can we neglect the multi layer structure of functional networks functional networks i e networks representing dynamic relationships between the components of a complex system have been instrumental for our understanding of among others the human brain due to limited data availability the multi layer nature of numerous functional networks has hitherto been neglected and nodes are endowed with a single type of links even when multiple relationships coexist at different physical levels a relevant problem is the assessment of the benefits yielded by studying a multi layer functional network against the simplicity guaranteed by the reconstruction and use of the corresponding single layer projection here i tackle this issue by using as a test case the functional network representing the dynamics of delay propagation through european airports neglecting the multi layer structure of a functional network has dramatic consequences on our understanding of the underlying system a fact to be taken into account when a projection is the only available information

enhancing synchronizability of diffusively coupled dynamical networks a survey in this paper we review the literature on enhancing synchronizability of diffusively coupled dynamical networks with identical nodes the last decade has witnessed intensive investigations on the collective behavior over complex networks and synchronization of dynamical systems is the most common form of collective behavior for many applications it is desired that the synchronizability the ability of networks in synchronizing activity of their individual dynamical units is enhanced there are a number of methods for improving the synchronization properties of dynamical networks through structural perturbation in this paper we survey such methods including adding removing nodes and or edges rewiring the links and graph weighting these methods often try to enhance the synchronizability through minimizing the eigenratio of the laplacian matrix of the connection graph a synchronizability measure based on the master stability function formalism we also assess the performance of the methods by numerical simulations on a number of real world networks as well as those generated through models such as preferential attachment watts strogatz and erdos renyi

nonintegrable dynamics of the triplet triplet spatiotemporal interaction in this paper we examine the coupling of two wave triplets sharing two common modes the analysis is performed in the solitonic sector of the parameter space where uncoupled solutions departing from linearly unstable homogeneous initial conditions evolve into a collection of regularly interspersed spatiotemporally localized spikes the uncoupled system is integrable but coupling destroys integrability as coupling grows energy transfer to smaller spatial scales does appear and becomes faster not only in linearly unstable but also in linearly stable cases chaos in low dimensional subsystems appears to be responsible for the transfer we perform a series of numerical tests to verify this idea

spectral characteristics of network redundancy many real world complex networks contain a significant amount of structural redundancy in which multiple vertices play identical topological roles such redundancy arises naturally from the simple growth processes which form and shape many real world systems since structurally redundant elements may be permuted without altering network structure redundancy may be formally investigated by examining network automorphism symmetry groups here we use a group theoretic approach to give a complete description of spectral signatures of redundancy in undirected networks in particular we describe how a network s automorphism group may be used to directly associate specific eigenvalues and eigenvectors with specific network motifs

m tensors and nonsingular m tensors abstract the m matrix is an important concept in matrix theory and has many applications recently this concept has been extended to higher order tensors 18 in this paper we establish some important properties of m tensors and nonsingular m tensors an m tensor is a z tensor we show that a z tensor is a nonsingular m tensor if and only if it is semi positive thus a nonsingular m tensor has all positive diagonal entries an m tensor regarding as the limit of a sequence of nonsingular m tensors has all nonnegative diagonal entries we introduce even order monotone tensors and present their spectral properties in matrix theory a z matrix is a nonsingular m matrix if and only if it is monotone this is no longer true in the case of higher order tensors we show that an even order monotone z tensor is an even order nonsingular m tensor but not vice versa an example of an even order nontrivial monotone z tensor is also given

link influence entropy abstract in this paper we propose a new metric link influence entropy line which describes importance of each node based on the influence of each link present in a network influence of a link can neither be effectively estimated using betweenness centrality nor using degree based probability measures the proposed line metric which provides an effective way to estimate the influence of a link in the network and incorporates this influence to identify nodal characteristics performs better compared to degree based entropy we found that line can differentiate various network types which degree based or betweenness centrality based node influence metrics cannot our findings show that spatial wireless networks and regular grid networks respectively have lowest and highest line values finally performance analysis of line is carried out on a real world network as well as on a wireless mesh network testbed to study the influence of our metric as well as influence stability of nodes in dynamic networks

