design of capacity approaching irregular low density parity check codes we design low density parity check ldpc codes that perform at rates extremely close to the shannon capacity the codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides our theoretical analysis of the codes is based on the work of richardson and urbanke see ibid vol 47 no 2 p 599 618 2000 assuming that the underlying communication channel is symmetric we prove that the probability densities at the message nodes of the graph possess a certain symmetry using this symmetry property we then show that under the assumption of no cycles the message densities always converge as the number of iterations tends to infinity furthermore we prove a stability condition which implies an upper bound on the fraction of errors that a belief propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution our codes are found by optimizing the degree structure of the underlying graphs we develop several strategies to perform this optimization we also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds

a polynomial algorithm of edge neighbor scattering number of trees abstract the edge neighbor scattering number ens is an alternative invulnerability measure of networks such as the vertices represent spies or virus carriers let g v e be a graph and e be any edge in g the open edge neighborhood of e is n e f e g f e e and f are adjacent and the closed edge neighborhood of e is n e n e e an edge e in g is said to be subverted when n e is deleted from g an edge set x e g is called an edge subversion strategy of g if each of the edges in x has been subverted from g the survival subgraph is denoted by g x an edge subversion strategy x is called an edge cut strategy of g if the survival subgraph g x is disconnected or is a single vertex or is the ens of a graph g is defined as e n s g max x e g g x x where x is any edge cut strategy of g g x is the number of the components of g x it is proved that the problem of computing the ens of a graph is np complete in this paper we give a polynomial algorithm of ens of trees

the effect of calmness on the solution set of systems of nonlinear equations we address the problem of solving a continuously differentiable nonlinear system of equations under the condition of calmness this property also called upper lipschitz continuity in the literature can be described by a local error bound and is being widely used as a regularity condition in optimization indeed it is known to be significantly weaker than classic regularity assumptions that imply that solutions are isolated we prove that under this condition the rank of the jacobian of the function that defines the system of equations must be locally constant on the solution set in addition we prove that locally the solution set must be a differentiable manifold our results are illustrated by examples and discussed in terms of their theoretical relevance and algorithmic implications

a shamanskii like self adaptive levenberg marquardt method for nonlinear equations abstract in this paper we are concerned with the shamanskii like self adaptive levenberg marquardt methods for nonlinear equations we consider two choices of levenberg marquardt parameter one of them is the standard self adaptive levenberg marquardt parameter the other is nonmonotone self adaptive levenberg marquardt parameter by using the nonmonotone technique of grippo lampariello and lucidi under the error bound condition which is weaker than nonsingularity we show that the shamanskii like self adaptive levenberg marquardt methods converge with q order m 1 numerical experiments show the new algorithms are efficient and promising

information processing in complex networks graph entropy and information functionals this paper introduces a general framework for defining the entropy of a graph our definition is based on a local information graph and on information functionals derived from the topological structure of a given graph more precisely an information functional quantifies structural information of a graph based on a derived probability distribution such a probability distribution leads directly to an entropy of a graph then the structural information content of a graph will be is interpreted and defined as the derived graph entropy another major contribution of this paper is the investigation of relationships between graph entropies in addition to this we provide numerical results demonstrating not only the feasibility of our method which has polynomial time complexity but also its usefulness with regard to practical applications aiming to an understanding of information processing in complex networks

sparse null basis computations in structural optimization we study the computation of sparse null bases of equilibrium matrices in the context of the force method in structural optimization two classes of structural problems are considered for the class of rigid jointed skeletal structures we use a partitioning method suggested by henderson and maunder to partition the problem into a set of independent null basis computations for the class of structures represented by a continuum we compute a sizable fraction of the null vectors in a basis from a consideration of the finite element formulation and the bipartite graph of the equilibrium matrix the remaining null vectors are computed by the triangular algorithm in 6 the new algorithms find sparser bases than the triangular algorithm and are also faster

inertia of complex unit gain graphs let t z c z 1 be a subgroup of the multiplicative group of all nonzero complex numbers c a t gain graph is a triple g t consisting of a graph g v e the circle group t and a gain function e t such that e i j e j i 1 e j i the adjacency matrix a of the t gain graph g of order n is an n n complex matrix aij where a i j e i j if v i is adjacent to v j 0 otherwise evidently this matrix is hermitian the inertia of is defined to be the triple i n i i i 0 where i i i 0 are numbers of the positive negative and zero eigenvalues of a including multiplicities respectively in this paper we investigate some properties of inertia of t gain graph

analyzing lattice networks through substructures abstract analyzing the topology of network structures is an important topic studied from many different aspects of science and mathematics the wiener polarity index number of unordered pairs of vertices at distance 3 from each other is one of the representative descriptors of graph structures it was computed for several lattice networks by chen et al 11 in an effort to understand the properties of these networks the wiener polarity index is a variation of the classic distance based graph invariant the wiener index sum of distances between all pairs of vertices which is known to be closely related to the number of substructures in this paper we examine the numbers of various subgraphs of order 4 for these lattice networks in addition to confirming their symmetric nature comparing the numbers of various substructures leads to insights on other less trivial characteristics of these network structures of common interest

a note on extremal trees with degree conditions abstract a fundamental question in the study of graph invariants asks for the extremal structures under certain constraints that maximize or minimize a graph invariant in this note we summarize some recent work on the extremal trees of distance based and degree based graph invariants under various degree conditions we note that many of such extremal structures turned out to be identical for different but similar invariants such common extremal structures are investigated through the greedy trees and majorization between degree sequences we show that many of the known extremal results can be obtained through this line of arguments we also introduce some new extremal results as immediate consequences

hermitian laplacian matrix and positive of mixed graphs a mixed graph is obtained from an undirected graph by orienting a subset of its edges the hermitian adjacency matrix of a mixed graph m of order n is the n n matrix h m h k l where h k l h l k i i 1 if there exists an orientation from vk to vl and h k l h l k 1 if there exists an edge between vk and vl but not exist any orientation and h k l 0 otherwise the value of a mixed walk w v 1 v 2 v 3 v l is h w h 12 h 23 h l 1 l a mixed walk is positive negative if h w 1 h w 1 a mixed cycle is called positive if its value is 1 a mixed graph is positive if each of its mixed cycle is positive in this work we firstly present the necessary and sufficient conditions for the positive of a mixed graph secondly we introduce the incident matrix and hermitian laplacian matrix of a mixed graph and derive some results about the hermitian laplacian spectrum

