new approach to study splines by blossoming method and application to the construction of a bivariate c 1 quartic quasi interpolant this work is a contribution in the approximation theory for studying and analyzing piecewise polynomial functions splines which uses the blossoming approach some existing results in the literature are reformulated such as the smoothness conditions between polynomials of a spline by using the affinity property of the blossom some definitions of sub splines are proposed which can be very useful in the study and the construction of splines such as macro elements or quasi interpolants as an application of the proposed results a c 1 quartic spline quasi interpolant with optimal approximation order is defined without using any mask for smoothness or b spline basis numerical results are presented and compared with other methods given in the literature

degree dispersion increases the rate of rare events in population networks there is great interest in predicting rare and extreme events in complex systems and in particular understanding the role of network topology in facilitating such events in this letter we show that degree dispersion the fact that the number of local connections in networks varies broadly increases the probability of large rare fluctuations in population networks generically we perform explicit calculations for two canonical and distinct classes of rare events network extinction and switching when the distance to threshold is held constant and hence stochastic effects are fairly compared among networks we show that there is a universal exponential increase in the rate of rare events proportional to the variance of a network s degree distribution over its mean squared

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

approximation orders and shape preserving properties of the multiquadric trigonometric b spline quasi interpolant the purpose of the paper is to derive some properties of the multiquadric trigonometric b spline quasi interpolant firstly the paper captures its approximation orders for high order derivatives based on the error estimate one can choose the shape parameter properly such that the quasi interpolant gives optimal approximations to high order derivatives moreover the approximation orders also show that from the theoretical point of view the considered quasi interpolant can be applied when the approximation of high order derivatives is needed i e numerical solution of some pdes construction of lyapunov function etc secondly the paper derives some shape preserving properties of the quasi interpolant these properties suggest that the quasi interpolant may be used in the geometric modeling cad cam for instance that requires shape preservation finally to illustrate the validity of the results some numerical examples are presented both theoretical and numerical results demonstrate that the quasi interpolant cannot only provide excellent approximations to high order derivatives but also preserve the shapes well

unified index to quantifying heterogeneity of complex networks abstract although recent studies have revealed that degree heterogeneity of a complex network has significant impact on the network performance and function a unified definition of the heterogeneity of a network with any degree distribution is absent in this paper we define a heterogeneity index 0 h 1 to quantify the degree heterogeneity of any given network we analytically show the existence of an upper bound of h 0 5 for exponential networks thus explain why exponential networks are homogeneous on the other hand we also analytically show that the heterogeneity index of an infinite power law network is between 1 and 0 5 if and only if its degree exponent is between 2 and 2 5 we further show that for any power law network with a degree exponent greater than 2 5 there always exists an exponential network such that both networks have the same heterogeneity index this may help to explain why 2 5 is a critical degree exponent for some dynamic behaviors on power law networks

langevin approach for the dynamics of the contact process on annealed scale free networks we study the dynamics of the contact process one of the simplest nonequilibrium stochastic processes taking place on a scale free network we consider the network topology as annealed i e all links are rewired at each microscopic time step so that no dynamical correlation can build up this is a practical implementation of the absence of correlations assumed by mean field approaches we present a detailed analysis of the contact process in terms of a langevin equation including explicitly the effects of stochastic fluctuations in the number of particles in finite networks this allows us to determine analytically the survival time for spreading experiments and the density of active sites in surviving runs the fluctuations in the topological structure induce anomalous scaling effects with respect to the system size when the degree distribution has a hard upper bound when the upper bound is soft the presence of outliers with huge connectivity perturbs the picture even more inducing an apparent shift of the critical point in light of these findings recent theoretical and numerical results in the literature are critically reviewed

self avoiding walks on scale free networks several kinds of walks on complex networks are currently used to analyze search and navigation in different systems many analytical and computational results are known for random walks on such networks self avoiding walks saw s are expected to be more suitable than unrestricted random walks to explore various kinds of real life networks here we study long range properties of random saw s on scale free networks characterized by a degree distribution p k ensuremath sim k ensuremath ensuremath gamma in the limit of large networks system size n ensuremath rightarrow ensuremath infty the average number s n of saw s starting from a generic site increases as ensuremath mu n with ensuremath mu k 2 k ensuremath 1 for finite n s n is reduced due to the presence of loops in the network which causes the emergence of attrition of the paths for kinetic growth walks the average maximum length l increases as a power of the system size l ensuremath sim n ensuremath alpha with an exponent ensuremath alpha increasing as the parameter ensuremath gamma is raised we discuss the dependence of ensuremath alpha on the minimum allowed degree in the network a similar power law dependence is found for the mean self intersection length of nonreversal random walks simulation results support our approximate analytical calculations

zero temperature glauber dynamics on small world networks the zero temperature glauber dynamics of the ferromagnetic ising model on small world networks rewired from a two dimensional square lattice has been studied by numerical simulations for increasing disorder in finite networks the nonequilibrium dynamics becomes faster so that the ground state is found more likely for any finite value of the rewiring probability p the likelihood of reaching the ground state goes to zero in the thermodynamic limit similarly to random networks the spin correlation r is found to decrease with distance as r exp r being a correlation length scaling with p as p 0 73 these results are compared with those obtained earlier for addition type small world networks