detection of topological patterns in complex networks correlation profile of the internet abstract a general scheme for detecting and analyzing topological patterns in large complex networks is presented in this scheme the network in question is compared with its properly randomized version that preserves some of its low level topological properties statistically significant deviation of any topological property of a network from this null model likely reflects its design principles and or evolutionary history we illustrate this basic scheme using the example of the correlation profile of the internet quantifying correlations between degrees of its neighboring nodes this profile distinguishes the internet from previously studied molecular networks with a similar scale free degree distribution we finally demonstrate that the clustering in a network is very sensitive to both the degree distribution and its correlation profile and compare the clustering in the internet to the appropriate null model

mixing patterns in networks we study assortative mixing in networks the tendency for vertices in networks to be connected to other vertices that are like or unlike them in some way we consider mixing according to discrete characteristics such as language or race in social networks and scalar characteristics such as age as a special example of the latter we consider mixing according to vertex degree i e according to the number of connections vertices have to other vertices do gregarious people tend to associate with other gregarious people we propose a number of measures of assortative mixing appropriate to the various mixing types and apply them to a variety of real world networks showing that assortative mixing is a pervasive phenomenon found in many networks we also propose several models of assortatively mixed networks both analytic ones based on generating function methods and numerical ones based on monte carlo graph generation techniques we use these models to probe the properties of networks as their level of assortativity is varied in the particular case of mixing by degree we find strong variation with assortativity in the connectivity of the network and in the resilience of the network to the removal of vertices

maximal planar networks with large clustering coefficient and power law degree distribution in this article we propose a simple rule that generates scale free networks with very large clustering coefficient and very small average distance these networks are called random apollonian networks rans as they can be considered as a variation of apollonian networks we obtain the analytic results of power law exponent ensuremath gamma 3 and clustering coefficient c frac 46 3 ensuremath 36 phantom rule 0 2em 0ex mathrm ln phantom rule 0 2em 0ex frac 3 2 ensuremath approx 0 74 which agree with the simulation results very well we prove that the increasing tendency of average distance of rans is a little slower than the logarithm of the number of nodes in rans since most real life networks are both scale free and small world networks rans may perform well in mimicking the reality the rans possess hierarchical structure as c k ensuremath sim k ensuremath 1 that are in accord with the observations of many real life networks in addition we prove that rans are maximal planar networks which are of particular practicability for layout of printed circuits and so on the percolation and epidemic spreading process are also studied and the comparisons between rans and barab asi albert ba as well as newman watts nw networks are shown we find that when the network order n the total number of nodes is relatively small as n ensuremath sim 10 4 the performance of rans under intentional attack is not sensitive to n while that of ba networks is much affected by n and the diseases spread slower in rans than ba networks in the early stage of the suseptible infected process indicating that the large clustering coefficient may slow the spreading velocity especially in the outbreaks

optimization of network robustness to waves of targeted and random attacks we study the robustness of complex networks to multiple waves of simultaneous sid targeted attacks in which the highest degree nodes are removed and siid random attacks sor failuresd in which fractions pt and pr respectively of the nodes are removed until the network collapses we find that the network design which optimizes network robustness has a bimodal degree distribution with a fraction r of the nodes having degree k2 skkl 1 rd r and the remainder of the nodes having degree k1 1 where kkl is the average degree of all the nodes we find that the optimal value ofr is of the order of pt pr for pt pr 1

attack vulnerability of complex networks we study the response of complex networks subject to attacks on vertices and edges several existing complex network models as well as real world networks of scientific collaborations and internet traffic are numerically investigated and the network performance is quantitatively measured by the average inverse geodesic length and the size of the largest connected subgraph for each case of attacks on vertices and edges four different attacking strategies are used removals by the descending order of the degree and the betweenness centrality calculated for either the initial network or the current network during the removal procedure it is found that the removals by the recalculated degrees and betweenness centralities are often more harmful than the attack strategies based on the initial network suggesting that the network structure changes as important vertices or edges are removed furthermore the correlation between the betweenness centrality and the degree in complex networks is studied

