrate that both exhibit properties that are indicative of nearly maximally entropic ensembles

symmetric laplacians quantum density matrices and their von neumann entropy abstract we show that the normalized symmetric laplacian of a simple graph can be obtained from the partial trace over a pure bipartite quantum state that resides in a bipartite hilbert space one part corresponding to the vertices the other corresponding to the edges this suggests an interpretation of the symmetric laplacian s von neumann entropy as a measure of bipartite entanglement present between the two parts of the state we then study extreme values for a connected graph s generalized renyi p entropy specifically we show that 1 the complete graph achieves maximum entropy 2 the 2 regular graph a achieves minimum renyi 2 entropy among all k regular graphs b is within log 4 3 of the minimum renyi 2 entropy and log 4 2 3 of the minimum von neumann entropy among all connected graphs c achieves a von neumann entropy less than the star graph point 2 contrasts sharply with similar work applied to normalized combinatorial laplacians where it has been shown that the star graph almost always achieves minimum von neumann entropy in this work we find that the star graph achieves maximum entropy in the limit as the number of vertices grows without bound

entropies of complex networks with hierarchically constrained topologies the entropy of a hierarchical network topology in an ensemble of sparse random networks with hidden variables associated with its nodes is the log likelihood that a given network topology is present in the chosen ensemble we obtain a general formula for this entropy which has a clear interpretation in some simple limiting cases the results provide keys with which to solve the general problem of fitting a given network with an appropriate ensemble of random networks

spectral similarity for barabasi albert and chung lu models abstract in the paper we have analyzed spectral similarity between barabasi albert and chung lu models we have shown the similarity of spectral distribution for sufficiently large barabasi albert parameter value contrary extreme eigenvalues and principal eigenvector are not similar for those model we provide applications of obtained results related to the spectral graph theory and efficiency of quantum spatial search

entropy rate of diffusion processes on complex networks we introduce the concept of entropy rate to characterize a diffusion process on a complex network the entropy rate represents the minimal amount of information necessary to describe the diffusion on the network and is a quantity extremely sensitive to the network topology and dynamics by opportunely tuning the kind of diffusion the entropy rate allows one to extract different properties of the network structure moreover entropy maximization indicates how to design optimal diffusion processes providing a new theoretical tool with applications to social technological and communication systems

entropy measures for networks toward an information theory of complex topologies the quantification of the complexity of networks is today a fundamental problem in the physics of complex systems a possible roadmap to solve the problem is via extending key concepts of information theory to networks in this rapid communication we propose how to define the shannon entropy of a network ensemble and how it relates to the gibbs and von neumann entropies of network ensembles the quantities we introduce here will play a crucial role for the formulation of null models of networks through maximum entropy arguments and will contribute to inference problems emerging in the field of complex networks

on the identification of the optimal partition of second order cone optimization problems this paper discusses the identification of the optimal partition of second order cone optimization soco by giving some condition numbers which only depend on the soco problem itself we derive some bounds on the magnitude of the blocks of variables along the central path and prove that the optimal partition mathcal b mathcal n mathcal r and mathcal t for soco problems can be identified along the central path when the barrier parameter mu is small enough then we generalize the results to a specific neighborhood of the central path

tailored graph ensembles as proxies or null models for real networks i tools for quantifying structure we study the tailoring of structured random graph ensembles to real networks with the objective of generating precise and practical mathematical tools for quantifying and comparing network topologies macroscopically beyond the level of degree statistics our family of ensembles can produce graphs with any prescribed degree distribution and any degree degree correlation function its control parameters can be calculated fully analytically and as a result we can calculate asymptotically formulae for entropies and complexities and for information theoretic distances between networks expressed directly and explicitly in terms of their measured degree distribution and degree correlations