correlations in interacting systems with a network topology we study pair correlations in interacting systems placed on complex networks we show that usually in these systems pair correlations between interacting objects e g spins separated by a distance ensuremath ell decay on average faster than 1 ensuremath ell z ensuremath ell here z ensuremath ell is the mean number of the ensuremath ell th nearest neighbors of a vertex in a network this behavior in particular leads to a dramatic weakening of correlations between second and more distant neighbors on networks with fat tailed degree distributions which have a divergent number z 2 in the infinite network limit in large networks of this kind only pair correlations between the nearest neighbors are actually observable we find the pair correlation function of the ising model on a complex network this exact result is confirmed by a phenomenological approach

belief propagation algorithm and the ising model on networks with arbitrary distributions of motifs we generalize the belief propagation algorithm to sparse random networks with arbitrary distributions of motifs triangles loops etc each vertex in these networks belongs to a given set of motifs generalization of the configuration model these networks can be treated as sparse uncorrelated hypergraphs in which hyperedges represent motifs here a hypergraph is a generalization of a graph where a hyperedge can connect any number of vertices these uncorrelated hypergraphs are treelike hypertrees which crucially simplifies the problem and allows us to apply the belief propagation algorithm to these loopy networks with arbitrary motifs as natural examples we consider motifs in the form of finite loops and cliques we apply the belief propagation algorithm to the ferromagnetic ising model with pairwise interactions on the resulting random networks and obtain an exact solution of this model we find an exact critical temperature of the ferromagnetic phase transition and demonstrate that with increasing the clustering coefficient and the loop size the critical temperature increases compared to ordinary treelike complex networks however weak clustering does not change the critical behavior qualitatively our solution also gives the birth point of the giant connected component in these loopy networks

exact solution of ising model on a small world network also the complete thermodynamics at all temperatures as a function of p and of the strength of long range interactions i though in this paper we only present the results for the free energy and specific heat in zero field the form of finite size scaling corrections to the thermodynamics can also be derived from our solution the following sec ii presents a description of our specific model and the basic algorithm for its exact solution the combinatorial treatment of its constituents is given is sec iii in sec iv we analyze the resulting thermodynamics and conclude on the mean field character of the ordering transition section v shows how the finite size scaling properties can be obtained within the same approach finally the relation between our results and those of dorogovtsev et al is discussed in sec vi

biased percolation on scale free networks biased degree dependent percolation was recently shown to provide strategies for turning robust networks fragile and vice versa here we present more detailed results for biased edge percolation on scale free net works we assume a network in which the probability for an edge between nodes i and j to be retained is proportional to kikjwith ki and kj the degrees of the nodes we discuss two methods of network recon struction sequential and simultaneous and investigate their properties by analytical and numerical means the system is examined away from the percolation transition where the size of the giant cluster is obtained and close to the transition where nonuniversal critical exponents are extracted using the generating functions method the theory is found to agree quite well with simulations by presenting an extension of the fortuin kasteleyn construction we find that biased percolation is well described by the q 1 limit of the q state potts model with inhomogeneous couplings

finite size effects for the ising model on random graphs with varying dilution abstract we investigate the finite size corrections to the equilibrium magnetization of an ising model on a random graph with n nodes and n edges with 1 2 by conveniently rescaling the coupling constant the free energy is made extensive as expected the system displays a phase transition of the mean field type for all the considered values of at the transition temperature of the fully connected curie weiss model finite size corrections are investigated for different values of the parameter using two different approaches a replica based finite n expansion and a cavity method numerical simulations are compared with theoretical predictions the cavity based analysis is shown to agree better with numerics

ferromagnetic transition in a simple variant of the ising model on multiplex networks abstract multiplex networks consist of a fixed set of nodes connected by several sets of edges which are generated separately and correspond to different networks layers here a simple variant of the ising model on multiplex networks with two layers is considered with spins located in the nodes and edges corresponding to ferromagnetic interactions between them critical temperatures for the ferromagnetic transition are evaluated for the layers in the form of random erdos renyi graphs or heterogeneous scale free networks using the mean field approximation and the replica method from the replica symmetric solution both methods require the use of different partial magnetizations associated with different layers of the multiplex network and yield qualitatively similar results if the layers are strongly heterogeneous the critical temperature differs noticeably from that for the ising model on a network being a superposition of the two layers evaluated in the mean field approximation neglecting the effect of the underlying multiplex structure on the correlations between the degrees of nodes the critical temperature evaluated from the replica symmetric solution depends sensitively on the correlations between the degrees of nodes in different layers and shows satisfactory quantitative agreement with that obtained from monte carlo simulations the critical behavior of the magnetization for the model with strongly heterogeneous layers can depend on the distributions of the degrees of nodes and is then determined by the properties of the most heterogeneous layer

a general method for constructing quasi interpolants from b splines a general method for constructing quasi interpolation operators based on b splines is developed given a b spline f in r s s 1 normalized by i z email protected i 1 the classical structure q f email protected i z email protected i f i for a quasi interpolation operator q is considered a minimization problem is derived from an estimate of the quasi interpolation error associated with q when lf is a linear combination of values of f at points in some neighbourhood of the support of f or a linear combination of values of f and some of its derivatives at some points in this set or a linear combination of weighted mean values of the function to be approximated that linear functional is defined to produce a quasi interpolant exact on the space of polynomials of maximal total degree included in the space spanned by the integer translates of f the solution of that minimization problem is characterized in terms of specific splines which do not depend on l but only on f

