the convergence of spline collocation for strongly elliptic equations on curves most boundary element methods for two dimensional boundary value problems are based on point collocation on the boundary and the use of splines as trial functions here we present a unified asymptotic error analysis for even as well as for odd degree splines subordinate to uniform or smoothly graded meshes and prove asymptotic convergence of optimal order the equations are collocated at the breakpoints for odd degree and the internodal midpoints for even degree splines the crucial assumption for the generalized boundary integral and integro differential operators is strong ellipticity our analysis is based on simple fourier expansions in particular we extend results by j saranen and w l wendland from constant to variable coefficient equations our results include the first convergence proof of midpoint collocation with piecewise constant functions i e the panel method for solving systems of cauchy singular integral equations

synchronization in scale free networks with degree correlation in this paper we study a model of synchronization process on scale free networks with degree degree correlations this model was already studied on this kind of networks without correlations by pastore y piontti et al a l pastore y piontti p a macri l a braunstein phys rev e 76 2007 046117 here we study the effects of the degree degree correlation on the behavior of the load fluctuationsw s in the steady state we found that for assortative networks there exist a specific correlation where the system is optimally synchronized in addition we found that close to this optimal value the fluctuations does not depend on the system size and therefore the system becomes fully scalable this result could be very important for some technological applications on the other hand far from the optimal correlation w s scales logarithmically with the system size

nonlinear scattering theory at low energy abstract we study the scattering theory of a conservative nonlinear one parameter group of operators on a hilbert space x relative to a group of linear unitary operators under certain hypotheses the scattering operator carries a neighborhood of 0 in x into x the theory is designed to apply to the semilinear schrodinger and klein gordon equations

network clustering coefficient without degree correlation biases the clustering coefficient quantifies how well connected are the neighbors of a vertex in a graph in real networks it decreases with the vertex degree which has been taken as a signature of the network hierarchical structure here we show that this signature of hierarchical structure is a consequence of degree correlation biases in the clustering coefficient definition we introduce a definition in which the degree correlation biases are filtered out and provide evidence that in real networks the clustering coefficient is constant or decays logarithmically with vertex degree

synchronization in weighted scale free networks with degree degree correlation abstract we study synchronization phenomena in scale free networks of asymmetrically coupled dynamical systems featuring degree degree correlation in the topology of the connection wiring we show that when the interaction is dominant from the high degree to the low degree or from the low degree to the high degree nodes synchronization is enhanced for assortative or disassortative degree degree correlations

boundary element collocation methods using splines with multiple knots we extend the theory of boundary element collocation methods by allowing reduced inter element smoothness or in other words by allowing trial functions that are splines with multiple knots our convergence analysis is based on a recurrence relation for the fourier coefficients of the numerical solution and so is restricted to uniform grids on smooth closed curves superconvergence is possible with special choices of the collocation points numerical experiments with a model problem confirm the convergence rates predicted by our theory

local clustering in scale free networks with hidden variables we investigate the presence of triangles in a class of correlated random graphs in which hidden variables determine the pairwise connections between vertices the class rules out self loops and multiple edges we focus on the regime where the hidden variables follow a power law with exponent 2 3 so that the degrees have infinite variance the natural cutoff h c characterizes the largest degrees in the hidden variable models and a structural cutoff h s introduces negative degree correlations disassortative mixing due to the infinite variance degrees we show that local clustering decreases with the hidden variable or degree we also determine how the average clustering coefficient c scales with the network size n as a function of h s and h c for scale free networks with exponent 2 3 and the default choices h s n 1 2 and h c n 1 1 this gives c n 2 lnn for the universality class at hand we characterize the extremely slow decay of c when 2 and show that for 2 1 say clustering starts to vanish only for networks as large as n 10 9

numerical solution of linear volterra integral equations of the second kind with sharp gradients collocation methods are a well developed approach for the numerical solution of smooth and weakly singular volterra integral equations in this paper we extend these methods through the use of partitioned quadrature based on the qualocation framework to allow the efficient numerical solution of linear scalar volterra integral equations of the second kind with smooth kernels containing sharp gradients in this case the standard collocation methods may lose computational efficiency despite the smoothness of the kernel we illustrate how the qualocation framework can allow one to focus computational effort where necessary through improved quadrature approximations while keeping the solution approximation fixed the computational performance improvement introduced by our new method is examined through several test examples the final example we consider is the original problem that motivated this work the problem of calculating the probability density associated with a continuous time random walk in three dimensions that may be killed at a fixed lattice site to demonstrate how separating the solution approximation from quadrature approximation may improve computational performance we also compare our new method to several existing gregory sinc and global spectral methods where quadrature approximation and solution approximation are coupled