analyticity of weighted central paths and error bounds for semidefinite programming the purpose of this paper is two fold firstly we show that every cholesky based weighted central path for semidefinite programming is analytic under strict complementarity this result is applied to homogeneous cone programming to show that the central paths defined by the known class of optimal self concordant barriers are analytic in the presence of strictly complementary solutions secondly we consider a sequence of primal dual solutions that lies within a prescribed neighborhood of the central path of a pair of primal dual semidefinite programming problems and converges to the respective optimal faces under the additional assumption of strict complementarity we derive two necessary and sufficient conditions for the sequence of primal dual solutions to converge linearly with their duality gaps

approximate entropy of network parameters we study the notion of approximate entropy within the framework of network theory approximate entropy is an uncertainty measure originally proposed in the context of dynamical systems and time series we first define a purely structural entropy obtained by computing the approximate entropy of the so called slide sequence this is a surrogate of the degree sequence and it is suggested by the frequency partition of a graph we examine this quantity for standard scale free and erdos renyi networks by using classical results of pincus we show that our entropy measure often converges with network size to a certain binary shannon entropy as a second step with specific attention to networks generated by dynamical processes we investigate approximate entropy of horizontal visibility graphs visibility graphs allow us to naturally associate with a network the notion of temporal correlations therefore providing the measure a dynamical garment we show that approximate entropy distinguishes visibility graphs generated by processes with different complexity the result probes to a greater extent these networks for the study of dynamical systems applications to certain biological data arising in cancer genomics are finally considered in the light of both approaches

approximate von neumann entropy for directed graphs in this paper we develop an entropy measure for assessing the structural complexity of directed graphs although there are many existing alternative measures for quantifying the structural properties of undirected graphs there are relatively few corresponding measures for directed graphs to fill this gap in the literature we explore an alternative technique that is applicable to directed graphs we commence by using chung s generalization of the laplacian of a directed graph to extend the computation of von neumann entropy from undirected to directed graphs we provide a simplified form of the entropy which can be expressed in terms of simple node in degree and out degree statistics moreover we find approximate forms of the von neumann entropy that apply to both weakly and strongly directed graphs and that can be used to characterize network structure we illustrate the usefulness of these simplified entropy forms defined in this paper on both artificial and real world data sets including structures from protein databases and high energy physics theory citation networks

solution of the two star model of a network the p star model or exponential random graph is among the oldest and best known of network models here we give an analytic solution for the particular case of the two star model which is one of the most fundamental of exponential random graphs we derive expressions for a number of quantities of interest in the model and show that the degenerate region of the parameter space observed in computer simulations is a spontaneously symmetry broken phase separated from the normal phase of the model by a conventional continuous phase transition

network ensemble comparisons with stochastic rewiring and von neumann entropy assessing whether a given network is typical or atypical for a random network ensemble i e network ensemble comparison has widespread applications ranging from null model selection and hypothesis testing to clustering and classifying networks we develop a framework for network ensemble comparison by subjecting the network to stochastic rewiring we study two rewiring processes uniform and degree preserved rewiring which yield random network ensembles that converge to the erdos renyi and configuration model ensembles respectively we study convergence through von neumann entropy vne a network summary statistic measuring information content based on the spectra of a laplacian matrix and develop a perturbation analysis for the expected effect of rewiring on vne our analysis yields an estimate for how many rewires are required for a given network to resemble a typical network from an ensemble offering a computationally efficient quantity for network ensemble comparison that does not require

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

extension of hoffman s error bound to polynomial systems given any linear system defined by linear inequalities equalities in re n hoffman s error bound says that the distance from any point x in re n to the solution set of the linear system is bounded by a constant independent of x times a certain residual function evaluated at x this paper considers arbitrary possibly nonconvex polynomial systems defined by polynomial inequalities equalities it is shown that the solution sets of such systems are in general holder continuous as the right hand side changes in contrast to the lipschitzian continuity for the linear case also for any convex quadratic inequality system possessing an interior solution the authors show that its solution set behaves in a lipschitzian manner as the right hand side varies this result sharpens the earlier error bound results by mangasarian and robinson for general convex differentiable inequality systems which in addition to the interiority assumption require either an asymptotic constraint qualification con

