stochastic and mixed flower graphs stochasticity is introduced to a well studied class of recursively grown graphs u v flower nets which have power law degree distributions as well as small world properties when u 1 the stochastic variant interpolates between different deterministic flower graphs thus adding flexibility to the model the random multiplicative growth process involved however leads to a spread ensemble of networks with finite variance for the number of links nodes and loops nevertheless the degree exponent and loopiness exponent attain unique values in the thermodynamic limit of infinitely large graphs we also study a class of mixed flower networks closely related to the stochastic flowers but which are grown recursively in a deterministic way the deterministic growth of mixed flower nets eliminates ensemble spreads and their recursive growth allows for exact analysis of their uniquely defined mixed properties

on the modelling of incompressibility in linear and non linear elasticity with the master slave approach this is the pre peer reviewed version of the following article munoz jose j on the modelling of incompressibility in linear and non linear elasticity with the master slave approach international journal for numerical methods in engineering abril 2008 vol 74 num 2 p 269 293 which has been published in final form at http www3 interscience wiley com journal 116323442 abstract

evolving apollonian networks with small world scale free topologies we propose two types of evolving networks evolutionary apollonian networks eans and general deterministic apollonian networks gdans established by simple iteration algorithms we investigate the two networks by both simulation and theoretical prediction analytical results show that both networks follow power law degree distributions with distribution exponents continuously tuned from 2 to 3 the accurate expression of clustering coefficient is also given for both networks moreover the investigation of the average path length of ean and the diameter of gdan reveals that these two types of networks possess small world feature in addition we study the collective synchronization behavior on some limitations of the ean

a stabilized mixed finite element method for darcy flow we develop new stabilized mixed finite element methods for darcy flow stability and an a priori error estimate in the stability norm are established a wide variety of convergent finite elements present themselves unlike the classical galerkin formulation which requires highly specialized elements an interesting feature of the formulation is that there are no mesh dependent parameters numerical tests confirm the theoretical results 2002 elsevier science b v all rights reserved

a deterministic small world network created by edge iterations abstract small world networks are ubiquitous in real life systems most previous models of small world networks are stochastic the randomness makes it more difficult to gain a visual understanding on how do different nodes of networks interact with each other and is not appropriate for communication networks that have fixed interconnections here we present a model that generates a small world network in a simple deterministic way our model has a discrete exponential degree distribution we solve the main characteristics of the model

a coupled multipoint stress multipoint flux mixed finite element method for the biot system of poroelasticity abstract we present a mixed finite element method for a five field formulation of the biot system of poroelasticity that reduces to a cell centered pressure displacement system on simplicial and quadrilateral grids a mixed stress displacement rotation formulation for elasticity with weak stress symmetry is coupled with a mixed velocity pressure darcy formulation the spatial discretization is based on combining the multipoint stress mixed finite element msmfe method for elasticity and the multipoint flux mixed finite element mfmfe method for darcy flow it uses the lowest order brezzi douglas marini mixed finite element spaces for the poroelastic stress and darcy velocity piecewise constant displacement and pressure and continuous piecewise linear or bilinear rotation a vertex quadrature rule is applied to the velocity stress and stress rotation bilinear forms which block diagonalizes the corresponding matrices and allows for local velocity stress and rotation elimination this leads to a cell centered positive definite system for pressure and displacement at each time step we perform error analysis for the semidiscrete and fully discrete formulations establishing first order convergence for all variables in their natural norms the numerical tests confirm the theoretical convergence rates and illustrate the locking free property of the method

topological properties of integer networks abstract inspired by pythagoras s belief that numbers represent the reality we study the topological properties of networks of composite numbers in which the vertices represent the numbers and two vertices are connected if and only if there exists a divisibility relation between them the network has a fairly large clustering coefficient c 0 34 which is insensitive to the size of the network the average distance between two nodes is shown to have an upper bound that is independent of the size of the network in contrast to the behavior in small world and ultra small world networks the out degree distribution is shown to follow a power law behavior of the form k 2 in addition these networks possess hierarchical structure as c k k 1 in accord with the observations of many real life networks