generalized tensor equations with leading structured tensors abstract systems of tensor equations tes have received much considerable attention in the recent literature in this paper we consider a class of generalized tensor equations gtes an important difference between gtes and tes is that gtes can be regarded as a system of non homogenous polynomial equations whereas tes is a homogenous one such a difference usually make the theoretical and algorithmic results tailored for tes not necessarily applicable to gtes to study properties of the solution set of gtes we first introduce a new class of so named z tensors which includes the set of all p tensors as a proper subset with the help of degree theory we prove that the system of gtes with a leading coefficient z tensor has at least one solution for any right hand side vector moreover we study the local error bounds under some appropriate conditions finally we employ a levenberg marquardt algorithm to find a solution to gtes and report some preliminary numerical results

variability analysis of complex networks measures based on stochastic distances abstract complex networks can model the structure and dynamics of different types of systems it has been shown that they are characterized by a set of measures in this work we evaluate the variability of complex network measures face to perturbations and for this purpose we impose controlled perturbations and quantify their effect we analyze theoretical models random small world and scale free and real networks a collaboration network and a metabolic networks along with the shortest path length vertex degree local cluster coefficient and betweenness centrality measures in such an analysis we propose the use of three stochastic quantifiers the kullback leibler divergence and the jensen shannon and hellinger distances the sensitivity of these measures was analyzed with respect to the following perturbations edge addition edge removal edge rewiring and node removal all of them applied at different intensities the results reveal that the evaluated measures are influenced by these perturbations additionally hypotheses tests were performed to verify the behavior of the degree distribution to identify the intensity of the perturbations that leads to break this property

mapping and discrimination of networks in the complexity entropy plane complex networks are usually characterized in terms of their topological spatial or information theoretic properties and combinations of the associated metrics are used to discriminate networks into different classes or categories however even with the present variety of characteristics at hand it still remains a subject of current research to appropriately quantify a network s complexity and correspondingly discriminate between different types of complex networks like infrastructure or social networks on such a basis here we explore the possibility to classify complex networks by means of a statistical complexity measure that has formerly been successfully applied to distinguish different types of chaotic and stochastic time series it is composed of a network s averaged per node entropic measure characterizing the network s information content and the associated jenson shannon divergence as a measure of disequilibrium we study 29 real world networks and show that networks of the same category tend to cluster in distinct areas of the resulting complexity entropy plane we demonstrate that within our framework connectome networks exhibit among the highest complexity while e g transportation and infrastructure networks display significantly lower values furthermore we demonstrate the utility of our framework by applying it to families of random scale free and watts strogatz model networks we then show in a second application that the proposed framework is useful to objectively construct threshold based networks such as functional climate networks or recurrence networks by choosing the threshold such that the statistical network complexity is maximized

the presence of an invariant manifold in the dissipative nonlinear four wave coupling abstract we examine the existence of an invariant manifold in the phase space structure presented by a dissipative four wave coupling we show that once in such manifold the system preserves two quantities as time evolves a quantity related to the energy of the system and another related to the exchange of energy between the waves we observe that the presence of this manifold allows the system to manifest stable solutions for a specific interval of the dissipation parameters out of this range the system brings up only uncoupled states no nonlinear saturated states the dissipation parameters affect only the transversal dynamics to the manifold changing how quickly the invariant manifold is reached in the invariant manifold the system exhibits a large number of coexisting periodic attractors limit cycles coupled states with an interwoven structure of basins of attraction the time series of conserved quantities and the properties of the lyapunov spectra are used to imply the existence of a lower dimensional invariant manifold since the dynamics in such invariant manifold preserves an energy like function we identify it as being conservative