riemannian geometric entropy for measuring network complexity a central issue in the science of complex systems is the quantitative characterization of complexity in the present work we address this issue by resorting to information geometry actually we propose a constructive way to associate with a in principle any network a differentiable object a riemannian manifold whose volume is used to define the entropy the effectiveness of the latter in measuring network complexity is successfully proved through its capability of detecting a classical phase transition occurring in both random graphs and scale free networks as well as of characterizing small exponential random graphs configuration models and real networks

an exact duality theory for semidefinite programming and its complexity implications in this paper we present a new and more complete duality for semidefinite programming sdp with the following features begin itemize item this dual is an explicit semidefinite program whose number of variables and the coefficient bitlengths are polynomial in those of the primal item if the primal is feasible then it is bounded if and only if the dual is feasible item the duality gap ie the difference between the primal and the dual objective function values is zero whenever the primal is feasible and bounded also in this case the dual attains its optimum item it yields a precise farkas lemma for semidefinite feasibility systems ie characterization of the it infeasibility of a semidefinite inequality in terms of the it feasibility of another polynomial size semidefinite inequality end itemize note that the standard duality for linear programming satisfies all of the above features but no such duality theory was previously known for sdp without slater like conditions being assumed then we apply the dual to derive certain complexity results for semidefinite programming problems the decision problem of semidefinite feasibility sdfp ie that of determining if a given semidefinite inequality system is feasible is the central problem of interest the complexity of sdfp is unknown but we show the following 1 in the turing machine model sdfp is not np complete unless np co np 2 in the real number model of blum shub and smale cite bss sdfp is in np cap co np we then give polynomial reductions from the following problems to sdfp 1 checking whether an sdp is bounded 2 checking whether a feasible and bounded sdp attains the optimum 3 checking the optimality of a feasible solution

interior point methods for the monotone semidefinite linear complementarity problem in symmetric matrices the sdlcp semidefinite linear complementarity problem in symmetric matrices introduced in this paper provides a unified mathematical model for various problems arising from systems and control theory and combinatorial optimization it is defined as the problem of finding a pair x y of n times n symmetric positive semidefinite matrices which lies in a given n n 1 2 dimensional affine subspace fc of sc 2 and satisfies the complementarity condition x bullet y 0 where sc denotes the n n 1 2 dimensional linear space of symmetric matrices and x bullet y the inner product of x and y the problem enjoys a close analogy with the lcp in the euclidean space in particular the central trajectory leading to a solution of the problem exists under the nonemptiness of the interior of the feasible region and a monotonicity assumption on the affine subspace fc the aim of this paper is to establish a theoretical basis of interior point methods with the use of newton directions toward the central trajectory for the monotone sdlcp

a note on the von neumann entropy of random graphs abstract in this note we consider the von neumann entropy of a density matrix obtained by normalizing the combinatorial laplacian of a graph by its degree sum we prove that the von neumann entropy of the typical erdos renyi random graph saturates its upper bound since connected regular graphs saturate this bound as well our result highlights a connection between randomness and regularity a general interpretation of the von neumann entropy of a graph is an open problem

robust self testing of quantum systems via noncontextuality inequalities characterizing unknown quantum states and measurements is a fundamental problem in quantum information processing in this letter we provide a novel scheme to self test local quantum systems using noncontextuality inequalities our work leverages the graph theoretic framework for contextuality introduced by cabello severini and winter combined with tools from mathematical optimization that guarantee the unicity of optimal solutions as an application we show that the celebrated klyachko can binicio ifmmode breve g else u g fi lu shumovsky inequality and its generalization to contextuality scenarios with odd n cycle compatibility relations admit robust self testing