lmi based criteria for synchronization of complex dynamical networks this paper considers the problem of controlling a complex dynamical network by means of pinning we point out that the synchronization criterion given in the form of matrix eigenvalues can be equivalent to a linear matrix inequality criterion we further investigate network synchronization via pinning and prove several linear matrix inequality theorems in particular we theoretically provide two typical pinning strategies based on whether the graph which is made up of unpinned nodes and edges between them is irreducible or not numerical simulations including k regular networks star shaped networks and ba scale free networks are shown for illustration and verification

families and clustering in a natural numbers network we develop a network in which the natural numbers are the vertices the decomposition of natural numbers by prime numbers is used to establish the connections we perform data collapse and show that the degree distribution of these networks scales linearly with the number of vertices we explore the families of vertices in connection with prime numbers decomposition we compare the average distance of the network and the clustering coefficient with the distance and clustering coefficient of the corresponding random graph in case we set connections among vertices each time the numbers share a common prime number the network has properties similar to a random graph if the criterion for establishing links becomes more selective only prime numbers greater than p l are used to establish links where the network has high clustering coefficient

comment on families and clustering in a natural numbers network corso phys rev e 69 036106 2004 constructs a family of graphs from subsets of the natural numbers and numerically estimates diameter degree and clustering we give exact asymptotic formulas for these quantities and thereby argue that number theory is a more appropriate tool than simulation

general random pseudofractal networks we introduce a general random pseudofractal network model by assigning fitness to each edge in this model continuous growth and attachment determined by their fitness of already existing edges are the two ingredients we obtain the analytical results that our model exhibits a power law degree distribution with exponent 2 m 1 m 1 where m and are tunable parameters we also show that a general random pseudofractal network has a large clustering coefficient and a small average distance leading to a small world effect these theoretical results agree well with numerical simulations

decoupling mixed finite elements on hierarchical triangular grids for parabolic problems in this paper we propose a numerical method for the solution of time dependent flow problems in mixed form such problems can be efficiently approximated on hierarchical grids obtained from an unstructured coarse triangulation by using a regular refinement process inside each of the initial coarse elements if these elements are considered as subdomains we can formulate a non overlapping domain decomposition method based on the lowest order raviart thomas elements properly enhanced with lagrange multipliers on the boundaries of each subdomain excluding the dirichlet edges a suitable choice of mixed finite element spaces and quadrature rules yields a cell centered scheme for the pressures with a local 10 point stencil the resulting system of differential algebraic equations is integrated in time by the crank nicolson method which is known to be a stiffly accurate scheme as a result we obtain independent subdomain linear systems that can be solved in parallel the behavior of the algorithm is illustrated on a variety of numerical experiments

high dimensional random apollonian networks abstract we propose a simple algorithm which produces a new category of networks high dimensional random apollonian networks with small world and scale free characteristics we derive analytical expressions for their degree distributions and clustering coefficients which are determined by the dimension of the network the values obtained for these parameters are in good agreement with simulation results and comparable to those coming from real networks we estimate also analytically that the average path length of the networks increases at most logarithmically with the number of vertices

local world evolving networks with tunable clustering abstract we propose an extended local world evolving network model including a triad formation tf step in the process of network evolution random fluctuation in the number of new edges is involved we derive analytical expressions for degree distribution clustering coefficient and average path length our model can unify the generic properties of real life networks scale free degree distribution high clustering and small inter node separation moreover in our model the clustering coefficient is tunable simply by changing the expected number of tf steps after a single local preferential attachment step

a geometric growth model interpolating between regular and small world networks we propose a geometric growth model which interpolates between one dimensional linear graphs and small world networks the model undergoes a transition from large to small worlds we study the topological characteristics by both theoretical predictions and numerical simulations which are in good accordance with each other our geometrically growing model is a complementarity for the static ws model