tensor methods for nonlinear equations a new class of methods for solving systems of nonlinear equations called tensor methods is introduced tensor methods are general purpose methods intended especially for problems where the jacobian matrix at the solution is singular or ill conditioned they base each iteration on a quadratic model of the nonlinear function the standard linear model augmented by a simple second order term the second order term is selected so that the model interpolates function values from several previous iterations as well as the current function value and jacobian the tensor method requires no more function and derivative information per iteration and hardly more storage or arithmetic per iteration than a standard method based on newton s method in extensive computational tests a tensor algorithm is significantly more efficient than a similar algorithm based on the standard linear model both on standard nonsingular test problems and on problems where the jacobian at the solution is singular

the joint graphical lasso for inverse covariance estimation across multiple classes we consider the problem of estimating multiple related gaussian graphical models from a high dimensional data set with observations belonging to distinct classes we propose the joint graphical lasso which borrows strength across the classes in order to estimate multiple graphical models that share certain characteristics such as the locations or weights of nonzero edges our approach is based upon maximizing a penalized log likelihood we employ generalized fused lasso or group lasso penalties and implement a fast admm algorithm to solve the corresponding convex optimization problems the performance of the proposed method is illustrated through simulated and real data examples

on graph entropy measures based on the number of independent sets and matchings abstract in this paper we consider the graph entropy measures based on the number of independent sets and matchings the reason to study these measures relates to the fact that the independent set and matching problem is computationally demanding however these features can be calculated for smaller networks in case one can establish efficient estimations those measures may be also used for larger graphs so we establish some upper and lower bounds as well as some information inequalities for these information theoretic quantities in order to give further evidence we also generate numerical results to study these measures such as list the extremal graphs for these entropies those results reveal the two entropies possess some new features

degree based entropies of networks revisited studies on the information content of graphs and networks have been initiated in the late fifties based on the seminal work due to shannon and rashevsky various graph parameters have been used for the construction of entropy based measures to characterize the structure of complex networks based on shannon s entropy in cao et al extremality of degree based graph entropies inform sci 278 2014 22 33 we studied graph entropies which are based on vertex degrees by using so called information functionals as a matter of fact there has been very little work to find their extremal values when considered shannon entropy based graph measures we pursue with this line of research by proving further extremal properties of the degree based graph entropies

second order randic index of fluoranthene type benzenoid systems fluoranthene type benzenoid system are polycyclic conjugated molecules consisting of two benzenoid fragments connected by two carbon carbon bonds so as to form a five membered ring fluoranthene type benzenoid system and its congeners are compounds which in view of their chemical and physical properties belong among benzenoid hydrocarbons let f denotes a fluoranthene type benzenoid system whose second order randic index is denoted by r2 f in this paper we give the expression of r2 f in terms of their inlet features and we find the minimal and maximal value of the second order randic index over the set of cata catacondensed fluoranthene type benzenoid systems and characterize their corresponding graphs

entropy bounds for dendrimers abstract many graph invariants have been used for the construction of entropy based measures to characterize the structure of complex networks when considering shannon entropy based graph measures there has been very little work to find their extremal values a reason for this might be the fact that shannon s entropy represents a multivariate function and all probability values are not equal to zero when considering graph entropies dehmer and kraus proved some extremal results for graph entropies which are based on information functionals and express some conjectures generated by numerical simulations to find extremal values of graph entropies dehmer and kraus discussed the extremal values of entropies for dendrimers in this paper we continue to study the extremal values of graph entropy for dendrimers which has most interesting applications in molecular structure networks and also in the pharmaceutical and biomedical area among all dendrimers with n vertices we obtain the extremal values of graph entropy based on different well known information functionals numerical experiments verifies our results

a novel method for analyzing inverse problem of topological indices of graphs using competitive agglomeration a novel method for analyzing the inverse problem of topological indices of graph is proposed in this paper which applies the data mining technique to classification of graphs according to topological indices differing from the way of using topological indices to categorize graphs into different isomorphic classes this method aims at dividing molecular graphs with n vertices into several compact classes according to their topological indices then the chemical and physical features of each kind of graph can be researched and new compounds can be found which has great significance on bio medical filed in the experiment three classes of simple connected graphs with 5 6 and 7 vertices respectively are investigated and analyzed using the proposed method the experimental results show the validity of it however there are still some problems needed to be researched further such as selecting the vector of topological indices for clustering selecting the distance between the vectors of topological indices choosing the aspects to analyze the properties of each kind of graph after clustering etc

extremality of degree based graph entropies abstract many graph invariants have been used for the construction of entropy based measures to characterize the structure of complex networks based on shannon s entropy we study graph entropies which are based on vertex degrees by using so called information functionals when considering shannon entropy based graph measures there has been very little work to find their extremal values the main contribution of this paper is to prove some extremal values for the underlying graph entropy of certain families of graphs and to find the connection between the graph entropy and the sum of degree powers further conjectures to determine extremal values of graph entropies are given

the general connectivity indices of fluoranthene type benzenoid systems the general connectivity index r g of a graph g v e is defined as r g u v e g d u d v where u v is an edge of g du and dv denote the degrees of the vertices u and v respectively and 0 is an arbitrary real number in this paper we give the expressions for computing the general connectivity indices of fluoranthene type benzenoid systems and we determine the extremal values of r g in f benzenoid systems with h hexagons for some real number especially the extremal values of r g in cata catacondensed fluoranthene type benzenoid systems with h hexagons were studied

a recursive approach to low complexity codes a method is described for constructing long error correcting codes from one or more shorter error correcting codes referred to as subcodes and a bipartite graph a graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes both the encoders and decoders proposed are shown to take advantage of the code s explicit decomposition into subcodes to decompose and simplify the associated computational processes bounds on the performance of two specific decoding algorithms are established and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed the proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver e g reliability information without greatly increasing the number of computations required it is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst error correction abilities the construction principles