depth based hypergraph complexity traces from directed line graphs in this paper we aim to characterize the structure of hypergraphs in terms of structural complexity measure measuring the complexity of a hypergraph in a straightforward way tends to be elusive since the hyperedges of a hypergraph may exhibit varying relational orders we thus transform a hypergraph into a line graph which not only accurately reflects the multiple relationships exhibited by the hyperedges but is also easier to manipulate for complexity analysis to locate the dominant substructure within a line graph we identify a centroid vertex by computing the minimum variance of its shortest path lengths a family of centroid expansion subgraphs of the line graph is then derived from the centroid vertex we compute the depth based complexity traces for the hypergraph by measuring either the directed or undirected entropies of its centroid expansion subgraphs the resulting complexity traces provide a flexible framework that can be applied to both hypergraphs and graphs we perform hyper graph classification in the principal component space of the complexity trace vectors experiments on hyper graph datasets abstracted from bioinformatics and computer vision data demonstrate the effectiveness and efficiency of the complexity traces highlightswe aim to characterize hypergraphs in terms of structural complexity measures straightforwardly measuring the complexity of a hypergraph tends to be elusive we transform a hypergraph into a line graph for measuring the complexities we compute depth based complexity traces for the hypergraph on line graphs the complexity traces provide a flexible framework for hypergraphs and graphs

a unified approach to error bounds for structured convex optimization problems error bounds which refer to inequalities that bound the distance of vectors in a test set to a given set by a residual function have proven to be extremely useful in analyzing the convergence rates of a host of iterative methods for solving optimization problems in this paper we present a new framework for establishing error bounds for a class of structured convex optimization problems in which the objective function is the sum of a smooth convex function and a general closed proper convex function such a class encapsulates not only fairly general constrained minimization problems but also various regularized loss minimization formulations in machine learning signal processing and statistics using our framework we show that a number of existing error bound results can be recovered in a unified and transparent manner to further demonstrate the power of our framework we apply it to a class of nuclear norm regularized loss minimization problems and establish a new error bound for this class under a strict complementarity type regularity condition we then complement this result by constructing an example to show that the said error bound could fail to hold without the regularity condition we believe that our approach will find further applications in the study of error bounds for structured convex optimization problems

a note on alternating projections for ill posed semidefinite feasibility problems we observe that sturm s error bounds readily imply that for semidefinite feasibility problems the method of alternating projections converges at a rate of mathcal o big k frac 1 2 d 1 2 big o k 12d 1 2 where d is the singularity degree of the problem the minimal number of facial reduction iterations needed to induce slater s condition consequently for almost all such problems in the sense of lebesgue measure alternating projections converge at a worst case rate of mathcal o big frac 1 sqrt k big o 1k

an interior point method for semidefinite programming we propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices

gibbs entropy of network ensembles by cavity methods the gibbs entropy of a microcanonical network ensemble is the logarithm of the number of network configurations compatible with a set of hard constraints this quantity characterizes the level of order and randomness encoded in features of a given real network here we show how to relate this entropy to large deviations of conjugated canonical ensembles we derive exact expression for this correspondence using the cavity methods for the configuration model for the ensembles with constraint degree sequence and community structure and for the ensemble with constraint degree sequence and number of links at a given distance

fast depth based subgraph kernels for unattributed graphs in this paper we investigate two fast subgraph kernels based on a depth based representation of graph structure both methods gauge depth information through a family of k layer expansion subgraphs rooted at a vertex 1 the first method commences by computing a centroid based complexity trace for each graph using a depth based representation rooted at the centroid vertex that has minimum shortest path length variance to the remaining vertices 2 this subgraph kernel is computed by measuring the jensen shannon divergence between centroid based complexity entropy traces the second method on the other hand computes a depth based representation around each vertex in turn the corresponding subgraph kernel is computed using isomorphisms tests to compare the depth based representation rooted at each vertex in turn for graphs with n vertices the time complexities for the two new kernels are o n 2 and o n 3 in contrast to o n 6 for the classic gartner graph kernel 3 key to achieving this efficiency is that we compute the required shannon entropy of the random walk for our kernels with o n 2 operations this computational strategy enables our subgraph kernels to easily scale up to graphs of reasonably large sizes and thus overcome the size limits arising in state of the art graph kernels experiments on standard bioinformatics and computer vision graph datasets demonstrate the effectiveness and efficiency of our new subgraph kernels highlightswe investigate two fast subgraph kernels based on a depth based representation our kernels gauge depth information rooted at a vertex for a graph the time complexities for the two kernels are o n 2 and o n 3 we evaluate the performance of our subgraph kernels on standard graph datasets we demonstrate the effectiveness of the proposed subgraph kernels