self similar disk packings as model spatial scale free networks the network of contacts in space filling disk packings such as the apollonian packing is examined these networks provide an interesting example of spatial scale free networks where the topology reflects the broad distribution of disk areas a wide variety of topological and spatial properties of these systems is characterized their potential as models for networks of connected minima on energy landscapes is discussed

flexible construction of hierarchical scale free networks with general exponent extensive studies have been done to understand the principles behind architectures of real networks recently evidence for hierarchical organization in many real networks has also been reported here we present a hierarchical model that reproduces the main experimental properties observed in real networks scale free of degree distribution p k frequency of the nodes that are connected to k other nodes decays as a power law p k ensuremath sim k ensuremath ensuremath gamma and power law scaling of the clustering coefficient c k ensuremath sim k ensuremath 1 the major points of our model can be summarized as follows a the model generates networks with scale free distribution for the degree of nodes with general exponent ensuremath gamma g2 and arbitrarily close to any specified value being able to reproduce most of the observed hierarchical scale free topologies in contrast previous models cannot obtain values of ensuremath gamma g2 58 b our model has structural flexibility because i it can incorporate various types of basic building blocks e g triangles tetrahedrons and in general fully connected clusters of n nodes and ii it allows a large variety of configurations i e the model can use more than n ensuremath 1 copies of basic blocks of n nodes the structural features of our proposed model might lead to a better understanding of architectures of biological and nonbiological networks

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

a petrov galerkin enriched method a mass conservative finite element method for the darcy equation starting from the non stable p1 p0 discretization we build enhanced methods for the darcy equation which are stable and locally mass conservative the methods are derived in a petrov galerkin framework where both velocity and pressure trial spaces are enriched with multiscale functions these functions solve local problems correcting the residuals of the strong equations in each element and interior edge which leads to a velocity space enhanced with functions belonging to the lowest order raviart thomas space several numerical tests validate the methods

two direction poly scale refinability the concept of two direction poly scale refinable functions with dilation factor a and shift invariant spaces is introduced the existence of solutions of two direction poly scale refinable equations is investigated in particular a necessary and sufficient condition for the orthonormality of the solutions is established some properties of two direction poly scale refinable functions are discussed finally two examples of two direction poly scale refinable functions are given

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

tutte polynomial of pseudofractal scale free web the tutte polynomial of a graph is a 2 variable polynomial which is quite important in both combinatorics and statistical physics it contains various numerical invariants and polynomial invariants such as the number of spanning trees the number of spanning forests the number of acyclic orientations the reliability polynomial chromatic polynomial and flow polynomial in this paper we study and obtain a recursive formula for the tutte polynomial of pseudofractal scale free web psfw and thus logarithmic complexity algorithm to calculate the tutte polynomial of the psfw is obtained although it is np hard for general graph by solving the recurrence relations derived from the tutte polynomial the rigorous solution for the number of spanning trees of the psfw is obtained therefore an alternative approach to determine explicitly the number of spanning trees of the psfw is given furthermore we analyze the all terminal reliability of the psfw and compare the results with those of the sierpinski gasket which has the same number of nodes and edges as the psfw in contrast with the well known conclusion that inhomogeneous networks e g scale free networks are more robust than homogeneous networks i e networks in which each node has approximately the same number of links with respect to random deletion of nodes the sierpinski gasket which is a homogeneous network as our results show is more robust than the psfw which is an inhomogeneous network with respect to random edge failures

recursive graphs with small world scale free properties we discuss a category of graphs recursive clique trees which have small world and scale free properties and allow a fine tuning of the clustering and the power law exponent of their discrete degree distribution we determine relevant characteristics of those graphs the diameter degree distribution and clustering parameter the graphs have also an interesting recursive property and generalize recent constructions with fixed degree distributions

high dimensional apollonian networks we propose a simple algorithm which produces high dimensional apollonian networks with both small world and scale free characteristics we derive analytical expressions for the degree distribution the clustering coefficient and the diameter of the networks which are determined by their dimension