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

entropy optimization of scale free networks robustness to random failures abstract many networks are characterized by highly heterogeneous distributions of links which are called scale free networks and the degree distributions follow p k ck we study the robustness of scale free networks to random failures from the character of their heterogeneity entropy of the degree distribution can be an average measure of a network s heterogeneity optimization of scale free networks robustness to random failures with average connectivity constant is equivalent to maximizing the entropy of the degree distribution by examining the relationship of the entropy of the degree distribution scaling exponent and the minimal connectivity we get the optimal design of scale free networks to random failures we conclude that the entropy of the degree distribution is an effective measure of network s resilience to random failures

network quotients structural skeletons of complex systems a defining feature of many large empirical networks is their intrinsic complexity however many networks also contain a large degree of structural repetition an immediate question then arises can we characterize essential network complexity while excluding structural redundancy in this article we utilize inherent network symmetry to collapse all redundant information from a network resulting in a coarse graining which we show to carry the essential structural information of the parent network in the context of algebraic combinatorics this coarse graining is known as the quotient we systematically explore the theoretical properties of network quotients and summarize key statistics of a variety of real world quotients with respect to those of their parent networks in particular we find that quotients can be substantially smaller than their parent networks yet typically preserve various key functional properties such as complexity heterogeneity and hub vertices and communication diameter and mean geodesic distance suggesting that quotients constitute the essential structural skeletons of their parent networks we summarize with a discussion of potential uses of quotients in analysis of biological regulatory networks and ways in which using quotients can reduce the computational complexity of network algorithms

partial synchronization and partial amplitude death in mesoscale network motifs we study the interplay between network topology and complex space time patterns and introduce a concept to analytically predict complex patterns in networks of stuart landau oscillators with linear symmetric and instantaneous coupling based solely on the network topology these patterns consist of partial amplitude death and partial synchronization and are found to exist in large variety for all undirected networks of up to 5 nodes the underlying concept is proved to be robust with respect to frequency mismatch and can also be extended to larger networks in addition it directly links the stability of complete in phase synchronization to only a small subset of topological eigenvalues of a network

symmetry based structure entropy of complex networks abstract precisely quantifying the heterogeneity or disorder of network systems is important and desired in studies of behaviors and functions of network systems although various degree based entropies have been available to measure the heterogeneity of real networks heterogeneity implicated in the structures of networks can not be precisely quantified yet hence we propose a new structure entropy based on automorphism partition analysis of extreme cases shows that entropy based on automorphism partition can quantify the structural heterogeneity of networks more precisely than degree based entropies we also summarized symmetry and heterogeneity statistics of many real networks finding that real networks are more heterogeneous in the view of automorphism partition than what have been depicted under the measurement of degree based entropies and that structural heterogeneity is strongly negatively correlated to symmetry of real networks

a note on the levenberg marquardt parameter abstract in c ma l jiang some research on levenberg marquardt method for the nonlinear equations appl math comput 184 2007 1032 1040 the lm parameter at the kth iteration is chosen as k f k 2 1 j k t f k 2 where f is the residual function j is the jacobi of f and 0 1 is a constant in this note we point out that the lm parameter can be any combination of f k 2 and j k t f k 2 provided it is positive furthermore we give a more general choice of the lm parameter and show that the lm method still preserves the quadratic convergence under the local error condition which is weaker than nonsingularity

emergence of symmetry in complex networks many real networks have been found to have a rich degree of symmetry which is a universal structural property of complex networks yet has been rarely studied so far one of the fascinating problems related to symmetry is exploration of the origin of symmetry in real networks for this purpose we summarized the statistics of local symmetric motifs that contribute to local symmetry of networks analysis of these statistics shows that the symmetry of complex networks is a consequence of similar linkage pattern which means that vertices with similar degrees tend to share common neighbors an improved version of the baraba i albert model integrating similar linkage pattern successfully reproduces the symmetry of real networks indicating that similar linkage pattern is the underlying ingredient that is responsible for the emergence of symmetry in complex networks