susceptibility of the ising model on the scale free network with a cayley tree like structure abstract we derive the exact expression for the zero field susceptibility of each spin of the ising model on the scale free sf network having the degree distribution p k k with the cayley tree like structure the system shows that i the zero field susceptibility of a spin in the interior part diverges below the transition temperature of the sf network with the bethe lattice like structure t c for 3 while it diverges at any finite temperature for 3 and ii the surface part diverges below the divergence temperature of the sf network with the cayley tree like structure t s for 3 while it diverges at any finite temperature for 3

near minimally normed spline quasi interpolants on uniform partitions spline quasi interpolants qis are local approximating operators for functions or discrete data we consider the construction of discrete and integral spline qis on uniform partitions of the real line having small infinity norms we call them near minimally normed qis they are exact on polynomial spaces and minimize a simple upper bound of their infinity norms we give precise results for cubic and quintic qis also the qi error is considered as well as the advantage that these qis present when approximating functions with isolated discontinuities

in plane free vibration of circular annular disks abstract this paper presents a generalized formulation for the in plane modal characteristics of circular annular disks under combinations of all possible classical boundary conditions the in plane free vibration of an elastic and isotropic disk is studied on the basis of the two dimensional linear plane stress theory of elasticity the boundary characteristic orthogonal polynomials are employed in the rayleigh ritz method to obtain the natural frequencies and associated mode shapes two approaches have been used to represent a clamped boundary condition the first approach assumes a polynomial expression that satisfies the clamped conditions while the second approach uses a disk with free boundary supported on artificial springs with stiffness tending to infinity the natural frequencies are tabulated and compared with data available in the literature mode shapes are presented to illustrate the free vibration behavior of the disk

minimizing the quasi interpolation error for bivariate discrete quasi interpolants we define a class of discrete quasi interpolants based on bivariate box splines by imposing the exactness on a space of polynomials of total degree depending on the box spline and minimizing a constant appearing in the leading term of an appropriate quasi interpolation error estimate we give some c 1 quadratic and c 2 quartic examples and compare them with other well known quasi interpolants

ising models on power law random graphs we study a ferromagnetic ising model on random graphs with a power law degree distribution and compute the thermodynamic limit of the pressure when the mean degree is finite degree exponent 2 for which the random graph has a tree like structure for this we closely follow the analysis by dembo and montanari ann appl probab 20 2 565 592 2010 which assumes finite variance degrees 3 adapting it when necessary and also simplifying it when possible our results also apply in cases where the degree distribution does not obey a power law

computing the range of values of real functions using b spline form abstract in this paper we present an easy and efficient method for computing the range of a function by using spline quasi interpolation we exploit the close relationship between the spline function and its control polygon and use tight subdivision technique in order to obtain monotonic splines which make the range of the spline easy to compute the proposed method is useful in case of given scattered data generated by some unknown function f or scientific measurements several numerical examples are given for cubic and quintic quasi interpolant approximant to illustrate the efficiency and the performance of our method

synchronization landscapes in small world connected computer networks motivated by a synchronization problem in distributed computing we studied a simple growth model on regular and small world networks embedded in one and two dimensions we find that the synchronization landscape corresponding to the progress of the individual processors exhibits kardar parisi zhang like kinetic roughening on regular networks with short range communication links although the processors on average progress at a nonzero rate their spread the width of the synchronization landscape diverges with the number of nodes desynchronized state hindering efficient data management when random communication links are added on top of the one and two dimensional regular networks resulting in a small world network large fluctuations in the synchronization landscape are suppressed and the width approaches a finite value in the large system size limit synchronized state in the resulting synchronization scheme the processors make close to uniform progress with a nonzero rate without global intervention we obtain our results by simulating the simulations based on the exact algorithmic rules supported by coarse grained arguments

broad edge of chaos in strongly heterogeneous boolean networks the dynamic stability of the boolean networks representing a model for the gene transcriptional regulation kauffman model is studied by calculating analytically and numerically the hamming distance between two evolving configurations this turns out to behave in a universal way close to the phase boundary only for in degree distributions with a finite second moment in degree distributions of the form pd k k with 2 3 thus having a diverging second moment lead to a slower increase of the hamming distance when moving towards the unstable phase and to a broadening of the phase boundary for finite n with decreasing we conclude that the heterogeneous regulatory network connectivity facilitates the balancing between robustness and evolvability in living organisms

resilience to damage of graphs with degree correlations the existence or nonexistence of a percolation threshold on power law correlated graphs is a fundamental question for which a general criterion is lacking in this work we investigate the problems of site and bond percolation on graphs with degree correlations and their connection with spreading phenomena we obtain some general expressions that allow the computation of the transition thresholds or their bounds using these results we study the effects of assortative and disassortative correlations on the resilience to damage of networks