local behavior of an iterative framework for generalized equations with nonisolated solutions an iterative framework for solving generalized equations with nonisolated solutions is presented for generalized equations with the structure where is a multifunction and f is single valued the framework covers methods that at each step solve subproblems of the type the multifunction approximates f around s besides a condition on the quality of this approximation two other basic assumptions are employed to show q superlinear or q quadratic convergence of the iterates to a solution a key assumption is the upper lipschitz continuity of the solution set map of the perturbed generalized equation moreover the solvability of the subproblems is required conditions that ensure these assumptions are discussed in general and by means of several applications they include monotone mixed complementarity problems karush kuhn tucker systems arising from nonlinear programs and nonlinear equations particular results deal with error bounds and upper lipschitz continuity properties for these problems

finding a shortest cycle in a subspace of the cycle space of a graph thomassen s 3 path condition shows that it is relatively easy for one to find a shortest cycle in a collection of cycles beyond a subspace of the cycle space of a connected graph and the real challenge is to find a shortest cycle contained in a given subspace of the cycle space of a graph in this article we investigate the shorter cycle structures in a given subspace of a graph and find a set of cycles in a given graph containing much information about short cycles we show that for a large range of subspaces of a graph satisfying a parity condition there exists a polynomial time algorithm to find a shortest cycle in these subspaces this makes a unified treatment of several famous algorithms finally we provide lower bounds of some types of short cycles in embedded graphs

message passing algorithms for counting short cycles in a graph a message passing algorithm for counting short cycles in a graph is presented for bipartite graphs which are of particular interest in coding the algorithm is capable of counting cycles of length g g 2 2g 2 where g is the girth of the graph for a general non bipartite graph cycles of length g g 1 2g 1 can be counted the algorithm is based on performing integer additions and subtractions in the nodes of the graph and passing extrinsic messages to adjacent nodes the complexity of the proposed algorithm grows as o g e 2 where e is the number of edges in the graph for sparse graphs the proposed algorithm significantly outperforms the existing algorithms tailored for counting em short cycles in terms of computational complexity and memory requirements we also discuss a more generic and basic approach of counting short cycles which is based on matrix multiplication and provide a message passing interpretation for such an approach we then demonstrate that an efficient implementation of the matrix multiplication approach has essentially the same complexity as the proposed message passing algorithm

design of ldpc codes based on multipath emd strategies for progressive edge growth low density parity check ldpc codes are capable of achieving excellent performance and provide a useful alternative for the high performance applications however at medium to high signal to noise ratios an observable error floor arises from the loss of independence of messages passed under iterative graph based decoding in this paper the error floor performance of the short block length codes is improved by the use of a novel candidate selection metric in code graph construction the proposed multipath extrinsic message degree emd approach avoids harmful structures in the graph by evaluating certain properties of the cycles introduced in each edge placement we present multipath emd based designs for several structured ldpc codes including quasi cyclic and the irregular repeat accumulate codes in addition an extended class of the diversity achieving codes on the challenging block fading channel is proposed and considered with the multipath emd design this combined approach is demonstrated to provide the gains in decoder convergence and error rate performance a simulation study evaluates the performance of the proposed and existing state of the art methods

expander codes using expander graphs we construct a new family of asymptotically good linear error correcting codes these codes have linear time sequential decoding algorithms and logarithmic time parallel decoding algorithms that use a linear number of processors we present both randomized and explicit constructions of these codes experimental results demonstrate the good performance of the randomly chosen codes

joint estimation of multiple dependent gaussian graphical models with applications to mouse genomics gaussian graphical models are widely used to represent conditional dependence among random variables in this paper we propose a novel estimator for data arising from a group of gaussian graphical models that are themselves dependent a motivating example is that of modeling gene expression collected on multiple tissues from the same individual here the multivariate outcome is affected by dependencies acting not only at the level of the specific tissues but also at the level of the whole body existing methods that assume independence among graphs are not applicable in this case to estimate multiple dependent graphs we decompose the problem into two graphical layers the systemic layer which affects all outcomes and thereby induces cross graph dependence and the category specific layer which represents graph specific variation we propose a graphical em technique that estimates both layers jointly establish estimation consistency and selection sparsistency of the proposed estimator and confirm by simulation that the em method is superior to a simple one step method we apply our technique to mouse genomics data and obtain biologically plausible results

on a problem analogues to lehmer s and hurwitz zeta function the main purpose of this paper is to introduce a problem analogous to lehmer s and study a mean value of its error part with the weight of hurwitz zeta function by using the properties of character sums and the analytic method a sharp asymptotic formula was given and a connection between the problem analogous to lehmer s and hurwitz zeta function was established

a new approach to the analytical and numerical form finding of tensegrity structures we develop a new formulation for the form finding of tensegrity structures in which the primary variables are the cartesian components of element lengths both an analytical and a numerical implementation of the formulation are described each require a description of the connectivity of the tensegrity with the iterative numerical method also requiring a random starting vector of member force densities the analytical and numerical form finding of tensegrity structures is demonstrated through six examples and the results obtained are compared and contrasted with those available in the literature to verify the accuracy and viability of the suggested methods

on the degeneracy of the randic entropy and related graph measures abstract numerous quantitative graph measures have been defined and applied in various disciplines such measures may be differentiated according to whether they are information theoretic or non information theoretic in this paper we examine an important property of randic entropy an information theoretic measure and examine some related graph measures based on random roots in particular we investigate the degeneracy of these structural graph measures and discuss numerical results finally we draw some conclusions about the measures applicability to deterministic and non deterministic networks

bounds for degree based network entropies in this paper we continue studying degree based entropies for networks the quantities represent information theoretic network measures which are based on using information functionals involving vertex degrees we prove bounds for entropies which are based on information functionals using degree powers and come up with interrelations between different measures such interrelations are important to study connections between the measures required to understand the measures in depth

extended bayesian information criteria for model selection with large model spaces summary the ordinary bayesian information criterion is too liberal for model selection when the model space is large in this paper we re examine the bayesian paradigm for model selection and propose an extended family of bayesian information criteria which take into account both the number of unknown parameters and the complexity of the model space their consistency is established in particular allowing the number of covariates to increase to infinity with the sample size their performance in various situations is evaluated by simulation studies it is demonstrated that the extended bayesian information criteria incur a small loss in the positive selection rate but tightly control the false discovery rate a desirable property in many applications the extended bayesian information criteria are extremely useful for variable selection in problems with a moderate sample size but with a huge number of covariates especially in genome wide association studies which are now an active area in genetics research