d eigenvalues of diffusion kurtosis tensors diffusion kurtosis imaging dki is a new model in medical engineering where a diffusion kurtosis dk tensor is involved a dk tensor is a fourth order three dimensional fully symmetric tensor in this paper we introduce d eigenvalues for a dk tensor the largest the smallest and the average d eigenvalues of a dk tensor correspond with the largest the smallest and the average apparent kurtosis coefficients akc of a water molecule in the space respectively we present their computational methods and discuss related anisotropy value of akc

neural network approach for solving nonsingular multi linear tensor systems abstract the main propose of this paper is to develop two neural network models for solving nonsingular multi linear tensor system theoretical analysis shows that each of the neural network models ensures the convergence performance for possible hardware implementation of the proposed neural network models based on digital circuits we adopt the euler type difference rule to discretize the corresponding gradient neural network gnn models the computer simulation results further substantiate that the models can solve a multi linear system with nonsingular tensors

onset of spatiotemporal chaos in a nonlinear system we describe the onset of spatiotemporal chaos in a spatially extended nonlinear dynamical system as a result of the loss of transversal stability of an invariant manifold representing a spatially homogeneous and temporally chaotic state the onset of spatiotemporal chaos is characterized by the switching between spatially homogeneous and nonhomogeneous states with statistical properties of on off intermittency

degree based attacks are not optimal for desynchronization in general networks suppressing harmful synchronization in complex networked systems is receiving increasing interest in various contexts including epileptic seizure and the internet and traffic congestion it is traditionally believed that by disrupting the most important nodes i e those nodes of high degrees synchronization can be effectively mitigated however this approach can perform poorly in more general situations such as homogeneous random or small world networks in this article we investigate how topological properties such as the heterogeneity of the network can have an impact on its desynchronization in particular we propose a topology aware scheme that chooses topologically sufficiently diversified contrarian nodes by taking into account the patterns of connectivity among them hence a maximal number of nodes in the network can be influenced leading to a more global impact and greater disruption to synchronization our scheme demonstrates significantly improved performance on various networked systems including homogeneous networks small world and even scale free networks

network robustness and random processes abstract we introduce two new measures of network robustness and apply them to four different strategies the measures are independent from the number of nodes in the network and have the strong potential to cover a large portfolio of applications using the monte carlo methods we demonstrate how to approximate the measures the methods are based on random interruption of links with suitable constraints which represent the above mentioned strategies we introduce two networks with obvious varying robustness to demonstrate the measures we also demonstrate how to employ the measures in order to improve the robustness of the networks by adding one new link we further indicate that the measures are able to identify the infrequently connected parts of the network and suggest the most appropriate improvement we also discuss the consequences of the obtained results and the possible applications of the measures

structure based graph distance measures of high degree of precision in recent years evaluating graph distance has become more and more important in a variety of real applications and many graph distance measures have been proposed among all of those measures structure based graph distance measures have become the research focus due to their independence of the definition of cost functions however existing structure based graph distance measures have low degree of precision because only node and edge information of graphs are employed in these measures to improve the precision of graph distance measures we define substructure abundance vector sav to capture more substructure information of a graph furthermore based on sav we propose unified graph distance measures which are generalization of the existing structure based graph distance measures in general the unified graph distance measures can evaluate graph distance in much finer grain we also show that unified graph distance measures based on occurrence mapping and some of their variants are metrics finally we apply the unified graph distance metric and its variants to the population evolution analysis and construct distance graphs of marker networks in three populations which reflect the single nucleotide polymorphism snp linkage disequilibrium ld differences among these populations

two state on off intermittency caused by unstable dimension variability in periodically forced drift waves certain high dimensional dynamical systems present two or more attractors characterized by different energy branches for some parameter values the dynamics oscillates between these two branches in a seemingly random fashion a phenomenon called two state on off intermittency in this work we show that the dynamical mechanism underlying this intermittency involves the severe breakdown of hyperbolicity of the attractors through a mechanism known as unstable dimension variability we characterize the parametric evolution of this variability using statistical properties of the finite time lyapunov exponents as a model system that exhibits this behavior we consider periodically forced and damped drift waves in this spatiotemporal example there is a low dimensional chaotic attractor that is created by an interior crisis already presenting unstable dimension variability