quasi interpolants associated with h splines on a three direction mesh h splines are b splines with regular hexagonal supports on a three direction mesh of the plane this family contains in particular classical box splines which are convolution powers of the piecewise affine pyramid in this paper we construct discrete or integral qua si interpolants having the best possible approximation order for the smooth functions the construction is based on properties of hexagonal sequences and associated difference operators and of discrete laplacian schemes

exact solutions of in plane natural vibration of a circular plate with outer edge restrained elastically abstract we solve exactly the equations of in plane natural vibration for a circular plate whose outer edge is restrained elastically the mode shapes are represented by trigonometric functions with a number of nodal diameters in the circumferential direction and mode functions in the radial direction we present the exact frequency equations and mode functions and tabulate the frequency parameters satisfying the frequency equations the corresponding mode functions and two dimensional mode shapes are illustrated when both radial and tangential stiffness are zero free edge infinity clamped edge or medium comparisons with previous reported results confirm the accuracy of the present work

the influence of preferential attachment on evolving networks abstract with the consideration of local events and preferential attachment the extended model proposed by albert and barabasi is studied firstly the extended model is built only by local events it is found that without preferential attachment the networks connectivity distribution decays exponentially for large connectivity and the decaying exponent varies with different probabilities of the addition of new links and rewiring secondly the extended model with partial preferential attachment is studied when preferential attachment only presents in the addition of new links or new nodes there is scale free regime in the phase diagram while preferential attachment is brought into the rewiring there coexist two regimes i e the exponential regime and the scale free regime thirdly fitness is introduced to the extended model with complete preferential attachment it is obtained that the network s connectivity distribution is a result affected by many factors including the fitness the local events and preferential attachment

a general spline differential quadrature method based on quasi interpolation the differential quadrature method is a numerical discretization technique for the approximation of derivatives the classical method is polynomial based and there is a natural restriction in the number of grid points involved a general spline based method is proposed to avoid this problem for any degree a lagrangian spline interpolant is defined having a fundamental function with small support a quasi interpolant is used to achieve the optimal approximation order that two stage scheme is detailed for the cubic quartic quintic and sextic cases and compared with another methods that appear in the literature

on the emergence of scaling in weighted networks abstract general conditions for the appearance of the power law distribution of total weights concentrated in vertices of complex network systems are established by use of the rate equation approach for networks evolving by connectivity governed attachment of every new node to p 1 exiting nodes and by ascription to every new link a weight taken from algebraic distributions independent of network topologies it is shown that the distribution of the total weight w asymptotically follows the power law p w w with the exponent 0 2 the power law dependence of the weight distribution is also proved to hold for asymptotically large w in the case of networks in which a link between nodes i and j carries a load w ij determined by node degrees k i and k j at the final stage of the network growth according to the relation w ij k i k j with 1 0 for this class of networks the scaling exponent describing the weight distribution is found to satisfy the relationship 1 where is the scaling index characterizing the distribution of node degrees n k k

increasing the approximation order of spline quasi interpolants abstract in this paper we show how by a very simple modification of bivariate spline discrete quasi interpolants we can construct a new class of quasi interpolants which have remarkable properties such as high order of regularity and polynomial reproduction more precisely given a spline discrete quasi interpolation operator q d which is exact on the space p m of polynomials of total degree at most m we first propose a general method to determine a new differential quasi interpolation operator q r d which is exact on p m r q r d uses the values of the function to be approximated at the points involved in the linear functional defining q d as well as the partial derivatives up to the order r at the same points from this result we then construct and study a first order differential quasi interpolant based on the c 1 cubic b spline on the equilateral triangulation with a hexagonal support when the derivatives are not available or extremely expensive to compute we approximate them by appropriate finite differences to derive new discrete quasi interpolants q d we estimate with small constants the quasi interpolation errors f q r d f and f q d f in the infinity norm finally numerical examples are used to analyze the performance of the method

replicated transfer matrix analysis of ising spin models on small world lattices we calculate equilibrium solutions for ising spin models on small world lattices which are constructed by superimposing random and sparse poissonian graphs with finite average connectivity c onto a one dimensional ring the nearest neighbour bonds along the ring are ferromagnetic whereas those corresponding to the poissonian graph are allowed to be random our models thus generally contain quenched connectivity and bond disorder within the replica formalism calculating the disorder averaged free energy requires the diagonalization of replicated transfer matrices in to addition developing the general replica symmetric theory we derive phase diagrams and calculate effective field distributions for two specific cases that of uniform sparse long range bonds i e small world magnets and that of j random sparse long range bonds i e small world spin glasses

finite size scaling in complex networks a finite size scaling fss theory is proposed for various models in complex networks in particular we focus on the fss exponent which plays a crucial role in analyzing numerical data for finite size systems based on the droplet excitation hyperscaling argument we conjecture the values of the fss exponents for the ising model the susceptible infected susceptible model and the contact process all of which are confirmed reasonably well in numerical simulations