strong duality for semidefinite programming it is well known that the duality theory for linear programming lp is powerful and elegant and lies behind algorithms such as simplex and interior point methods however the standard lagrangian for nonlinear programs requires constraint qualifications to avoid duality gaps semidefinite linear programming sdp is a generalization of lp where the nonnegativity constraints are replaced by a semidefiniteness constraint on the matrix variables there are many applications e g in systems and control theory and combinatorial optimization however the lagrangian dual for sdp can have a duality gap we discuss the relationships among various duals and give a unified treatment for strong duality in semidefinite programming these duals guarantee strong duality i e a zero duality gap and dual attainment this paper is motivated by the recent paper by ramana where one of these duals is introduced

on the low rank solutions for linear matrix inequalities in this paper we present a polynomial time procedure to find a low rank solution for a system of linear matrix inequalities lmi the existence of such a low rank solution was shown in the work of au yeung and poon and the work of barvinok in the approach of au yeung and poon an earlier unpublished manuscript of bohnenblust played an essential role both proofs in the work of au yeung and poon and that of barvinok are nonconstructive in nature the aim of this paper is to provide a polynomial time constructive procedure to find such a low rank solution approximatively extensions of our new results and their relations to some of the known results in the literature are discussed

spectral properties of random reactance networks and random matrix pencils our goal is to study the statistical properies of dielectric resonances which are poles of conductance of a large random lc network such poles are a particular example of eigenvalues n of matrix pencils h w with w being a positive definite matrix and h a random real symmetric one we first consider spectra of the matrix pencils with independent identically distributed entries of h then we concentrate on an infinite range full connectivity version of a random lc network in all cases we calculate the mean eigenvalue density and the two point correlation function in the framework of efetov s supersymmetry approach fluctuations in spectra turn out to be the same as those provided by the wigner dyson theory of usual random matrices

duality between equilibrium and growing networks in statistical physics any given system can be either at an equilibrium or away from it networks are not an exception most network models can be classified as either equilibrium or growing here we show that under certain conditions there exists an equilibrium formulation for any growing network model and vice versa the equivalence between the equilibrium and nonequilibrium formulations is exact not only asymptotically but even for any finite system size the required conditions are satisfied in random geometric graphs in general and causal sets in particular and to a large extent in some real networks

large deviations analysis of some recursive algorithms with state dependent noise on presente une approche generale pour demontrer des theoremes du type grande deviation pour des systemes intervenant en theorie de la communication et en commande

stochastic approximation methods for systems over an infinitehorizon the paper develops efficient and general stochastic approximation sa methods for improving the operation of parametrized systems of either the continuous or discrete event dynamical systems types and which are of interest over a long time period for example one might wish to optimize or improve the stationary or average cost per unit time performance by adjusting the systems parameters the number of applications and the associated literature are increasing at a rapid rate this is partly due to the increasing activity in computing pathwise derivatives and adapting them to the average cost problem although the original motivation and the examples come from an interest in the infinite horizon problem the techniques and results are of general applicability in sa we present an updating and review of powerful ordinary differential equation type methods in a fairly general context and based on weak convergence ideas the results and proof techniques are applicable to a wide variety of applications exploiting the full potential of these ideas can greatly simplify and extend much current work their breadth as well as the relative ease of using the basic ideas are illustrated in detail via typical examples drawn from discrete event dynamical systems piecewise deterministic dynamical systems and a stochastic differential equations model in these particular illustrations we use either infinitesimal perturbation analysis type estimators mean square derivative type estimators or finite difference type estimators markov and non markov models are discussed the algorithms for distributed asynchronous updating as well as the fully synchronous schemes are developed

analyticity of the central path at the boundary point in semidefinite programming abstract in this paper we study the limiting behavior of the central path for semidefinite programming sdp we show that the central path is an analytic function of the barrier parameter even at the limit point provided that the semidefinite program has a strictly complementary solution a consequence of this property is that the derivatives of any order of the central path have finite limits as the barrier parameter goes to zero