a posteriori error control in adaptive qualocation boundary element analysis for a logarithmic kernel integral equation of the first kind sobolev slobodeckij norms on small overlapping domains of two neighboring elements serve as a posteriori error estimators and mesh refining indicators in adaptive boundary element methods this paper is concerned with two error estimators eta f and mu f the first variant eta f is efficient and the second mu f is reliable that is up to multiplicative constants and numerical quadrature errors they are lower or upper error bounds faermann recently established reliability and efficiency of eta f for the galerkin boundary element method and considered mu f this work approaches the two estimators eta f le mu f for the galerkin qualocation and collocation boundary element methods for a single layer operator integral equation of the first kind upper and lower bounds are established theoretically and validated numerically numerical experiments support the estimators accuracies and the efficiencies of proposed adaptive mesh refining algorithms even in energy norms for qualocation and collocation schemes difficulties for alpha le 1 2 are caused by the lack of a sobolev embedding h alpha gamma hookrightarrow mathcal c gamma hence for the latter schemes equivalence of error and estimators in h alpha 1 gamma can be proven only for alpha 1 2 numerical evidence conjectures equivalence for alpha 1 2 as well

some discrete methods for boundary integral equations on smooth closed curves in this paper discrete trigonometric methods as well as quadrature methods for solution of boundary integral equations on smooth closed curves are presented and analyzed the analysis applies the general approach based on the consistency stability and convergence properties the trigonometric methods including e g variants based on product integration turn out to have high convergence even exponential convergence the family of quadrature methods covers operators with the even symbol and of order beta 0 they can be considered as fully discretized petrov galerkin methods that employ approximations both for the integral operator and the inner product the operator is approximated by using the trapezoidal rule either directly or after subtraction of the kernel singularity for instance for symm s equation beta 1 a method with the rate o h 5 of convergence is presented furthermore the quadrature solution of the cauchy singular integral equation beta 0 and the hypersingular

nonlinear small data scattering for the wave and klein gordon equation on etudie l operateur diffusion qui appartient a la paire d equations u tt au f u 0 et u tt au 0 a represente l operateur j 1 n 2 x j 2 m 2 m r et f u u 1 u ou r 1

percolation and epidemics in random clustered networks the social networks that infectious diseases spread along are typically clustered because of the close relation between percolation and epidemic spread the behavior of percolation in such networks gives insight into infectious disease dynamics a number of authors have studied percolation or epidemics in clustered networks but the networks often contain preferential contacts in high degree nodes we introduce a class of random clustered networks and a class of random unclustered networks with the same preferential mixing percolation in the clustered networks reduces the component sizes and increases the epidemic threshold compared to the unclustered networks

epidemic spreading in correlated complex networks we study a dynamical model of epidemic spreading on complex networks in which there are explicit correlations among the node s connectivities for the case of markovian complex networks showing only correlations between pairs of nodes we find an epidemic threshold inversely proportional to the largest eigenvalue of the connectivity matrix that gives the average number of links which from a node with connectivity k go to nodes with connectivity k ensuremath numerical simulations on a correlated growing network model provide support for our conclusions

mesh grading for integral equations of the first kind with logarithmic kernel when the galerkin method is used to solve the first kind integral equations with logarithmic kernel on polygons or open arcs the singularities of the solution degrade the rates of convergence the aim of this paper is to show for the particular case of piecewise constant approximation spaces that in some cases the o h 3 order of convergence can be restored by a suitable grading of the mesh

an a posteriori error estimate for a first kind integral equation in this paper we present a new a posteriori error estimate for the boundary element method applied to an integral equation of the first kind the estimate is local and sharp for quasi uniform meshes and so improves earlier work of ours the mesh dependence of the constants is analyzed and shown to be weaker than expected from our previous work besides the galerkin boundary element method the collocation method and the qualocation method are considered a numerical example is given involving an adaptive feedback algorithm

lp lp estimates for the schrodinger equation on the line and inverse scattering for the nonlinear schrodinger equation with a potential abstract in this paper i prove a l p l p estimate for the solutions to the one dimensional schrodinger equation with a potential in l 1 where in the generic case 3 2 and in the exceptional case i e when there is a half bound state of zero energy 5 2 i use this estimate to construct the scattering operator for the nonlinear schrodinger equation with a potential i prove moreover that the low energy limit of the scattering operator uniquely determines the potential and the coupling constant of the nonlinearity using a method that allows as well for the reconstruction of the potential and of the nonlinearity

on the numerical solution of a logarithmic integral equation of the first kind for the helmholtz equation we describe a quadrature method for the numerical solution of the logarithmic integral equation of the first kind arising from the single layer approach to the dirichlet problem for the two dimensional helmholtz equation in smooth domains we develop an error analysis in a sobolev space setting and prove fast convergence rates for smooth boundary data

time dependent method for non linear schrodinger equations in inverse scattering problems abstract we consider an inverse scattering problem for a non linear schrodinger equation with a non linearity of power type applying the time dependent technique developed by enss and weder we prove that the non linearity is uniquely determined from a scattering data defined via the scattering operator we also obtain a reconstruction formula which is valid for large scattering data