understanding the temporal pattern of spreading in heterogeneous networks theory of the mean infection time for a reliable prediction of an epidemic or information spreading pattern in complex systems well defined measures are essential in the susceptible infected model on heterogeneous networks the cluster of infected nodes in the intermediate time regime exhibits too large fluctuation in size to use its mean size as a representative value the cluster size follows quite a broad distribution which is shown to be derived from the variation of the cluster size with the time when a hub node was first infected on the contrary the distribution of the time taken to infect a given number of nodes is well concentrated at its mean suggesting the mean infection time is a better measure we show that the mean infection time can be evaluated by using the scaling behaviors of the boundary area of the infected cluster and use it to find a nonexponential but algebraic spreading phase in the intermediate stage on strongly heterogeneous networks such slow spreading originates in only small degree nodes left susceptible while most hub nodes are already infected in the early exponential spreading stage our results offer a way to detour around large statistical fluctuations and quantify reliably the temporal pattern of spread under structural heterogeneity

morphology of technological levels in an innovation propagation model we study the dynamical properties of the propagation of innovation on a two dimensional lattice random network scale free network and cayley tree in order to investigate the diversity of technological level we study the scaling property of width w n t which represents the root mean square of the technological level of agents here n is the total number of agents from the numerical simulations we find that the steady state value of w n t w sat n scales as w sat n n 1 2 when the system is in a flat ordered phase for d 2 in the flat ordered phase most of the agents have the same technological level on the other hand when the system is in a smooth disordered phase the value of w sat n does not depend on n these behaviors are completely different from those observed on a one dimensional 1d lattice by considering the effect of the underlying topology on the propagation dynamics for d 2 we also provide a mean field analysis for w sat n which agrees very well with the observed behaviors of w sat n this directly shows that the morphological properties in order disorder transition on a 1d lattice is completely different from that on higher dimensions it also provides an evidence that the upper critical dimension for the roughening transition of the propagation of innovation is d u 2

communication and correlation among communities given a network and a partition in communities we consider the issues how communities influence each other and when two given communities do communicate specifically we address these questions in the context of small world networks where an arbitrary quenched graph is given and long range connections are randomly added we prove that among the communities a superposition principle applies and gives rise to a natural generalization of the effective field theory already presented by m ostilli and j f f mendes phys rev e 78 031102 2008 n 1 which here n 1 consists in a sort of effective tap thouless anderson and palmer equations in which each community plays the role of a microscopic spin the relative susceptibilities derived from these equations calculated at finite or zero temperature where the method provides an effective percolation theory give us the answers to the above issues unlike the case n 1 asymmetries among the communities may lead via the tap like structure of the equations to many metastable states whose number in the case of negative shortcuts among the communities may grow exponentially fast with n as examples we consider the n viana bray communities model and the n one dimensional small world communities model despite being the simplest ones the relevance of these models in network theory as e g in social networks is crucial and no analytic solution were known until now connections between percolation and the fractal dimension of a network are also discussed finally as an inverse problem we show how from the relative susceptibilities a natural and efficient method to detect the community structure of a generic network arises

near best quasi interpolants associated with h splines on a three direction mesh spline quasi interpolants with optimal approximation orders and small norms are useful in several applications in this paper we construct the so called near best discrete and integral quasi interpolants based on h splines i e b splines with regular hexagonal supports on the uniform three directional mesh of the plane these quasi interpolants are obtained so as to be exact on some space of polynomials and to minimize an upper bound of their infinite norms which depend on a finite number of free parameters we show that this problem has always a solution which is not unique in general concrete examples of these types of quasi interpolants are given in the two last sections

ising model in small world networks the ising model in small world networks generated from two and three dimensional regular lattices has been studied monte carlo simulations were carried out to characterize the ferromagnetic transition appearing in these systems in the thermodynamic limit the phase transition has a mean field character for any finite value of the rewiring probability p which measures the disorder strength of a given network for small values of p both the transition temperature and critical energy change with p as a power law in the limit stackrel ensuremath rightarrow p 0 the heat capacity at the transition temperature diverges logarithmically in two dimensional 2d networks and as a power law in 3d

random field ising model on networks with inhomogeneous connections we study a zero temperature phase transition in the random field ising model on scale free networks with the degree exponent using an analytic mean field theory we find that the spins are alway

translates of multivarlate splines abstract we study spaces generated by translations of a fixed function over lattice points we provide some new algebraic and approximation properties for these spaces which show their applicability for finite element analysis

synchronization transition of heterogeneously coupled oscillators on scale free networks we investigate the synchronization transition of the modified kuramoto model where the oscillators form a scale free network with degree exponent ensuremath lambda an oscillator of degree k i is coupled to its neighboring oscillators with asymmetric and degree dependent coupling in the form of j k i ensuremath eta ensuremath 1 by invoking the mean field approach we find eight different synchronization transition behaviors depending on the values of ensuremath eta and ensuremath lambda and derive the critical exponents associated with the order parameter and the finite size scaling in each case the synchronization transition point j c is determined as being zero finite when ensuremath eta g ensuremath lambda ensuremath 2 ensuremath eta l ensuremath lambda ensuremath 2 the synchronization transition is also studied from the perspective of cluster formation of synchronized vertices the cluster size distribution and the largest cluster size as a function of the system size are derived for each case using the generating function technique our analytic results are confirmed by numerical simulations