on finding bipartite graphs with a small number of short cycles and large girth the problem of finding bipartite tanner graphs with given degree sequences that have large girth and few short cycles is of great interest in many applications including construction of good low density parity check ldpc codes in this paper we prove that for given integers alpha beta and gamma and degree sequences pi and pi the problem of determining whether there exists a simple bipartite graph with degree sequences pi pi that has at most alpha beta and gamma cycles of length four six and eight respectively is np complete this is proved by a two step polynomial time reduction from the 3 partition problem on the other hand using connections to linear hypergraphs we prove that given the degree sequence pi a polynomial time algorithm can be devised to determine whether there exists a bipartite graph whose degree sequence on one side of the bipartition is pi and has a girth of at least six in addition to these complexity results we devise a quasi polynomial time algorithm that can construct a bipartite graph with given irregular degree sequences and a girth that increases logarithmically with the size of the graph compared to the well known progressive edge growth peg algorithm the proposed method in some cases results in tanner graphs with larger girth particularly for scenarios where the degree sequences of the graph are strictly enforced

on the extremal values of general degree based graph entropies in this note we prove one part of the conjecture about upper and lower bounds of the degree based graph entropy ik t in the class of trees introduced in s cao m dehmer y shi extremality of degree based graph entropies information sciences 278 2014 22 33 using lagrange multipliers and jensen s inequality and disprove the other part by providing a family of counter examples our main result is the following the path pn is unique tree on n vertices that maximizes ik t for k 0 and the star sn is unique tree on n vertices that minimizes ik t for k 1

counting short cycles in bipartite graphs a fast technique algorithm and a hardness result in this paper we propose a new technique based on the so called breadth first search algorithm to count the short cycles of a bipartite graph for a general bipartite graph with v nodes and girth g our technique has a time complexity of mathcal o v 2 delta to count g cycles and g 2 cycles and a time complexity of mathcal o v 2 delta 2 to count g 4 cycles where delta is the maximum node degree in the graph moreover for bi regular bipartite graphs the latter complexity is further reduced to mathcal o v 2 delta compared to the fastest known algorithm which has a complexity mathcal o g v 2 delta 2 the proposed method always has a lower complexity for counting g cycles and g 2 cycles it also has a lower complexity for counting g 4 cycles in bi regular graphs and in scenarios where g is increased with the size of the graph related to the problem of counting short cycles we also demonstrate using a long standing conjecture that there is no algorithm with time complexity less than mathcal o left v 2 frac 2 1 i right that can determine whether a given sparse bipartite graph has a cycle of length 4i an important application of the results presented here is to count the short cycles of tanner graphs of low density parity check ldpc codes

a polynomial time algorithm to find the shortest cycle basis of a graph define the length of a basis of the cycle space of a graph to be the sum of the lengths of all cycles in the basis an algorithm is given that finds a cycle basis with the shortest possible length in o m 3 n operations where m is the number of edges and n is the number of vertices this is the first known polynomial time algorithm for this problem edges may be weighted or unweighted also the shortest cycle basis is shown to have at most 3 n 1 n 2 2 edges for the unweighted case o mn 2 algorithm to obtain a suboptimal cycle basis of length o n 2 for unweighted graphs is also given

further results on digraphs with completely real laplacian spectra let n be the digraph with vertex set v n 1 2 n and arc set e l k k l k 2 n k k 1 k 1 n 1 and n i j the digraph obtained by deleting one arc i j from n where 1 j

topological evolution in dense granular materials a complex networks perspective abstract one of the great challenges in the science of complex materials materials capable of emergent behavior such as self organized pattern formation is deciphering their inherent structural design principles as they deform in response to external loads we have been exploring the efficacy of techniques from complex networks to the study of dense granular materials as a means to i uncover such design principles and ii identify suitable metrics that quantify the evolution of structure during deformation herein we characterize the developing network structure and loss of connectivity in a quasistatically deforming granular medium from the perspective of complex networks attention is paid to the evolution of the contact and contact force networks at the local or mesoscopic level i e a particle and its immediate neighbors as well as the macroscopic level we explore network motifs and other topological properties at these multiple length scales in an attempt to find that which best correlates with the constitutive properties of nonaffine deformation and dissipation spatially and with respect to strain key processes or rearrangement events that cause loss of connectivity within the material domain e g microbanding and force chain buckling are investigated network statistics of these processes previously shown to be major sources of energy dissipation and nonaffine deformation are then tied to corresponding trends observed in the evolving macroscopic network it is shown that consideration of the unweighted contact network alone is insufficient to tie dissipation to loss of material connectivity

counterexamples to conjectures on graph distance measures based on topological indexes in this paper we disprove three conjectures from dehmer et al 2014 on graph distance measures based on topological indices by providing explicit classes of trees that do not satisfy proposed inequalities the constructions are based on the families of trees that have the same wiener index graph energy or randic index but different degree sequences

joint estimation of multiple precision matrices with common structures estimation of inverse covariance matrices known as precision matrices is important in various areas of statistical analysis in this article we consider estimation of multiple precision matrices sharing some common structures in this setting estimating each precision matrix separately can be suboptimal as it ignores potential common structures this article proposes a new approach to parameterize each precision matrix as a sum of common and unique components and estimate multiple precision matrices in a constrained l1 minimization framework we establish both estimation and selection consistency of the proposed estimator in the high dimensional setting the proposed estimator achieves a faster convergence rate for the common structure in certain cases our numerical examples demonstrate that our new estimator can perform better than several existing methods in terms of the entropy loss and frobenius loss an application to a glioblastoma cancer data set reveals some interesting gene networks across multiple cancer subtypes