scale free networks from varying vertex intrinsic fitness a new mechanism leading to scale free networks is proposed in this letter it is shown that in many cases of interest the connectivity power law behavior is neither related to dynamical properties nor to preferential attachment assigning a quenched fitness value x i to every vertex and drawing links among vertices with a probability depending on the fitnesses of the two involved sites gives rise to what we call a good get richer mechanism in which sites with larger fitness are more likely to become hubs i e to be highly connected

large deviations of random walks on random graphs we study the rare fluctuations or large deviations of time integrated functionals or observables of an unbiased random walk evolving on erdos renyi random graphs and construct a modified biased random walk that explains how these fluctuations arise in the long time limit two observables are considered the sum of the degrees visited by the random walk and the sum of their logarithm related to the trajectory entropy the modified random walk is used for both quantities to explain how sudden changes in degree fluctuations similar to dynamical phase transitions are related to localization transitions for the second quantity we also establish links between the large deviations of the trajectory entropy and the maximum entropy random walk

diffusion on hierarchical systems of weakly coupled networks abstract we analyzed diffusion dynamics on weakly coupled networks interconnected networks by means of separation of time scales using an adiabatic approximation we reduced the system dynamics to a markov chain with aggregated variables and derived a transport equation that is analogous to fick s first law and includes a driving force entropy production is a sum of microscopic entropy transport which results from the particle s migration between networks of different topologies and macroscopic entropy production of the markov chain equilibrium particles partition between different sub networks depends only on internal sub network parameters by changing structure of networks one can not only modify diffusion constants but can also induce or reverse the direction of the particles flow between different networks our framework confirmed by numerical simulations is also useful for considering diffusion in nested systems corresponding to hierarchical networks with several different time scales thus it can serve to uncover hidden hierarchy levels from observations of diffusion processes

entropy of network ensembles in this paper we generalize the concept of random networks to describe network ensembles with nontrivial features by a statistical mechanics approach this framework is able to describe undirected and directed network ensembles as well as weighted network ensembles these networks might have nontrivial community structure or in the case of networks embedded in a given space they might have a link probability with a nontrivial dependence on the distance between the nodes these ensembles are characterized by their entropy which evaluates the cardinality of networks in the ensemble in particular in this paper we define and evaluate the structural entropy i e the entropy of the ensembles of undirected uncorrelated simple networks with given degree sequence we stress the apparent paradox that scale free degree distributions are characterized by having small structural entropy while they are so widely encountered in natural social and technological complex systems we propose a solution to the paradox by proving that scale free degree distributions are the most likely degree distribution with the corresponding value of the structural entropy finally the general framework we present in this paper is able to describe microcanonical ensembles of networks as well as canonical or hidden variable network ensembles with significant implications for the formulation of network constructing algorithms

research on the search ability of brownian particles on networks with an adaptive mechanism abstract in this paper we focus on the search ability of brownian particles with an adaptive mechanism in the adaptive mechanism nodes are allowed to be able to change their own accepting probability according to their congestion states two searching traffic models the static one in which nodes have fixed accepting probability to the incoming particles and the adaptive one in which nodes have adaptive accepting probability to the incoming particles are presented for testing the adaptive mechanism instead of number of hops we use the traveling time which includes not only the number of hops for a particle to jump from the source node to the destination but also the time that the particle stays in the queues of nodes to evaluate the search ability of brownian particles we apply two models to different networks the experiment results show that the adaptive mechanism can decrease the network congestion and the traveling time of the first arriving particle furthermore we investigate the influence of network topologies on the congestion of networks by addressing several main properties degree distribution average path length and clustering coefficient we show the reason why random topologies are more able to deal with congested traffic states than others we also propose an absorption strategy to deal with the additional brownian particles in networks the experiment results on barabasi albert ba scale free networks show that the absorption strategy can increase the probability of a successful search and decrease the average per node particles overhead for our models