optimal bivariate c1 cubic quasi interpolation on a type 2 triangulation in a guessab o nouisser g schmeisser multivariate approximation by a combination of modified taylor polynomials j comput appl math 196 2006 162 179 a general method is proposed to increase the approximation order of approximation operators in this work by using these enhancement techniques we introduce and study new schemes based on a c 1 spline quasi interpolant on a type 2 triangulation they are designed for approximating real valued functions defined on r 2 the proposed method is based on the following idea from a discrete quasi interpolant defined by the quadratic box spline exact on p 2 and by judiciously choosing the first order taylor coefficients we derive a cubic differential quasi interpolant yielding optimal approximation order in addition when the derivatives are not available or are extremely expensive to compute we approximate them by finite difference approximations having the desired accuracy to derive a new class of discrete quasi interpolants as an essential difference to some of the existing methods we only use the given data values and then we do not modify the original triangulation finally we present some numerical tests which confirm the efficiency of the newly quasi interpolant and demonstrate good visual quality in particular we compare it with a differential quasi interpolant done by lai m j lai approximation order from bivariate c 1 cubics on a four directional mesh is full comput aided geom design 11 2 1994 215 223 which is also exact on p 3 but uses third order partial derivatives

robustness and fragility of the susceptible infected susceptible epidemic models on complex networks we analyze two alterations of the standard susceptible infected susceptible sis dynamics that preserve the central properties of spontaneous healing and infection capacity of a vertex increasing unlimitedly with its degree all models have the same epidemic thresholds in mean field theories but depending on the network properties simulations yield a dual scenario in which the epidemic thresholds of the modified sis models can be either dramatically altered or remain unchanged in comparison with the standard dynamics for uncorrelated synthetic networks having a power law degree distribution with exponent 5 2 differences are more remarkable for 3 where a finite threshold is found in the modified models in contrast with the vanishing threshold of the original one this duality is elucidated in terms of epidemic lifespan on star graphs we verify that the activation of the modified sis models is triggered in the innermost component of the network given by a k core decomposition for 3 the activation in the modified dynamics is collective involving essentially the whole network while it is triggered by hubs in the standard sis the duality also appears in the finite size scaling of the critical quantities where mean field behaviors are observed for the modified but not for the original dynamics our results feed the discussions about the most proper conceptions of epidemic models to describe real systems and the choices of the most suitable theoretical approaches to deal with these models

modal characteristics of in plane vibration of circular plates clamped at the outer edge the equations of in plane vibration in thin flat plates are solved for free vibration in circular plates clamped at the outer edge the mode shapes are represented by trigonometric functions in the circumferential direction and by series summation of bessel functions in the radial direction accuracy of the predictions of natural frequencies and mode shapes is assessed by comparisons with finite element predictions and with previously reported results the present solution gives very accurate predictions the work also highlights the nature of coupling between the different circumferential and radial modes and the response of different vibrational modes at the center of the plate it is shown that the center point of the plate vibrates only for modes with unity circumferential wave number number of nodal diameters nondimensional frequency parameters are listed and the radial mode shapes of natural vibration are depicted to illustrate the free vibration behavior in the frequency range of practical interest

flux distribution of metabolic networks close to optimal biomass production we study a statistical model describing the steady state distribution of the fluxes in a metabolic network the resulting model defined on continuous variables can be solved by the cavity method in particular analytical tractability is possible solving the cavity equation over an ensemble of networks with the same degree distribution as a real metabolic network the flux distribution that optimizes production of biomass has a fat tail with a power law exponent independent of the structural properties of the underlying network these results are in complete agreement with the flux balance analysis outcome and in qualitative agreement with the experimental results

spin models on random graphs with controlled topologies beyond degree constraints we study ising spin models on finitely connected random interaction graphs which are drawn from an ensemble in which not only the degree distribution p k can be chosen arbitrarily but which allows for further fine tuning of the topology via preferential attachment of edges on the basis of an arbitrary function q k k of the degrees of the vertices involved we solve these models using finite connectivity equilibrium replica theory within the replica symmetric ansatz in our ensemble of graphs phase diagrams of the spin system are found to depend no longer only on the chosen degree distribution but also on the choice made for q k k the increased ability to control interaction topology in solvable models beyond prescribing only the degree distribution of the interaction graph enables a more accurate modeling of real world interacting particle systems by spin systems on suitably defined random graphs

quasi interpolation by c1 quartic splines on type 1 triangulations abstract in this paper we construct two new families of c 1 quartic quasi interpolating splines on type 1 triangulations approximating regularly distributed data the splines are directly determined by setting their bernstein bezier coefficients to appropriate combinations of the given data values instead of defining the approximating splines as linear combinations of compactly supported bivariate spanning functions and do not use prescribed derivatives at any point of the domain the quasi interpolation operators provided by the proposed schemes interpolate the data values at the vertices of the triangulation reproduce cubic polynomials and yield approximation order four for smooth functions we also propose some numerical tests that confirm the theoretical results

on near best discrete quasi interpolation on a four directional mesh spline quasi interpolants are practical and effective approximation operators in this paper we construct qis with optimal approximation orders and small infinity norms called near best discrete quasi interpolants which are based on w splines i e b splines with octagonal supports on the uniform four directional mesh of the plane these quasi interpolants are exact on some space of polynomials and they minimize an upper bound of their infinity norms depending on a finite number of free parameters we show that this problem has always a solution in general nonunique concrete examples of such quasi interpolants are given in the last section