a nystrom flavored calderon calculus of order three for two dimensional waves time harmonic and transient in this paper we present and test a full discretization of all elements of the calderon calculus layer potentials and integral operators for the helmholtz equation in smooth closed curves in the plane the resulting integral equations provide approximations of order three for all variables involved tests are shown for a wide array of direct indirect and combined field integral equation at fixed frequency and for a convolution quadrature based approximation in the time domain

assortative model for social networks in this brief report we present a version of a network growth model generalized in order to describe the behavior of social networks the case of study considered is the preprint archive at cul arxiv org each node corresponds to a scientist and a link is present whenever two authors wrote a paper together this graph is a nice example of degree assortative network that is to say a network where sites with similar degree are connected to each other the model presented is one of the few able to reproduce such behavior giving some insight on the microscopic dynamics at the basis of the graph structure

corrected collocation methods for periodic pseudodifferential equations we propose an approximation method for periodic pseudodifferential equations which yields higher convergence rates in sobolev spaces with negative order than the collocation method the main idea consists in correcting the usual collocation solution in a certain way by the solution of a small galerkin system for the same equation both trigonometric and spline approximation methods are considered in most of the cases our convergence result even improves that of the qualocation method

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

properties of certain trigonometric series arising in numerical analysis h o f kdlsinn6 n l where 0 is real tx 0 and i 1 series of this kind have obvious connec tions with the riemann zeta function and with bernoulli polynomials and are important for fractional integration 6 xii sect 83 they have also arisen recently 2 4 in the study of certain numerical methods for bound ary integral equations on smooth curves in the plane for the applications in 2 we are interested firstly in knowing that g has a unique zero in the interval 0 rc so that a quadrature rule employing this zero is well defined and for later applications we need to know that this zero increases monotonically with c in the case where o 3 1 these questions were addressed in l as were the corresponding trickier questions about the partial sums for o b 1 we write g e p gol e b g e 4 me d fvbut fi f h e mi and for 0 p d f define k 4 b f p g 0 pi of crucial importance to the applications in 2 are the facts proved in theorem 2 below that h is nonnegative and h non positive and that k are both positive except in special cases in essence these properties ensure that the numerical method treated in 2 is stable the rate of convergence of the numerical methods of 2 is determined by the behaviour of g f i for small values of p the salient properties follow from 2 below in section 5 of the paper we indicate some extensions of our results to other classical orthogonal polynomials 2

on the convergence of collocation methods for symm s integral equation on open curves on etablit la convergence des methodes de collocation pour l equation integrale de symm sur des courbes ouvertes on confirme experimentalement les resultats theoriques sur les vitesses de convergence asymptotiques

the convergence of even degree spline collocation solution for potential problems in smooth domains of the plane in this article we derive new error estimates for collocation solution of potential type problems by using even degree smooth splines as trial functions it turns out that for smooth potentials the assured convergence is of the same order as by using splines of the odd degreed 1 some numerical examples which conform the theoretical results are presented

effect of node degree correlation on synchronization of identical pulse coupled oscillators we explore the effect of correlations between the in and out degrees of random directed networks on the synchronization of identical pulse coupled oscillators numerical experiments demonstrate that the proportion of initial conditions resulting in a globally synchronous state prior to a large but finite time is an increasing function of node degree correlation for those networks observed to globally synchronize both the mean and standard deviation of time to synchronization are decreasing functions of node degree correlation pulse coupled oscillator networks with negatively correlated node degree often exhibit multiple coherent attracting states with trajectories performing fast transitions between them these effects of node degree correlation on dynamics of pulse coupled oscillators are consistent with aspects of network topology e g the effect of node degree correlation on the eigenvalues of the laplacian matrix that have been shown to affect synchronization in other contexts

percolation on correlated networks we reconsider the problem of percolation on an equilibrium random network with degree degree correlations between nearest neighboring vertices focusing on critical singularities at a percolation threshold we obtain criteria for degree degree correlations to be irrelevant for critical singularities we present examples of networks in which assortative and disassortative mixing leads to unusual percolation properties and new critical exponents

the w k p continuity of the schrodinger wave operators on the line we prove that the wave operators for the schrodinger equation on the line are continuous on the sobolev spaces w k p 1 p moreover if the potential is exceptional and where f 1 x 0 is a jost solution at zero energy the wave operators are continuous on w k 1 and on w k

origin of degree correlations in the internet and other networks it has been argued that the observed anticorrelation between the degrees of adjacent vertices in the network representation of the internet has its origin in the restriction that no two vertices have more than one edge connecting them here we propose a formalism for modeling ensembles of graphs with single edges only and derive values for the exponents and correlation coefficients characterizing them our results confirm that the conjectured mechanism does indeed give rise to correlations of the kind seen in the internet although only a part of the measured correlation can be accounted for in this way