selection of parity check equations for the iterative message passing detection of m sequences we consider the joint detection and decoding of m sequences the receiver has to decide whether an m sequence is received and possibly to decode its initial state to do so it implements an iterative message passing decoding algorithm that operates on a parity check matrix built upon a number of reference parity check equations satisfied by the m sequence this matrix concatenates several elementary parity check matrices which are derived from reference equations unlike the conventional decoding case the detection problem imposes to consider false alarms that may occur when the decoder is only fed with noise while absorbing sets are known to be responsible for the error floor phenomenon of iterative message passing decoders we show that they may have a beneficial effect on the detection performance in that they may prevent the decoder to produce false alarms we further compute the number of hybrid cycles of length six and eight in the tanner graph of the decoder and use the minimization of this number as criterion to derive an algorithm for selecting the reference parity check equations this algorithm was found to be efficient for minimizing the probability of false alarm and decreases also the probability of wrong detection in the very small snr region this has been achieved at the cost of a reduction of the probability of correct detection

high dimensional semiparametric gaussian copula graphical models ing the spearman s rho and kendall s tau we prove that the nonparanormal skeptic achieves the optimal parametric rates of convergence for both graph recovery and parameter estimation this result suggests that the nonparanormal graphical models can be used as a safe replacement of the popular gaussian graphical models even when the data are truly gaussian besides theoretical analysis we also conduct thorough numerical simulations to compare the graph recovery performance of dierent estimators under both ideal and noisy settings the proposed methods are then applied on a largescale genomic dataset to illustrate their empirical usefulness the r package huge implementing the proposed methods is available on the comprehensive r archive network http cran r project org

comparisons of karci and shannon entropies and their effects on centrality of social networks abstract in order to measure the amount of different information in a system entropy concept can be used graph entropy measures nodes contribution to the entropy of the graph by this way the influential actors can be identified due to this case a new entropy based method was proposed to identify the influential actors karci entropy was applied to the social networks first time the alpha parameter allowed us to combine many different conditions together when measuring in the network the other important contribution of this paper is to predict the value of alpha parameter of karci entropy by using fuzzy logic after that karci and shannon entropies were compared based on experimental results moreover karci entropy was compared to traditional centrality measures if karci entropy definition is considered as a set of entropies shannon entropy can be regarded as an element of this set accordingly it can be concluded that karci entropy is superior to shannon entropy

a lower bound of revised szeged index of bicyclic graphs abstract the revised szeged index of a graph is defined as s z g e u v e n u e n 0 e 2 n v e n 0 e 2 where nu e and nv e are respectively the number of vertices of g lying closer to vertex u than to vertex v and the number of vertices of g lying closer to vertex v than to vertex u and n0 e is the number of vertices equidistant to u and v in the paper we identify the lower bound of revised szeged index among all bicyclic graphs and also characterize the extremal graphs that attain the lower bound

joint estimation of multiple graphical models gaussian graphical models explore dependence relationships between random variables through the estimation of the corresponding inverse covariance matrices in this paper we develop an estimator for such models appropriate for data from several graphical models that share the same variables and some of the dependence structure in this setting estimating a single graphical model would mask the underlying heterogeneity while estimating separate models for each category does not take advantage of the common structure we propose a method that jointly estimates the graphical models corresponding to the different categories present in the data aiming to preserve the common structure while allowing for differences between the categories this is achieved through a hierarchical penalty that targets the removal of common zeros in the inverse covariance matrices across categories we establish the asymptotic consistency and sparsity of the proposed estimator in the high dimensional case and illustrate its performance on a number of simulated networks an application to learning semantic connections between terms from webpages collected from computer science departments is included

a high order modified levenberg marquardt method for systems of nonlinear equations with fourth order convergence fan 2014 presented an accelerated modified levenberg marquardt method for nonlinear equations at every iteration the accelerated modified lm method computed not only a lm trial step but also an additional approximate lm step which employed a line search in this paper based on the accelerated modified lm method we compute the approximate lm step one more time at every iteration and obtain a high order accelerating modified levenberg marquardt method under the local error bound condition which is weaker than nonsingularity the convergence order of this new method is shown to be fourth a globally convergence is also given by the trust region technique numerical results show that the new method is efficient and could save many calculations of the jacobian

adaptive thresholding for sparse covariance matrix estimation in this article we consider estimation of sparse covariance matrices and propose a thresholding procedure that is adaptive to the variability of individual entries the estimators are fully data driven and demonstrate excellent performance both theoretically and numerically it is shown that the estimators adaptively achieve the optimal rate of convergence over a large class of sparse covariance matrices under the spectral norm in contrast the commonly used universal thresholding estimators are shown to be suboptimal over the same parameter spaces support recovery is discussed as well the adaptive thresholding estimators are easy to implement the numerical performance of the estimators is studied using both simulated and real data simulation results demonstrate that the adaptive thresholding estimators uniformly outperform the universal thresholding estimators the method is also illustrated in an analysis on a dataset from a small round blue cell tumor microarray experiment a supplement to this ar

note on two generalizations of the randic index for a given graph g the well known randic index of g introduced by milan randic in 1975 is defined as r g u v e g d u d v 1 2 where the sum is taken over all edges uv and du denotes the degrees of u bollobas and erdos generalized this index by replacing 1 2 with any real number which is called the general randic index dvo ak et al introduced a modified version of randic index r g u v e g max d u d v 1 based on this recently knor et al introduced two generalizations r g u v e g min d u d v and r g u v e g max d u d v for any real number clearly the former is a lower bound for the general randic index and the latter is its upper bound knor et al studied extremal values of r g and r g and concluded some open problems in this paper we consider the open problems and give some comments and results some results for chemical trees are obtained

on the distance based graph entropies in this note we prove a sharp upper bound for the distance based graph entropy introduced in chen et al 2014 furthermore we resolve one part of the conjecture proposed in the same paper and provide counterexamples for the other part

regular and irregular progressive edge growth tanner graphs we propose a general method for constructing tanner graphs having a large girth by establishing edges or connections between symbol and check nodes in an edge by edge manner called progressive edge growth peg algorithm lower bounds on the girth of peg tanner graphs and on the minimum distance of the resulting low density parity check ldpc codes are derived in terms of parameters of the graphs simple variations of the peg algorithm can also be applied to generate linear time encodeable ldpc codes regular and irregular ldpc codes using peg tanner graphs and allowing symbol nodes to take values over gf q q 2 are investigated simulation results show that the peg algorithm is a powerful algorithm to generate good short block length ldpc codes