a small world network of prime numbers abstract according to goldbach conjecture any even number can be broken up as the sum of two prime numbers n p q we construct a network where each node is a prime number and corresponding to every even number n we put a link between the component primes p and q in most cases an even number can be broken up in many ways and then we chose one decomposition with a probability p q through computation of average shortest distance and clustering coefficient we conclude that for 1 8 the network is of small world type and for 1 8 it is of regular type we also present a theoretical justification for such behaviour

magnetic models on apollonian networks thermodynamic and magnetic properties of ising models defined on the triangular apollonian network are investigated this and other similar networks are inspired by the problem of covering a euclidian domain with circles of maximal radii maps for the thermodynamic functions in two subsequent generations of the construction of the network are obtained by formulating the problem in terms of transfer matrices numerical iteration of this set of maps leads to very precise values for the thermodynamic properties of the model different choices for the coupling constants between only nearest neighbors along the lattice are taken into account for both ferromagnetic and antiferromagnetic constants long range magnetic ordering is obtained with exception of a size dependent effective critical behavior of the correlation length no evidence of asymptotic criticality was detected

a multipoint flux mixed finite element method on distorted quadrilaterals and hexahedra in this paper we develop a new mixed finite element method for elliptic problems on general quadrilateral and hexahedral grids that reduces to a cell centered finite difference scheme a special non symmetric quadrature rule is employed that yields a positive definite cell centered system for the pressure by eliminating local velocities the method is shown to be accurate on highly distorted rough quadrilateral and hexahedral grids including hexahedra with non planar faces theoretical and numerical results indicate first order convergence for the pressure and face fluxes

exact scaling properties of a hierarchical network model we report on the exact results for the degree k the diameter d the clustering coefficient c and the betweenness centrality b of a hierarchical network model with a replication factor m such quantities are calculated exactly with the help of recursion relations using the results we show that i the degree distribution follows a power law p k approximately k gamma with gamma 1 ln m ln m 1 ii the diameter grows logarithmically as d approximately ln n with the number of nodes n iii the clustering coefficient of each node is inversely proportional to its degree c proportional variant 1 k and the average clustering coefficient is nonzero in the infinite n limit and iv the betweenness centrality distribution follows a power law p b approximately b 2 we discuss a classification scheme of scale free networks into the universality class with the clustering property and the betweenness centrality distribution

exactly solvable scale free network model we study a deterministic scale free network recently proposed by barabasi ravasz and vicsek we find that there are two types of nodes the hub and rim nodes which form a bipartite structure of the network we first derive the exact numbers p k of nodes with degree k for the hub and rim nodes in each generation of the network respectively using this we obtain the exact exponents of the distribution function p k of nodes with k degree in the asymptotic limit of k infinity we show that the degree distribution for the hub nodes exhibits the scale free nature p k proportional to k gamma with gamma ln 3 ln 2 1 584 962 while the degree distribution for the rim nodes is given by p k proportional to e gamma k with gamma ln 3 2 0 405 465 second we analytically calculate the second order average degree of nodes d third we numerically as well as analytically calculate the spectra of the adjacency matrix a for representing topology of the network we also analytically obtain the exact number of degeneracies at each eigenvalue in the network the density of states i e the distribution function of eigenvalues exhibits the fractal nature with respect to the degeneracy fourth we study the mathematical structure of the determinant of the eigenequation for the adjacency matrix fifth we study hidden symmetry zero modes and its index theorem in the deterministic scale free network finally we study the nature of the maximum eigenvalue in the spectrum of the deterministic scale free network we will prove several theorems for it using some mathematical theorems thus we show that most of all important quantities in the network theory can be analytically obtained in the deterministic scale free network model of barabasi ravasz and vicsek therefore we may call this network model the exactly solvable scale free network