reconstruction of a potential on the line that is a priori known on the half line in this paper the authors study the inverse schrodinger scattering on the real line a method is given that allows unique reconstruction of a potential that is a priori known on the half line from the knowledge of the reflection coefficient and the bound state energies in particular no information on the forming constants is required the method is based on an appropriate trace formula and on the solution of the nonlinear ordinary differential equation that is obtained when the potential is replaced by its trace formula in the schrodinger equation the deift trubowitz approach to inverse scattering is followed the main new point is the way in which bound states are treated in addition to its mathematical interest the case when the potential is a priori known on the half line is particularly interesting in many applications one can consider for example a potential that has compact support or that it is zero on a half line

on space time means and everywhere defined scattering operators for nonlinear klein gordon equations on demontre l existence d operateurs de diffusion definis partout pour certaines equations de klein gordon non lineaires

numerical evaluation of some trigonometric series we present a method for the accurate numerical evaluation of a family of trigonometric series arising in the design of special purpose quadrature rules for boundary element methods the series converge rather slowly but can be expressed in terms of fourier chebyshev series that converge rapidly let r be a positive integer and define the functions gr and hr by gr t 2 e r cos 2rmt m 1

product integration collocation methods for noncompact integral operator equations we discuss the numerical solution of a class of second kind integral equations in which the integral operator is not compact such equations arise for example when boundary integral methods are applied to potential problems in a two dimensional domain with corners in the boundary we are able to prove the optimal orders of convergence for the usual collocation and product integration methods on graded meshes provided some simple modifications are made to the underlying basis functions these are sufficient to ensure stability but do not damage the rate of convergence numerical experiments show that such modifications are necessary in certain circumstances

uncovering disassortativity in large scale free networks mixing patterns in large self organizing networks such as the internet the world wide web and social and biological networks are often characterized by degree degree dependencies between neighboring nodes in this paper we propose a new way of measuring degree degree dependencies one of the problems with the commonly used assortativity coefficient is that in disassortative networks its magnitude decreases with the network size we mathematically explain this phenomenon and validate the results on synthetic graphs and real world network data as an alternative we suggest to use rank correlation measures such as spearman s rho our experiments convincingly show that spearman s rho produces consistent values in graphs of different sizes but similar structure and it is able to reveal strong positive or negative dependencies in large graphs in particular we discover much stronger negative degree degree dependencies in web graphs than was previously thought rank correlations allow us to compare the assortativity of networks of different sizes which is impossible with the assortativity coefficient due to its genuine dependence on the network size we conclude that rank correlations provide a suitable and informative method for uncovering network mixing patterns

asymptotic properties of degree correlated scale free networks the possible correlation profiles of networks with a given scale free degree distribution are restricted and bounded by maximally correlated configurations dissortative networks consist of nested bilayers in which low degree vertices are connected to high degree vertices the number of these bilayers attains a constant value for large network size n assortative networks exhibit monolayers of low degree vertices the number of which grows monotonously with n analytical relations for the pearson correlation coefficient r of these extremal configurations are derived and shown to provide lower and upper bounds on the possible r values both bounds are found to vanish for large networks

spline qualocation methods for variable coefficient elliptic equations on curves here the stability and convergence results of oqualocation methods providing additional orders of convergence are extended from the special class of pseudodifferential equations with constant coefficient symbols to general classical pseudodifferential equations of strongly and of oddly elliptic type the analysis exploits localization in the form of frozen coefficients pseudohomogeneous asymptotic symbol representation of classical pseudodifferential operators a decisive commutator property of the qualocation projection and requires qualocation rules which provide sufficiently many additional degrees of precision for the numerical integration of smooth remainders numerical examples show the predicted high orders of convergence

identification of intrinsic long range degree correlations in complex networks many real world networks exhibit degree degree correlations between nodes separated by more than one step such long range degree correlations lrdcs can be fully described by one joint and four conditional probability distributions with respect to degrees of two randomly chosen nodes and shortest path distance between them while lrdcs are induced by nearest neighbor degree correlations nndcs between adjacent nodes some networks possess intrinsic lrdcs which cannot be generated by nndcs here we develop a method to extract intrinsic lrdc in a correlated network by comparing the probability distributions for the given network with those for nearest neighbor correlated random networks we also demonstrate the utility of our method by applying it to several real world networks

correlation analysis of short text based on network model abstract correlation of words in the text is of great importance in text analysis like text retrieval keywords extraction and text clustering for short text because of the limited information of text content it is difficult to catch the correlation well among words in this paper we propose an algorithm based on the complex network to calculate the correlation of words in short texts a new variable edge degree is proposed and used in studying the network model of texts by using fluctuation analysis we give the condition that edge degree correlation between words exists beyond nearest neighbors further analysis shows that numerical results of the fluctuation function of edge degree act a power law distribution and that the scaling exponent diverges at a long distance under the finite size effect and varies in different texts the fluctuation function separates the words in a text into different clusters and this property is used to measure inner correlation of different words hub nodes act a significant influence on the long range edge degree correlation through changing the linear trend of the fluctuation function in a log log plot