quic quadratic approximation for sparse inverse covariance estimation the l1 regularized gaussian maximum likelihood estimator mle has been shown to have strong statistical guarantees in recovering a sparse inverse covariance matrix or alternatively the underlying graph structure of a gaussian markov random field from very limited samples we propose a novel algorithm for solving the resulting optimization problem which is a regularized log determinant program in contrast to recent state of the art methods that largely use first order gradient information our algorithm is based on newton s method and employs a quadratic approximation but with some modifications that leverage the structure of the sparse gaussian mle problem we show that our method is superlinearly convergent and present experimental results using synthetic and real world application data that demonstrate the considerable improvements in performance of our method when compared to previous methods

the bipartite unicyclic graphs with the first n 3 4 largest matching energies the theory of matching energy of graphs since been proposed by gutman and wagner in 2012 has attracted more and more attention it is denoted by b n m the class of bipartite graphs with order n and size m in particular b n n denotes the set of bipartite unicyclic graphs which is an interesting class of graphs in this paper for odd n we characterize the bipartite unicyclic graphs with the first n 3 4 largest matching energies there is an interesting correspondence we conclude that the graph with the second maximal matching energy in b n n for odd n 11 is p n 6 which is the only graph attaining the maximum value of the energy among all the bipartite unicyclic graphs for n 16

model selection and estimation in the gaussian graphical model summary we propose penalized likelihood methods for estimating the concentration matrix in the gaussian graphical model the methods lead to a sparse and shrinkage estimator of the concentration matrix that is positive definite and thus conduct model selection and estimation simultaneously the implementation of the methods is nontrivial because of the positive definite constraint on the concentration matrix but we show that the computation can be done effectively by taking advantage of the efficient maxdet algorithm developed in convex optimization we propose a bic type criterion for the selection of the tuning parameter in the penalized likelihood methods the connection between our methods and existing methods is illustrated simulations and real examples demonstrate the competitive performance of the new methods

the energy of a graph abstract the energy e g of a simple graph g is defined to be the sum of the absolute values of the eigen values of g if g is a k regular graph on n vertices then e g k k n 1 n k b 2 and this bound is sharp it is shown that for each 0 there exist infinitely many n for each of which there exists a k regular graph g of order n with k e g b 2 two graphs with the same number of vertices are equienergetic if they have the same energy we show that for any positive integer n 3 there exist two equienergetic graphs of order 4n that are not cospectral

partial correlation estimation by joint sparse regression models this article features online supplementary material in this article we propose a computationally efficient approach space sparse partial correlation estimation for selecting nonzero partial correlations under the high dimension low sample size setting this method assumes the overall sparsity of the partial correlation matrix and employs sparse regression techniques for model fitting we illustrate the performance of space by extensive simulation studies it is shown that space performs well in both nonzero partial correlation selection and the identification of hub variables and also outperforms two existing methods we then apply space to a microarray breast cancer dataset and identify a set of hub genes that may provide important insights on genetic regulatory networks finally we prove that under a set of suitable assumptions the proposed procedure is asymptotically consistent in terms of model selection and parameter estimation

sparsistency and rates of convergence in large covariance matrix estimation this paper studies the sparsistency and rates of convergence for estimating sparse covariance and precision matrices based on penalized likelihood with nonconvex penalty functions here sparsistency refers to the property that all parameters that are zero are actually estimated as zero with probability tending to one depending on the case of applications sparsity priori may occur on the covariance matrix its inverse or its cholesky decomposition we study these three sparsity exploration problems under a unified framework with a general penalty function we show that the rates of convergence for these problems under the frobenius norm are of order s n log p n n 1 2 where s n is the number of nonzero elements p n is the size of the covariance matrix and n is the sample size this explicitly spells out the contribution of high dimensionality is merely of a logarithmic factor the conditions on the rate with which the tuning parameter n goes to 0 have been made explicit and compared under different penalties as a result for the l 1 penalty to guarantee the sparsistency and optimal rate of convergence the number of nonzero elements should be small sn o pn at most among o pn2 parameters for estimating sparse covariance or correlation matrix sparse precision or inverse correlation matrix or sparse cholesky factor where sn is the number of the nonzero elements on the off diagonal entries on the other hand using the scad or hard thresholding penalty functions there is no such a restriction

properties of gomory hu co cycle bases let d v a be a directed graph with non negative arc weights we study the problem of computing certain special co cycle bases of d in particular a minimum weight weakly fundamental co cycle basis a co cycle in d corresponds to a cut in the underlying undirected graph and a 1 0 1 arc incidence vector is associated with each co cycle where the 1 coordinates are used for the arcs crossing the cut the weight of a co cycle c is the sum of the weights of those arcs a such that c a 1 the vector space over q generated by the arc incidence vectors of the co cycles is the co cycle space of d the co cycle space of d can also be defined as the orthogonal complement of the cycle space of d a set of linearly independent co cycles that span the co cycle space is a co cycle basis of d the problem of computing a co cycle basis c 1 c k such that the sum of weights of the co cycles in the basis is the least possible is the minimum co cycle basis problem a co cycle basis c 1 c k is weakly fundamental if for every i there is an arc a i such that c i a i 1 while c j a i 0 for j i the minimum cycle basis problem in directed and undirected graphs is a well studied problem and while polynomial time algorithms are known for these problems the problem of computing a minimum weight weakly fundamental cycle basis has recently been shown to be apx hard we show that the co cycle basis corresponding to the cuts of a gomory hu tree t of the underlying undirected graph of d is a minimum weight weakly fundamental co cycle basis of d this is in fact a minimum co cycle basis of d and it is also totally unimodular thus this is a special co cycle basis that simultaneously answers several questions in the domain of co cycle bases it is known that there is no such special cycle basis for general graphs

a history of graph entropy measures this survey seeks to describe methods for measuring the entropy of graphs and to demonstrate the wide applicability of entropy measures setting the scene with a review of classical measures for determining the structural information content of graphs we discuss graph entropy measures which play an important role in a variety of problem areas including biology chemistry and sociology in addition we examine relationships between selected entropy measures illustrating differences quantitatively with concrete examples