entropy of tournament digraphs abstract the renyi entropy h of complete antisymmetric directed graphs i e tournaments is explored we optimize h when 2 and 3 and find that as increases h s sensitivity to what we refer to as regularity increases as well a regular tournament on n vertices is one with each vertex having out degree n 1 2 but there is a lot of diversity in terms of structure among the regular tournaments for example a regular tournament may be such that each vertex s out set induces a regular tournament a doubly regular tournament or a transitive tournament a rotational tournament as increases on the set of regular tournaments h has maximum value on doubly regular tournaments and minimum value on rotational tournaments the more regular the higher the entropy we show however that among all tournaments on a fixed number of vertices h 2 and h 3 are maximized by any regular tournament on that number of vertices we also provide a calculation that is equivalent to the von neumann entropy but may be applied to any directed or undirected graph and shows that the von neumann entropy is a measure of how quickly a random walk on the graph or directed graph settles

analysis of adaptive step size sa algorithms for parameter tracking we present proofs and data for adaptive step size algorithms for tracking time varying parameters when recursive stochastic approximation type algorithms are used a classical problem in adaptive control and communication theory concerns the tracking of the best fit of a given form when the statistics or the parameters change slowly a major and yet unresolved problem has been the choice of the step sizes in the tracking algorithm an algorithm for adapting the step size using the same system measurements which are used for the tracking was suggested by benveniste and various examples worked out by brossier the numerical results were very encouraging but proofs were lacking these proofs are supplied here together with supporting numerical data the proofs are based on recent results in stochastic approximation the adaptive step size technique works very well indeed much supporting analysis is presented particularly concerning the interpretation of certain stationary processes as stationary pathwise derivatives finite difference forms are also treated these are mathematically simpler and can be applied to a wide variety of systems even when the system is not well modeled the data shows that they work well

shannon and von neumann entropy of random networks with heterogeneous expected degree entropic measures of complexity are able to quantify the information encoded in complex network structures several entropic measures have been proposed in this respect here we study the relation between the shannon entropy and the von neumann entropy of networks with given expected degree sequence we find in different examples of network topologies that when the degree distribution contains some heterogeneity an intriguing correlation emerges between the two entropic quantities this results seems to suggest that heterogeneity in the expected degree distribution is implying an equivalence between a quantum and a classical description of networks which respectively corresponds to the von neumann and the shannon entropy

eigenvalue constrained faces we characterize the exposed faces of convex sets l of symmetric matrices invariant under orthogonal similarity ut u for all orthogonal u such sets fl are exactly those determined by eigenvalue constraints typical examples are the positive semidefinite cone and the unit balls of common matrix norms the set 9 of all diagonal matrices in is known to be convex if and only if 59 is and 9 is invariant under the group of permutations acting on diagonal entries we show how any exposed face of g is naturally associated with an exposed face of 9 by relating the stabilizer groups of the two faces 0 1998 elsevier science inc

entropic origin of disassortativity in complex networks why are most empirical networks with the prominent exception of social ones generically degree degree anticorrelated to answer this long standing question we define the ensemble of correlated networks and obtain the associated shannon entropy maximum entropy can correspond to either assortative correlated or disassortative anticorrelated configurations but in the case of highly heterogeneous scale free networks a certain disassortativity is predicted offering a parsimonious explanation for the question above our approach provides a neutral model from which in the absence of further knowledge regarding network evolution one can obtain the expected value of correlations when empirical observations deviate from the neutral predictions as happens for social networks one can then infer that there are specific correlating mechanisms at work

class of correlated random networks with hidden variables we study a class of models of correlated random networks in which vertices are characterized by hidden variables controlling the establishment of edges between pairs of vertices we find analytical expressions for the main topological properties of these models as a function of the distribution of hidden variables and the probability of connecting vertices the expressions obtained are checked by means of numerical simulations in a particular example the general model is extended to describe a practical algorithm to generate random networks with an a priori specified correlation structure we also present an extension of the class to map nonequilibrium growing networks to networks with hidden variables that represent the time at which each vertex was introduced in the system