measuring mixing patterns in complex networks by spearman rank correlation coefficient abstract in this paper we utilize spearman rank correlation coefficient to measure mixing patterns in complex networks compared with the widely used pearson coefficient spearman coefficient is rank based nonparametric and size independent thus it is more effective to assess linking patterns of diverse networks especially for large size networks we demonstrate this point by testing a variety of empirical and artificial networks moreover we show that normalized spearman ranks of stubs are subject to an interesting linear rule where the correlation coefficient is just the spearman coefficient this compelling linear relationship allows us to directly produce networks with any prescribed spearman coefficient our method apparently has an edge over the well known uncorrelated configuration model

self similar scale free networks and disassortativity self similar networks with scale free degree distribution have recently attracted much attention since these apparently incompatible properties were reconciled in c song s havlin and h a makse nature 433 392 2005 by an appropriate box counting method that enters the measurement of the fractal dimension we study two genetic regulatory networks saccharomyces cerevisiae n m luscombe m m babu h yu m snyder s teichmann and m gerstein nature 431 308 2004 and escherichia coli http www ccg unam mx computationalgenomics regulondb datasets regulonnetdatasets html and http www gbf de systemsbiology and show their self similar and scale free features in extension to the datasets studied by c song s havlin and h a makse nature 433 392 2005 moreover by a number of numerical results we support the conjecture that self similar scale free networks are not assortative from our simulations so far these networks seem to be disassortative instead we also find that the qualitative feature of disassortativity is scale invariant under renormalization but it appears as an intrinsic feature of the renormalization prescription as even assortative networks become disassortative after a sufficient number of renormalization steps

spline qualocation methods for boundary integral equations this paper further develops the qualocation method for the solution of integral equations on smooth closed curves qualocation is a petrov galerkin method in which the outer integrals are performed numerically by special quadrature rules here we allow the use of splines as trial and test functions and prove stability and convergence results for certain qualocation rules when the underlying petrov galerkin method is stable in the common case of a first kind equation with the logarithmic kernel a typical qualocation method employs piecewise linear splines and performs the outer integrals using just two points per interval this method with appropriate choice of the quadrature points and weights has ordero h 5 convergence in a suitable negative norm numerical experiments suggest that these rates of convergence are maintained for integral equations on open intervals when graded meshes are used to cope with unbounded solutions

on the asymptotic convergence of collocation methods with spline functions of even degree we investigate the collocation of linear one dimensional strongly elliptic integro dif ferential or more generally pseudo differential equations on closed curves by even degree polynomial splines the equations are collocated at the respective midpoints subject to uniform nodal grids of the even degree s splines we prove quasioptimal and optimal order asymptotic error estimates in a scale of sobolev spaces the results apply in particular to boundary element methods used for numerical computations in engineering applications the equations considered include fredholm integral equations of the second and the first kind singular integral equations involving cauchy kernels and integro differential equations having convolutional or constant coefficient principal parts respectively the error analysis is based on an equivalence between the collocation and certain varia tional methods with different degree splines as trial and as test functions we further need to restrict our operators essentially to pseudo differential operators having convolutional prin cipal part this allows an explicit fourier analysis of our operators as well as of the spline spaces in terms of trigonometric polynomials providing babuska s stability condition based on strong ellipticity our asymptotic error estimates extend partly those obtained by d n arnold and w l wendland from the case of odd degree splines to the case of even degree splines 1 introduction in this paper we investigate the asymptotic convergence of the collocation method using even degree polynomial splines applied to strongly elliptic systems of pseudo differential equations on closed curves with convolutional prin cipal part the collocation here employs the gauss points of one point integration as collocation points which corresponds to the usual boundary element collocation in applications 5 12 17 19 26 31 34 this is in contrast to the method investigated by g schmidt in 29 where one collocates at the break points of even order splines the asymptotic convergence properties for the standard galerkin method with splines of arbitrary orders are well known 22 for the collocation method however asymptotic convergence up to now has been shown for strongly elliptic systems only in the case of odd order splines by d n arnold and w l wendland 7

robustness of onionlike correlated networks against targeted attacks recently it was found by schneider et al proc natl acad sci usa 108 3838 2011 using simulations that scale free networks with onion structure are very robust against targeted high degree attacks the onion structure is a network where nodes with almost the same degree are connected motivated by this work we propose and analyze based on analytical considerations an onionlike candidate for a nearly optimal structure against simultaneous random and targeted high degree node attacks the nearly optimal structure can be viewed as a set of hierarchically interconnected random regular graphs the degrees and populations of whose nodes are specified by the degree distribution this network structure exhibits an extremely assortative degree degree correlation and has a close relationship to the onion structure after deriving a set of exact expressions that enable us to calculate the critical percolation threshold and the giant component of a correlated network for an arbitrary type of node removal we apply the theory to the cases of random scale free networks that are highly vulnerable against targeted high degree node removal our results show that this vulnerability can be significantly reduced by implementing this onionlike type of degree degree correlation without much undermining the almost complete robustness against random node removal we also investigate in detail the robustness enhancement due to assortative degree degree correlation by introducing a joint degree degree probability matrix that interpolates between an uncorrelated network structure and the onionlike structure proposed here by tuning a single control parameter the optimal values of the control parameter that maximize the robustness against simultaneous random and targeted attacks are also determined our analytical calculations are supported by numerical simulations