the extraction and complexity limits of graphical models for linear codes two broad classes of graphical modeling problems for codes can be identified in the literature constructive and extractive problems the former class of problems concern the construction of a graphical model in order to define a new code the latter class of problems concern the extraction of a graphical model for a fixed given code the design of a new low density parity check code for some given criteria e g target block length and code rate is an example of a constructive problem the determination of a graphical model for a classical linear block code that implies a decoding algorithm with desired performance and complexity characteristics is an example of an extractive problem this work focuses on extractive graphical model problems and aims to lay out some of the foundations of the theory of such problems for linear codes the primary focus of this work is a study of the space of all graphical models for a fixed given code the tradeoff between cyclic topology and complexity in this space is characterized via the introduction of a new bound the forest inducing cut set bound fi csb the proposed bound provides a more precise characterization of this tradeoff than that which can be obtained using existing tools e g the csb and can be viewed as a generalization of the square root bound for tail biting trellises to graphical models with arbitrary cyclic topologies searching the space of graphical models for a given code is then enabled by introducing a set of basic graphical model transformation operations that are shown to span this space finally heuristics for extracting novel graphical models for linear block codes using these transformations are investigated

a modified two steps levenberg marquardt method for nonlinear equations the modified levenberg marquardt mlm method to solve nonlinear equations was introduced by fan 2012 this method uses an addition of the levenberg marquardt step and an approximate lm step as the trial step at every iteration using a trust region technique the global and cubic convergence of the mlm method under the local error bound condition is proved fan 2012 recently fan proposed an accelerated mlm algorithm by using a line search strategy to generate a modified lm step and showed that the convergence rate of the algorithm is min 1 2 3 which results the cubic convergence for 1 fan 2014 in this paper by introducing an adaptive lm parameter for amlm algorithm we propose an efficient amlm algorithm the cubic convergence of the new algorithm is presented while numerical experiments show the new algorithm is promising

extremal values of matching energies of one class of graphs in 1978 gutman proposed the concept of graph energy defined as the sum of the absolute values of eigenvalues of the adjacency matrix of a molecular graph which is related to the energy of electrons in conjugated hydrocarbons recently gutman and wagner proposed the concept of matching energy and pointed out that the chemical applications of matching energy go back to the 1970s in this paper we study the extremal values of the matching energy and characterize the graphs with minimal matching energy among all tricyclic graphs with a given diameter our methods can help to find more extremal values for other classes of molecular networks and the results suggest the structures with extremal energies

an algorithm for counting short cycles in bipartite graphs let g u spl cup w e be a bipartite graph with disjoint vertex sets u and w edge set e and girth g this correspondence presents an algorithm for counting the number of cycles of length g g 2 and g 4 incident upon every vertex in u spl cup w the proposed cycle counting algorithm consists of integer matrix operations and its complexity grows as o gn sup 3 where n max u w

new highlights and a new centrality measure based on the adapted pagerank algorithm for urban networks a centrality measure with a practical application in urban network has been proposed the measure establishes a ranking of nodes focused on the topological data distribution the amount of data associated with the nodes of the network has a great importance the topology of the network and the information associated to the nodes are involved the adapted pagerank algorithm apa proposed by agryzkov et al provides us a method to establish a ranking of nodes in an urban network we can say that it constitutes a centrality measure in urban networks with the main characteristic that is able to consider the importance of data obtained from the urban networks in the process of computing the centrality of every node starting from the basic idea of this model we modify the construction of the matrix used for the classification of the nodes in order of importance in the apa model the data matrix is constructed from the original idea of pagerank vector given an equal chance to jump from one node to another regardless of the topological distance between nodes in the new model this idea is questioned a new matrix with the data network is constructed so that now the data from neighboring nodes are considered more likely than data from the nodes that are farther away in addition this new algorithm has the characteristic that depends on a parameter which allows us to decide the importance attached in the computation of the centrality to the topology of the network and the amount of data associated with the node various numerical experiments with a network of very small size are performed to test the influence of the data associated with the nodes depending always on the choice of the parameter also we check the differences between the values produced by the original apa model and the new one finally these measures are applied to a real urban network in which we perform a visual comparison of the results produced by the various measures calculated from the algorithms studied

on the automation of the force method in the optimal plastic design of frames it is well known that linear programming provides computationally the only solution to the optimal plastic design of plane frames it is also known that the formulation of the problem by the force method using kinematic variables requires for its solution by the simplex method much less computation than the corresponding program which uses for its formulation the displacement method however the force method is not easily automated this problem is circumvented in the paper by presenting a new algorithm which automatically selects a subminimal cycle base in any planar graph based on a shortest path technique between two points of a connected graph the algorithm forms cycles by finding the shortest paths between the end nodes of members of the graph the check for independence is done by fictitiously increasing the lengths of the members that form the cycle this leads to an easy computer implementation of the algorithm the algorithm is then used to provide a statical basis for any plane frame equilibrium with applied loads is satisfied by the use of cantilevers that follow the shortest load path of each load to the ground the whole process automates the data generation of the force method and makes possible to use in a computer program the same input data as in the displacement method a few examples of applications are given

on the construction of structured ldpc codes free of small trapping sets we present a method to construct low density parity check ldpc codes with low error floors on the binary symmetric channel codes are constructed so that their tanner graphs are free of certain small trapping sets these trapping sets are selected from the trapping set ontology for the gallager a b decoder they are selected based on their relative harmfulness for a given decoding algorithm we evaluate the relative harmfulness of different trapping sets for the sum product algorithm by using the topological relations among them and by analyzing the decoding failures on one trapping set in the presence or absence of other trapping sets we apply this method to construct structured ldpc codes to facilitate the discussion we give a new description of structured ldpc codes whose parity check matrices are arrays of permutation matrices this description uses latin squares to define a set of permutation matrices that have disjoint support and to derive a simple necessary and sufficient condition for the tanner graph of a code to be free of four cycles