eigenvalues and invariants of tensors abstract a tensor is represented by a supermatrix under a co ordinate system in this paper we define e eigenvalues and e eigenvectors for tensors and supermatrices by the resultant theory we define the e characteristic polynomial of a tensor an e eigenvalue of a tensor is a root of the e characteristic polynomial in the regular case a complex number is an e eigenvalue if and only if it is a root of the e characteristic polynomial we convert the e characteristic polynomial of a tensor to a monic polynomial and show that the coefficients of that monic polynomial are invariants of that tensor i e they are invariant under co ordinate system changes we call them principal invariants of that tensor the maximum number of principal invariants of mth order n dimensional tensors is a function of m and n we denote it by d m n and show that d 1 n 1 d 2 n n d m 2 m for m 3 and d m n m n 1 m for m n 3 we also define the rank of a tensor all real eigenvectors associated with nonzero e eigenvalues are in a subspace with dimension equal to its rank

two parameter bifurcation study of the regularized long wave equation we perform a two parameter bifurcation study of the driven damped regularized long wave equation by varying the amplitude and phase of the driver increasing the amplitude of the driver brings the system to the regime of spatiotemporal chaos stc a chaotic state with a large number of degrees of freedom several global bifurcations are found including codimension two bifurcations and homoclinic bifurcations involving three tori and the manifolds of steady waves leading to the formation of chaotic saddles in the phase space we identify four distinct routes to stc they depend on the phase of the driver and involve boundary and interior crises intermittency the ruelle takens scenario the feigenbaum cascade an embedded saddle node homoclinic and other bifurcations this study elucidates some of the recently reported dynamical phenomena

some properties of strong h tensors and general h tensors abstract h matrices matrices whose comparison matrix is an m matrix are well studied in matrix theory and have numerous applications e g linear complementarity problems and iterative methods for solving systems of linear equations in this article we establish some properties of their tensor counterparts strong h tensors and general h tensors

scale free trees the skeletons of complex networks we investigate the properties of the spanning trees of various real world and model networks the spanning tree representing the communication kernel of the original network is determined by maximizing the total weight of the edges whose weights are given by the edge betweenness centralities we find that a scale free tree and shortcuts organize a complex network especially in ubiquitous scale free networks it is found that the scale free spanning tree shows very robust betweenness centrality distributions and the remaining shortcuts characterize the properties of the original network such as the clustering coefficient and the classification of scale free networks by the betweenness centrality distribution

criterions for the positive definiteness of real supersymmetric tensors the positive definiteness of an even degree homogeneous polynomial form f x plays an important role in the stability study of nonlinear autonomous systems via lyapunov s direct method in automatic control and the positive definiteness of f x is equivalent to that of an even order supersymmetric tensor which defines f x in this paper we provide some criterions for identifying the positive definiteness of an even order real supersymmetric tensor moreover an iterative algorithm for identifying the positive definiteness of an even order real supersymmetric tensor is obtained numerical examples are given to verify the corresponding results

a sufficient descent nonlinear conjugate gradient method for solving m tensor equations abstract tensor equations is a kind of important tensor optimization problems with higher order nonlinear equations which are widely used in engineering and economics this paper is concerned with solving m tensor equations we transform m tensor equations to nonlinear unconstrained optimization problems then a sufficient descent nonlinear conjugate gradient method with inexact line search is proposed for solving the transformed unconstrained optimization problem under only one mild assumption the global convergence of the proposed method is proved finally to show the effectiveness of the proposed nonlinear conjugate gradient method we compare it with three term conjugate gradient method and newton method the numerical results show that the proposed nonlinear conjugate gradient method is potentially efficient

a levenberg marquardt method for solving semi symmetric tensor equations abstract in this paper we propose a levenberg marquardt lm method for solving tensor equations with semi symmetric coefficient tensor and prove its global convergence and local quadratic convergence under the local error bound condition which is weaker than non singularity as application we solve h eigenvalue of real semi symmetric tensor by the lm method at last some numerical examples are provided to illustrate the efficiency and validity of these methods proposed