nature of the epidemic threshold for the susceptible infected susceptible dynamics in networks we develop an analytical approach to the susceptible infected susceptible epidemic model that allows us to unravel the true origin of the absence of an epidemic threshold in heterogeneous networks we find that a delicate balance between the number of high degree nodes in the network and the topological distance between them dictates the existence or absence of such a threshold in particular small world random networks with a degree distribution decaying slower than an exponential have a vanishing epidemic threshold in the thermodynamic limit

on the stability of the nystrom method for the muskhelishvili equation on contours with corners the stability of the nystrom method for the muskhelishvili equation on piecewise smooth simple contours is studied it is shown that in the space l 2 the method is stable if and only if certain operators a j from an algebra of toeplitz operators are invertible the operators a j depend on the parameters of the equation considered on the opening angles j of the corner points j and on parameters of the approximation method mentioned numerical experiments show that there are opening angles where the operators a j are noninvertible therefore for contours with such corners the method under consideration is not stable otherwise the method is always stable numerical examples show an excellent convergence of the method

general formulation of long range degree correlations in complex networks we provide a general framework for analyzing degree correlations between nodes separated by more than one step i e beyond nearest neighbors in complex networks one joint and four conditional probability distributions are introduced to fully describe long range degree correlations with respect to degrees k and k of two nodes and shortest path length l between them we present general relations among these probability distributions and clarify the relevance to nearest neighbor degree correlations unlike nearest neighbor correlations some of these probability distributions are meaningful only in finite size networks furthermore as a baseline to determine the existence of intrinsic long range degree correlations in a network other than inevitable correlations caused by the finite size effect the functional forms of these probability distributions for random networks are analytically evaluated within a mean field approximation the utility of our argument is demonstrated by applying it to real world networks

percolation transition in networks with degree degree correlation we introduce an exponential random graph model for networks with a fixed degree distribution and a tunable degree degree correlation we then investigate the nature of the percolation transition in a correlated network with a poisson degree distribution it is found that negative correlation is irrelevant in that the percolation transition in the disassortative network belongs to the same universality class as in the uncorrelated network positive correlation turns out to be relevant the percolation transition in the assortative network is characterized by the nondiverging mean size of finite clusters and power law scalings of the density of the largest cluster and the cluster size distribution in the nonpercolating phase as well as at the critical point our results suggest that the unusual type of percolation transition in the growing network models reported recently may be inherited from the assortative degree degree correlation

final state problem for the cubic nonlinear klein gordon equation we study the final state problem for the nonlinear klein gordon equation utt u uxx u3 t r x r where r we prove the existence of solutions in the neighborhood of the approximate solutions 2 re u t w t where u t is the free evolution group defined by u t f 1e it f x 1 x2 f and f 1 are the direct and inverse fourier transformations respectively and w t x f 1 u e 3 2 i 2 u 2log t with a given final data u is a real valued function and 3 i u l is small

on scattering and everywhere defined scattering operators for nonlinear klein gordon equations asymptotic properties of solutions of the nonlinear klein gordon equation t2u u m2u f u 0 nlkg u 0 tu 0 are investigated which are inherited from the corresponding solutions v of the linear klein gordon equation t2v v m2v 0v 0 tv 0 kg in particular the finiteness of time integrals in lq over r of certain sobolevnorms in space of the solution is proved to be such a hereditary property together with a device by w a strauss and a weak decay result for the kg due to r s strichartz this is used to prove that under suitable restrictions on the nonlinearity the scattering operator for the nlkg is defined on all of l21 l2 for n 3

interplay of degree correlations and cluster synchronization we study the evolution of coupled chaotic dynamics on networks and investigate the role of degree degree correlation in the networks cluster synchronizability we find that an increase in the disassortativity can lead to an increase or a decrease in the cluster synchronizability depending on the degree distribution and average connectivity of the network networks with heterogeneous degree distribution exhibit significant changes in cluster synchronizability as well as in the phenomena behind cluster synchronization as compared to those of homogeneous networks interestingly cluster synchronizability of a network may be very different from global synchronizability due to the presence of the driven phenomenon behind the cluster formation furthermore we show how degeneracy at the zero eigenvalues provides an understanding of the occurrence of the driven phenomenon behind the synchronization in disassortative networks the results demonstrate the importance of degree degree correlations in determining cluster synchronization behavior of complex networks and hence have potential applications in understanding and predicting dynamical behavior of complex systems ranging from brain to social systems