identifying the borders of mathematical knowledge based on a divide and conquer approach knowledge about nature has been organized into a set of interrelated facts allowing a natural representation in terms of graphs each chunk of knowledge corresponds to a node while relationships between such chunks are expressed as edges this organization becomes particularly clear in the case of mathematical theorems with their intense cross implications and relationships we have derived a web of mathematical theorems from wikipedia and thanks to the powerful concept of entropy identified its more central and frontier elements our results also suggest that the central nodes are the oldest theorems while the frontier nodes are those recently added to the network the network communities have also been identified allowing further insights about the organization of this network such as its highly modular structure

regularizing the abstract convex program abstract characterizations of optimality for the abstract convex program inf p x g x s x p where s is an arbitrary convex cone in a finite dimensional space is a convex set and p and g are respectively convex and s convex on were given in 10 these characterizations hold without any constraint qualification they use the minimal cone s f of p and the cone of directions of constancy d g s f in the faithfully convex case these cones can be used to regularize p i e transform p into an equivalent program p r for which slater s condition holds we present an algorithm that finds both s f and d g s f the main step of the algorithm consists in solving a particular complementarity problem we also present a characterization of optimality for p in terms of the cone of directions of constancy of a convex functional d g rather than d g s f

using mapping entropy to identify node centrality in complex networks abstract the problem of finding the best strategy to attack a network or immunize a population with a minimal number of nodes has attracted much current research interest the assessment of node importance has been a fundamental issue in the research of complex networks in this paper we propose a new concept called mapping entropy me to identify the importance of a node in the complex network the concept is established according to the local information which considers the correlation among all neighbors of a node we evaluate the efficiency of the centrality by static and dynamic attacks on standard network models and real world networks the simulation result shows that the new centrality is more efficient than traditional attack strategies whether it is static or dynamic

coordinate shadows of semidefinite and euclidean distance matrices we consider the projected semidefinite and euclidean distance cones onto a subset of the matrix entries these two sets are precisely the input data defining feasible semidefinite and euclidean distance completion problems we classify when these sets are closed and use the boundary structure of these two sets to elucidate the krislock wolkowicz facial reduction algorithm in particular we show that under a chordality assumption the minimal cones of these problems admit combinatorial characterizations as a by product we record a striking relationship between the complexity of the general facial reduction algorithm singularity degree and facial exposedness of conic images under a linear mapping

maximal entropy random walks in complex networks with limited information maximization of the entropy rate is an important issue to design diffusion processes aiming at a well mixed state we demonstrate that it is possible to construct maximal entropy random walks with only local information on the graph structure in particular we show that an almost maximal entropy random walk is obtained when the step probabilities are proportional to a power of the degree of the target node with an exponent that depends on the degree degree correlations and is equal to 1 in uncorrelated graphs

optimization problems with perturbations a guided tour this paper presents an overview of some recent and significant progress in the theory of optimization problems with perturbations we put the emphasis on methods based on upper and lower estimates of the objective function of the perturbed problems these methods allow one to compute expansions of the optimal value function and approximate optimal solutions in situations where the set of lagrange multipliers is not a singleton may be unbounded or is even empty we give rather complete results for nonlinear programming problems and describe some extensions of the method to more general problems we illustrate the results by computing the equilibrium position of a chain that is almost vertical or horizontal

error bounds for linear matrix inequalities for iterative sequences that converge to the solution set of a linear matrix inequality we show that the distance of the iterates to the solution set is at most o epsilon 2 d the nonnegative integer d is the so called degree of singularity of the linear matrix inequality and epsilon denotes the amount of constraint violation in the iterate for infeasible linear matrix inequalities we show that the minimal norm of epsilon approximate primal solutions is at least 1 o epsilon 1 2 d 1 and the minimal norm of epsilon approximate farkas type dual solutions is at most o 1 epsilon 2 d 1 as an application of these error bounds we show that for any bounded sequence of epsilon approximate solutions to a semidefinite programming problem the distance to the optimal solution set is at most o epsilon 2 k where k is the degree of singularity of the optimal solution set