graph entropy based on the number of spanning forests of c cyclic graphs abstract graph entropies have been introduced to quantitatively measure the structural information content of graphs and networks they have plenty of applications in various fields utilizing the number of subgraphs to establish measures for determining the complexity of molecular graphs are also prevalent in the study of mathematical chemistry in this paper we develop a new graph entropy measure that is based on the number of spanning forests we prove explicit expressions for the entropy for trees unicyclic and bicyclic graphs and show that the cycle graph cn attains the maximal value of the entropy for unicyclic graphs with order n and large cycle lengths based on generating numerical results we conjecture extremal unicyclic graphs with respect to the entropy as well as we compare the values of our entropy for c cyclic graphs and generate graphs of bicyclic graphs and tricyclic graphs with 6 vertices for performing further research

network entropies based on independent sets and matchings entropy based measures have been used to characterize the structure of complex networks various graph entropies based on different graph parameters have been proposed and studied so it has been intricate to decide which measure is the right one for solving a particular problem in this paper we introduce graph entropy measures based on independent sets and matchings of graphs the values of entropies of some special graphs are calculated and we draw several conclusions regrading the usability of the measures based on our findings we conclude that it should be demanding studying extremal values of the novel entropies

on the general trigonometric sums weighted by character sums the main purpose of this paper is using the estimates for character sums and the mean value properties of dirichlet l function to study the hybrid mean value properties of the general trigonometric sums which is weighted by the 2kth and first power mean of character sums in 1 p 4 the two sharp asymptotic formulae are derived and some interesting connections between these sums are established

on steiner degree distance of trees let g be a connected graph and u v w its vertices by du is denoted the degree of the vertex u by d u v the ordinary distance of the vertices u and v and by d u v w the steiner distance of u v w the degree distance dd of g is defined as the sum of terms d u d v d u v over all pairs of vertices of g as early as in the 1990s a linear relation was discovered between dd of trees and the wiener index we now consider sdd the steiner distance generalization of dd defined as the sum of terms d u d v d w d u v w over all triples of vertices of g also in this case a linear relation between sdd and the wiener index could be established

factoring with two large primes the study of integer factoring algorithms and the design of faster factoring algorithms is a subject of great importance in cryptology cf 1 and a constant concern for cryptographers in this paper we present a new technique that proved to be extremely useful not only to achieve a considerable speed up of an older and widely studied factoring algorithm but also and more importantly to make practical application of a new factoring algorithm feasible while this first application does not pose serious threats to factorization based cryptosystems the consequences of the second application could be very encouraging from the cryptanalysts point of view

wiener polarity index of dendrimers abstract network structures have many applications in biological information theory physical and social sciences research on network measurements is more and more important recently a number of topological indices have been studied as descriptors of complex networks the wiener polarity index has been introduced for a long time and known to be related to the cluster coefficient of networks in this paper we study the wiener polarity index of dendrimers which has many applications in other fields

a comparative analysis of new graph distance measures and graph edit distance the problem of determining the structural similarity or distance between graphs got considerable attention during the last decades consequently various similarity and distance measures for networks have already been investigated dehmer etal studied a new distance measure for graphs which is based on topological indices an interesting problem is to compare our class of measures with the well known graph edit distance which has been studied extensively and often serves as a benchmark measure but note that ged is generally np hard in this paper we compare the edit distance with our graph distance measures in order to do so we use some well known topological indices such as graph energy wiener index randi index and graph entropy by using several special graph classes numerical results reveal that the graph distance measure based on graph energy approximates ged well this fact could trigger an important research direction for studying graph edit distance and other comparative network measures

model selection through sparse maximum likelihood estimation for multivariate gaussian or binary data we consider the problem of estimating the parameters of a gaussian or binary distribution in such a way that the resulting undirected graphical model is sparse our approach is to solve a maximum likelihood problem with an added l1 norm penalty term the problem as formulated is convex but the memory requirements and complexity of existing interior point methods are prohibitive for problems with more than tens of nodes we present two new algorithms for solving problems with at least a thousand nodes in the gaussian case our first algorithm uses block coordinate descent and can be interpreted as recursive l1 norm penalized regression our second algorithm based on nesterov s first order method yields a complexity estimate with a better dependence on problem size than existing interior point methods using a log determinant relaxation of the log partition function wainwright and jordan 2006 we show that these same algorithms can be used to solve an approximate sparse maximum likelihood problem for the binary case we test our algorithms on synthetic data as well as on gene expression and senate voting records data

three steps modified levenberg marquardt method with a new line search for systems of nonlinear equations three steps modified levenberg marquardt method for nonlinear equations was introduced by yang 2013 this method uses the addition of the levenberg marquardt lm step and two approximate lm steps as the trial step at every iteration using trust region technique the global and biquadratic convergence of the method is proved by yang the main aim of this paper is to introduce a new line search strategy while investigating the convergence properties of the method with this line search technique since the search direction of yang method may be not a descent direction standard line searches cannot be used directly in this paper we propose a new nonmonotone third order armijo type line search technique which guarantees the global convergence of this method while we use an adaptive lm parameter it is proved that the convergence order of the new method is biquadratic numerical results show the new algorithm is efficient and promising

on the tanner graph cycle distribution of random ldpc random protograph based ldpc and random quasi cyclic ldpc code ensembles in this paper we study the cycle distribution of random low density parity check ldpc codes randomly constructed protograph based ldpc codes and random quasi cyclic qc ldpc codes we prove that for a random bipartite graph with a given irregular degree distribution the distributions of cycles of different length tend to independent poisson distributions as the size of the graph tends to infinity we derive asymptotic upper and lower bounds on the expected values of the poisson distributions that are independent of the size of the graph and only depend on the degree distribution and the cycle length for a random lift of a bi regular protograph we prove that the asymptotic cycle distributions are essentially the same as those of random bipartite graphs as long as the degree distributions are identical for random qc ldpc codes however we show that the cycle distribution can be quite different from the other two categories in particular depending on the protograph and the value of c the expected number of cycles of length c in this case can be either theta n or theta 1 where n is the lifting degree code length we also provide numerical results that match our theoretical derivations our results provide a theoretical foundation for emperical results that were reported in the literature but were not well justified they can also be used for the analysis and design of ldpc codes and associated algorithms that are based on cycles

transactions letters random redundant iterative soft in soft out decoding this letter presents an iterative soft in soft out siso decoding algorithm based on redundant tanner graphs that is applicable to arbitrary linear block codes the proposed algorithm utilizes the permutation group of a code in order to efficiently and randomly generate redundant parity checks