reshuffling scale free networks from random to assortative to analyze the role of assortativity in networks we introduce an algorithm which produces assortative mixing to a desired degree this degree is governed by one parameter p changing this parameter one can construct networks ranging from fully random p 0 to totally assortative p 1 we apply the algorithm to a barabasi albert scale free network and show that the degree of assortativity is an important parameter governing the geometrical and transport properties of networks thus the average path length of the network increases dramatically with the degree of assortativity moreover the concentration dependences of the size of the giant component in the node percolation problem for uncorrelated and assortative networks are strongly different the behavior of the clustering coefficient is also discussed

quadrature methods for strongly elliptic equations of negative order on smooth closed curves in this article the authors introduce and analyze a family of simple quadrature methods for strongly elliptic equations of negative order the results cover all the classical pseudodifferential equations of this kind and hence the boundary integral equations appearing in applications the quadrature methods are based on the trapezoidal approximation of the operator with respect to a uniform mesh and on varepsilon collocation they are proved to have the stability and the convergence with the maximal rate o h beta of convergence in general and o h beta 1 for some special values of varepsilon here beta is the order of the operator the method is applicable to systems of boundary integral equations as well in this connection the authors have conducted some numerical experiments which confirm their theoretical results

averaging techniques for the effective numerical solution of symm s integral equation of the first kind averaging techniques for finite element error control occasionally called zz estimators for the gradient recovery enjoy a high popularity in engineering because of their striking simplicity and universality one does not even require a pde to apply the nonexpensive post processing routines recently averaging techniques have been mathematically proved to be reliable and efficient for various applications of the finite element method this paper establishes a class of averaging error estimators for boundary integral methods symm s integral equation of the first kind with a nonlocal single layer integral operator serves as a model equation studied both theoretically and numerically we introduce four new error estimators which are proven to be reliable and efficient up to terms of higher order the higher order terms depend on the regularity of the exact solution several numerical experiments illustrate the theoretical results and show that the normally unknown error is sharply estimated by the proposed estimators i e error and estimators almost coincide

a spline trigonometric galerkin method and an exponentially convergent boundary integral method we consider a galerkin method for functional equations in one space variable which uses periodic cardinal splines as trial functions and trigonometric polynomials as test functions we analyze the method applied to the integral equation of the first kind arising from a single layer potential formulation of the dirichlet problem in the interior or exterior of an analytic plane curve in constrast to ordinary spline galerkin methods we show that the method is stable and so provides quasioptimal approximation in a large family of hilbert spaces including all the sobolev spaces of negative order as a consequence we prove that the approximate solution to the dirichlet problem and all its derivatives converge pointwise with exponential rate

emergence of fractal scaling in complex networks some real world networks are shown to be fractal or self similar it is widespread that such a phenomenon originates from the repulsion between hubs or disassortativity here we show that this common belief fails to capture the causality our key insight to address it is to pinpoint links critical to fractality those links with small edge betweenness centrality bc constitute a special architecture called fractal reference system which gives birth to the fractal structure of those reported networks in contrast a small amount of links with high bc enable small world effects hiding the intrinsic fractality with enough of such links removed fractal scaling spontaneously arises from nonfractal networks our results provide a multiple scale view on the structure and dynamics and place fractality as a generic organizing principle of complex networks on a firmer ground

epidemic extinction paths in complex networks we study the extinction of long lived epidemics on finite complex networks induced by intrinsic noise applying analytical techniques to the stochastic susceptible infected susceptible model we predict the distribution of large fluctuations the most probable or optimal path through a network that leads to a disease free state from an endemic state and the average extinction time in general configurations our predictions agree with monte carlo simulations on several networks including synthetic weighted and degree distributed networks with degree correlations and an empirical high school contact network in addition our approach quantifies characteristic scaling patterns for the optimal path and distribution of large fluctuations both near and away from the epidemic threshold in networks with heterogeneous eigenvector centrality and degree distributions

tuning clustering in random networks with arbitrary degree distributions we present a generator of random networks where both the degree dependent clustering coefficient and the degree distribution are tunable following the same philosophy as in the configuration model the degree distribution and the clustering coefficient for each class of nodes of degree k are fixed ad hoc and a priori the algorithm generates corresponding topologies by applying first a closure of triangles and second the classical closure of remaining free stubs the procedure unveils an universal relation among clustering and degree degree correlations for all networks where the level of assortativity establishes an upper limit to the level of clustering maximum assortativity ensures no restriction on the decay of the clustering coefficient whereas disassortativity sets a stronger constraint on its behavior correlation measures in real networks are seen to observe this structural bound