a multinomial spline approximation scheme using spline quasi interpolants abstract this paper presents a multinomial spline approximation scheme based on spline quasi interpolants the scheme can be considered as an extension of the usual bernstein approximation for complex exponentials error estimates and numerical examples are given to show that this new scheme could produce highly accurate results

voter model dynamics in complex networks role of dimensionality disorder and degree distribution we analyze the ordering dynamics of the voter model in different classes of complex networks we observe that whether the voter dynamics orders the system depends on the effective dimensionality of the interaction networks we also find that when there is no ordering in the system the average survival time of metastable states in finite networks decreases with network disorder and degree heterogeneity the existence of hubs i e highly connected nodes in the network modifies the linear system size scaling law of the survival time the size of an ordered domain is sensitive to the network disorder and the average degree decreasing with both however it seems not to depend on network size and on the heterogeneity of the degree distribution

ising model on networks with an arbitrary distribution of connections we find the exact critical temperature t c of the nearest neighbor ferromagnetic ising model on an equilibrium random graph with an arbitrary degree distribution p k we observe an anomalous behavior of the magnetization magnetic susceptibility and specific heat when p k is fat tailed or loosely speaking when the fourth moment of the distribution diverges in infinite networks when the second moment becomes divergent t c approaches infinity the phase transition is of infinite order and size effect is anomalously strong

on the construction of trivariate near best quasi interpolants based on c 2 quartic splines on type 6 tetrahedral partitions the construction of new quasi interpolants qis having optimal approximation order and small infinity norm and based on a trivariate c 2 quartic box spline is addressed in this paper these quasi interpolants called near best qis are obtained in order to be exact on the space of cubic polynomials and to minimize an upper bound of their infinity norm which depends on a finite number of free parameters in a tetrahedral sequence defining the coefficients of the qis we show that this problem has always a unique solution which is explicitly given we also prove that the sequence of the resulting near best quasi interpolants converges in the infinity norm to the schoenberg operator

a characterization of multivariate quasi interpolation formulas and its applications let be a compactly supported function on s ands the linear space withgenerator that is s is the linear span of the multiinteger translates of it is well known that corresponding to a generator there are infinitely many quasi interpolation formulas a characterization of these formulas is presented which allows for their direct calculation in a variety of forms suitable to particular applications and in addition provides a clear formulation of the difficult problem of minimally supported quasi interpolants we introduce a generalization of interpolation called interpolation and a notion of higher order quasi interpolation called approximation a characterization of approximants similar to that of quasi interpolants is obtained with similar applications among these applications are estimating least squares approximants without matrix inversion surface fitting to incomplete or semi scattered discrete data and constructing generators with one point quasi interpolation formulas it will be seen that the exact values of the generator at the multi integers s facilitates the above study an algorithm to yield this information for box splines is discussed

superconvergent local quasi interpolants based on special multivariate quadratic spline space over a refined quadrangulation abstract in this paper we first recall some results concerning the construction and the properties of quadratic b splines over a refinement of a quadrangulation of a planar domain introduced recently by lamnii et al then we introduce the b spline representation of hermite interpolant in the special space s 2 1 0 of any polynomial or any piecewise polynomial over refined quadrangulation of after that we use this b representation for constructing several superconvergent discrete quasi interpolants the new results that we present in this paper are an improvement and a generalization of those developed in the above cited paper

partition function zeros for the ising model on complete graphs and on annealed scale free networks we analyse the partition function of the ising model on graphs of two different types complete graphs wherein all nodes are mutually linked and annealed scale free networks for which the degree distribution decays as p k k we are interested in zeros of the partition function in the cases of complex temperature or complex external field fisher and lee yang zeros respectively for the model on an annealed scale free network we find an integral representation for the partition function which in the case 5 reproduces the zeros for the ising model on a complete graph for 3 5 we derive the dependent angle at which the fisher zeros impact onto the real temperature axis this in turn gives access to the dependent universal values of the critical exponents and critical amplitudes ratios our analysis of the lee yang zeros reveals a difference in their behaviour for the ising model on a complete graph and on an annealed scale free network when 3 5 whereas in the former case the zeros are purely imaginary they have a non zero real part in latter case so that the celebrated lee yang circle theorem is violated

number of cliques in random scale free network ensembles abstract in this paper we calculate the average number of cliques in random scale free networks in the limit of large network size n 1 we consider first the hidden variable ensemble and subsequently the molloy reed ensemble in both cases we find that cliques i e fully connected subgraphs appear also when the average degree is finite this is in contrast to what happens in erdos and renyi graphs in which diverging average degree is required to observe cliques of size c 3 moreover we show that in random scale free networks the clique number i e the size of the largest clique present in the network diverges with the system size

scaling and percolation in the small world network model in this paper we study the small world network model of watts and strogatz which mimics some aspects of the structure of networks of social interactions we argue that there is one nontrivial length scale in the model analogous to the correlation length in other systems which is well defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero giving a normal critical point in this limit this length scale governs the crossover from large to small world behavior in the model as well as the number of vertices in a neighborhood of given radius on the network we derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex vertex distance on the network and using series expansion and pad e approximants find an approximate analytic form for the scaling function we calculate the effective dimension of small world graphs and show that this dimension varies as a function of the length scale on which it is measured in a manner reminiscent of multifractals we also study the problem of site percolation on small world networks as a simple model of disease propagation and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms in epidemiological terms the point at which an epidemic occurs the typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph all our analytic results are confirmed by extensive numerical simulations of the model