correlation of automorphism group size and topological properties with program size complexity evaluations of graphs and complex networks the systems biology dtc university of oxford uk dated february 25 2014 we show that numerical approximations of kolmogorov complexity k of graphs and networkscapture some group theoretic and topological properties of empirical networks ranging frommetabolic to social networks and of small synthetic networks that we have produced that k andthe size of the group of automorphisms of a graph are correlated opens up interesting connectionsto problems in computational geometry and thus connects several measures and concepts fromcomplexity science we derive these results via two di erent kolmogorov complexity approximationmethods applied to the adjacency matrices of the graphs and networks the methods used are thetraditional lossless compression approach to kolmogorov complexity and a normalised version of ablock decomposition method bdm based on algorithmic probability theory keywords kolmogorov complexity graph automorphism complex networks graph automor phisms algorithmic probability block decomposition method compressibility biological networks network biology i introduction

growing scale free small world networks with tunable assortative coefficient abstract in this paper we propose a simple rule that generates scale free small world networks with tunable assortative coefficient these networks are constructed by two stage adding process for each new node the model can reproduce scale free degree distributions and small world effect the simulation results are consistent with the theoretical predictions approximately interestingly we obtain the nontrivial clustering coefficient c and tunable degree assortativity r by adjusting the parameter the preferential exponent the model can unify the characterization of both assortative and disassortative networks

alternating projection method for a class of tensor equations abstract this paper considers how to solve a class of tensor equations arising from the unified definition of tensor vector products of special interest is the order 3 tensor equation whose solutions are the intersection of a group of quadrics from a geometric point of view inspired by the method of alternating projections for set intersection problems we develop a hybrid alternating projection algorithm for solving order 3 tensor equations the local linear convergence of the alternating projection method is established under suitable conditions some numerical experiments are conducted to evaluate the effect of the proposed algorithm

analysis of percolation behaviors of clustered networks with partial support dependence relations we carry out a study of percolation behaviors of clustered networks with partial support dependence relations by adopting two different attacking strategies attacking only one network and both networks which help to further understand real coupled networks for two different attacking strategies we find that the system changes from a second order phase transition to a first order phase transition as coupling strengthq increases we also notice that the first order region becomes smaller and the secondorder region becomes larger as average degree or clustering coefficient increases and as the average supported degree approaches infinity coupled clustered networks become independent and only the second order transition is observed which is similar toq 0 furthermore we find that clustering coefficient has a significant impact on robustness of the system for strong coupling strength but for weak coupling strength it has little influence especially for attacking both networks the study implies that we can obtain a more robust network by reducing clustering coefficient and increasing average degree for strong coupling strength however for weak coupling strength a more robust network is obtained only by increasing average degree for the same support average degree

an h tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms abstract identifying the positive definiteness of an even order homogeneous multivariate form is an important task due to its wide applications in such as medical imaging and the stability analysis of nonlinear autonomous systems via lyapunov s direct method in automatic control and multivariate network realizability analysis in this paper based on the equivalence of the positive definiteness of the form to that of the underlying tensor and the links between the positive definiteness of a tensor with strong h tensor we propose an h tensor based iterative scheme for identifying the positive definiteness of multivariate homogeneous forms the validity of the iterative scheme is guaranteed theoretically and the given numerical experiments show the efficiency of the scheme

vulnerability of complex networks under intentional attack with incomplete information we study the vulnerability of complex networks under intentional attack with incomplete information which means that one can only preferentially attack the most important nodes among a local region of a network the known random failure and the intentional attack are two extreme cases of our study using the generating function method we derive the exact value of the critical removal fraction fc of nodes for the disintegration of networks and the size of the giant component to validate our model and method we perform simulations of intentional attack with incomplete information in scale free networks we show that the attack information has an important effect on the vulnerability of scale free networks we also demonstrate that hiding a fraction of the nodes information is a cost efficient strategy for enhancing the robustness of complex networks