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

reproducing kernel method for the numerical solution of the brinkman forchheimer momentum equation we consider two efficient methods for the solution of the brinkman forchheimer momentum equation with boundary conditions on the square physically this model describes the flow of fully developed forced convection in a porous saturated rectangular duct after first demonstrating the existence and symmetry properties of a solution we apply the reproducing kernel method in order to solve the brinkman forchheimer momentum equation we then demonstrate the applicability of the method by considering several specific numerical examples which allow us to understand the variation of the physical solutions as one changes any of the several model parameters the numerical results demonstrate the utility of the reproducing kernel method for solving nonlinear elliptic partial differential equations on compact domains

inverse scattering on the line for the nonlinear klein gordon equation with a potential abstract in this paper we obtain a lp lp estimate for the klein gordon equation with a potential on the line this result follows from the previously known estimates in the case without potential and from the continuity on the sobolev spaces of the schrodinger wave operators that we proved in a previous paper using this estimate we construct the scattering operator for the nonlinear klein gordon equation with a potential moreover we prove that the small amplitude scattering operator uniquely determines both the potential and the nonlinearity our proof also gives a method for the reconstruction of the potential and of the nonlinearity

on the convergence of collocation methods for boundary integral equations on polygons the integral equations encountered in boundary element methods are frequently solved numerically using collocation with spline trial functions convergence proofs and error estimates for these approximation methods have been only available in the following cases fredholm integral equations of the second kind 4 7 one dimensional pseudodifferential equations and singular integral equations with piecewise smooth coefficients on smooth curves 2 3 17 26 29 and some special results on the classical neumann integral equation of potential theory for polygonal plane domains 5 8 9 here we give convergence proofs for collocation with piecewise linear trial functions for neumann s integral equation and symm s integral equation on plane curves with corners we derive asymptotic error estimates in sobolev norms and analyze the effect of graded meshes 0 introduction in this paper we give convergence proofs and asymptotic error estimates in sobolev norms for collocation with piecewise linear spline trial func tions applied to two basic integral equations of potential theory on plane polygons namely the integral equation of the second kind with the double layer potential neumann s integral equation and the integral equation of the first kind with the simple layer potential symm s integral equation we use an idea of arnold and wendland 2 namely considering dirac delta functions the test functions in the collocation method as second derivatives of piecewise linear functions therefore similar results as presented here should be possible for splines of higher odd order corresponding results for even order splines are not yet available thus for one of the simplest methods of numerically solving dirichlet s problem on a plane domain with corners the midpoint collocation with piecewise constant trial functions for the first kind integral equation with the simple layer potential convergence is still an open problem the method of fourier series that yields the convergence proof in the case of a smooth boundary 27 cannot be applied in the presence of corners we apply the method of local mellin transformation that has previously been used to derive error estimates for galerkin methods for a wide class of operators including those occurring in boundary element methods in acoustics electromag

correlations in weighted networks we develop a statistical theory to characterize correlations in weighted networks we define the appropriate metrics quantifying correlations and show that strictly uncorrelated weighted networks do not exist due to the presence of structural constraints we also introduce an algorithm for generating maximally random weighted networks with arbitrary p k s to be used as null models the application of our measures to real networks reveals the importance of weights in a correct understanding and modeling of these heterogeneous systems

low energy scattering for nonlinear klein gordon equations the scattering operator which belongs to a pair of pdes consisting of the klein gordon equation and a perturbation of it by a power like nonlinearity z hfl u is studied it is shown that this operator can be defined on a whole neighbourhood of the origin in energy space if z hfl u u p 1u or u p where 1 4 n 1 p 1 4 n 2 and the space dimension n 2 is arbitrary

higher order assortativity in complex networks abstract assortativity was first introduced by newman and has been extensively studied and applied to many real world networked systems since then assortativity is a graph metric and describes the tendency of high degree nodes to be directly connected to high degree nodes and low degree nodes to low degree nodes it can be interpreted as a first order measure of the connection between nodes i e the first autocorrelation of the degree degree vector even though assortativity has been used so extensively to the author s knowledge no attempt has been made to extend it theoretically indeed newman assortativity is about being adjacent but even though two nodes may not by connected through an edge they could have possibly a strong level of connectivity through a large number of walks and paths between them this is the scope of our paper we introduce for undirected and unweighted networks higher order assortativity by extending the newman index based on a suitable choice of the matrix driving the connections higher order assortativity be defined for paths shortest paths and random walks of a given length the newman assortativity is a particular case of each of these measures when the matrix is the adjacency matrix or in other words the autocorrelation is of order 1 our higher order assortativity indices help discriminating networks having the same newman index and may reveal new topological network features an application to airline network italy and us and to enron email network as well as examples and simulations are discussed

a quadrature based approach to improving the collocation method the collocation method is a popular method for the approximate solution of boundary integral equations but typically does not achieve the high order of convergence reached by the galerkin method in appropriate negative norms in this paper a quadrature based method for improving upon the collocation method is proposed and developed in detail for a particular case that case involves operators with even symbol such as the logarithmic potential operating on 1 periodic functions a smooth spline trial space of odd degree with constant mesh spacingh 1 n and a quadrature rule with 2n points where ann point quadrature rule would be equivalent to the collocation method in this setting it is shown that a special quadrature rule which depends on the degree of the splines and the order of the operator can yield a maximum order of convergence two powers ofh higher than the collocation method