criticality on networks with topology dependent interactions weighted scale free networks with topology dependent interactions are studied it is shown that the possible universality classes of critical behavior which are known to depend on topology can also be explored by tuning the form of the interactions at fixed topology for a model of opinion formation simple mean field and scaling arguments show that a mapping gamma gamma mu 1 mu describes how a shift of the standard exponent gamma of the degree distribution can absorb the effect of degree dependent pair interactions j ij proportional to k i k j mu where k i stands for the degree of vertex i this prediction is verified by extensive numerical investigations using the cavity method and monte carlo simulations the critical temperature of the model is obtained through the bethe peierls approximation and with the replica technique the mapping can be extended to nonequilibrium models such as those describing the spreading of a disease on a network

frequency equation for the in plane vibration of a clamped circular plate the in plane vibration of the circular plate structure is important in the transmission of high frequency vibration but none have produced an exact frequency equation of the in plane vibration of a clamped circular plate therefore this paper focuses on deriving the frequency equation for the in plane vibration of the clamped circular plate of uniform thickness with an isotropic material in the elastic range to derive the frequency equation for the clamped circular plate in the cylindrical coordinate kinetic and potential energy for in plane behavior were first obtained by using the stress strain displacement expressions and applying hamilton s principle which led to two sets of highly coupled differential equations for the equations of motion substitution of helmholtz decomposition for the coupled differential equations produced uncoupled equations of motion the assumption of a harmonic solution for the uncoupled equations led to wave equations using the separation of the variable the general solutions for the wave equations were obtained the solutions generated the in plane displacements in the r and directions finally the application of boundary conditions yielded the frequency equation for the clamped circular plate the derived frequency equation was validated by finite element analysis and by comparison of previously reported results

statistically consistent coarse grained simulations for critical phenomena in complex networks we propose a degree based coarse graining approach that not just accelerates the evaluation of dynamics on complex networks but also satisfies the consistency conditions for both equilibrium statistical distributions and nonequilibrium dynamical flows for the ising model and susceptible infected susceptible epidemic model we introduce these required conditions explicitly and further prove that they are satisfied by our coarse grained network construction within the annealed network approximation finally we numerically show that the phase transitions and fluctuations on the coarse grained network are all in good agreements with those on the original one

first and second order phase transitions in scale free networks we study first and second order phase transitions of ferromagnetic lattice models on scale free networks with a degree exponent gamma using the example of the q state potts model we derive a general self consistency relation within the frame of the weiss molecular field approximation which presumably leads to exact critical singularities depending on the value of gamma we have found three different regimes of the phase diagram as a general trend first order transitions soften with decreasing gamma and the critical singularities at the second order transitions are gamma dependent

voter models on heterogeneous networks we study simple interacting particle systems on heterogeneous networks including the voter model and the invasion process these are both two state models in which in an update event an individual changes state to agree with a neighbor for the voter model an individual imports its state from a randomly chosen neighbor here the average time t n to reach consensus for a network of n nodes with an uncorrelated degree distribution scales as n ensuremath mu 1 2 ensuremath mu 2 where ensuremath mu k is the k text th moment of the degree distribution quick consensus thus arises on networks with broad degree distributions we also identify the conservation law that characterizes the route by which consensus is reached parallel results are derived for the invasion process in which the state of an agent is exported to a random neighbor we further generalize to biased dynamics in which one state is favored the probability for a single fitter mutant located at a node of degree k to overspread the population char22 the fixation probability char22 is proportional to k for the voter model and to 1 k for the invasion process

quenched central limit theorems for the ising model on random graphs the main goal of the paper is to prove central limit theorems for the magnetization rescaled by sqrt n for the ising model on random graphs with n vertices both random quenched and averaged quenched measures are considered we work in the uniqueness regime beta beta c or beta 0 and b ne 0 where beta is the inverse temperature beta c is the critical inverse temperature and b is the external magnetic field in the random quenched setting our results apply to general tree like random graphs as introduced by dembo montanari and further studied by dommers and the first and third author and our proof follows that of ellis in mathbb z d for the averaged quenched setting we specialize to two particular random graph models namely the 2 regular configuration model and the configuration model with degrees 1 and 2 in these cases our proofs are based on explicit computations relying on the solution of the one dimensional ising models

effects of local population structure in a reaction diffusion model of a contact process on metapopulation networks we investigate the effects of local population structure in reaction diffusion processes representing a contact process cp on metapopulations represented as complex networks considering a model in which the nodes of a large scale network represent local populations defined in terms of a homogeneous graph we show by means of extensive numerical simulations that the critical properties of the reaction diffusion system are independent of the local population structure even when this one is given by a ordered linear chain this independence is confirmed by the perfect matching between numerical critical exponents and the results from a heterogeneous mean field theory suited in principle to describe situations of local homogeneous mixing the analysis of several variations of the reaction diffusion process allows us to conclude the independence from population structure of the critical properties of cp like models on metapopulations and thus of the universality of the reaction diffusion description of this kind of models