auxiliary space preconditioning for mixed finite element discretizations of richards equation abstract we propose an auxiliary space method for the solution of the indefinite problem arising from mixed method finite element discretizations of scalar elliptic problems the proposed technique uses conforming elements as an auxiliary space and utilizes special interpolation operators for the transfer of residuals and corrections between the spaces we show that the corresponding method provides optimal solver for the indefinite problem by only solving symmetric and positive definite auxiliary problems we apply this preconditioner to the mixed form discretization of richards equation linearized with the l scheme we provide numerical tests validating the theoretical estimates

a multipoint flux mixed finite element method on hexahedra we develop a mixed finite element method for elliptic problems on hexahedral grids that reduces to cell centered finite differences the paper is an extension of our earlier paper for quadrilateral and simplicial grids m f wheeler and i yotov siam j numer anal 44 2006 pp 2082 2106 the construction is motivated by the multipoint flux approximation method and it is based on an enhancement of the lowest order brezzi douglas duran fortin bddf mixed finite element spaces on hexahedra in particular there are four fluxes per face one associated with each vertex a special quadrature rule is employed that allows for local velocity elimination and leads to a symmetric and positive definite cell centered system for the pressures theoretical and numerical results indicate first order convergence for pressures and subface fluxes on sufficiently regular grids as well as second order convergence for pressures at the cell centers second order convergence for face fluxes is also observed computationally

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

statistics of cycles how loopy is your network we study the distribution of cycles of length h in large networks of size n 1 and find it to be an excellent ergodic estimator even in the extreme inhomogeneous case of scale free networks the distribution is sharply peaked around a characteristic cycle length h n our results suggest that h and the exponent might usefully characterize broad families of networks in addition to an exact counting of cycles in hierarchical nets we present a monte carlo sampling algorithm for approximately locating h and reliably determining our empirical results indicate that for small random scale free nets of degree exponent 1 1 and grows as the nets become larger

modular vs non modular preconditioners for fluid structure systems with large added mass effect in this article we address the numerical simulation of fluid structure interaction fsi problems featuring large added mass effect we analyze different preconditioners for the coupled system matrix obtained after space time discretization and linearization of the fsi problem the classical dirichlet neumann preconditioner has the advantage of modularity because it allows to reuse existing fluid and structure codes with minimum effort simple interface communication unfortunately its performance is very poor in case of large added mass effects alternatively we consider two non modular approaches the first one consists in preconditioning the coupled system with a suitable diagonal scaling combined with an ilut preconditioner the system is then solved by a krylov method the drawback of this procedure is that the combination of fluid and structure codes to solve the coupled system is not straightforward the second non modular approach we consider is a splitting technique based on an inexact block lu factorization of the linear fsi system the resulting algorithm computes the fluid velocity separately from the coupled pressure structure system at each iteration reducing the computational cost independently of the preconditioner the efficiency of semi implicit algorithms i e those that treat geometric and fluid nonlinearities in an explicit way is highlighted and their performance compared to the one of implicit algorithms all the methods are tested on three dimensional blood vessel systems the algorithm combining the non modular ilut preconditioner with krylov methods proved to be the fastest

spectral properties of the apollonian network abstract a first characterization of the spectral properties of the apollonian network is presented results are based on the numerically evaluated eigenvalue spectra of adjacency matrices for the first nine generations of the network the main features refer to the presence of several gaps on the eigenvalue axis and a very large number of degenerated eigenvalues the increase in the degeneracy of a family of eigenvalues is found to obey a very simple empirical relation

large scale topological and dynamical properties of the internet we study the large scale topological and dynamical properties of real internet maps at the autonomous system level collected in a 3 yr time interval we find that the connectivity structure of the internet presents statistical distributions settled in a well defined stationary state the large scale properties are characterized by a scale free topology consistent with previous observations correlation functions and clustering coefficients exhibit a remarkable structure due to the underlying hierarchical organization of the internet the study of the internet time evolution shows a growth dynamics with aging features typical of recently proposed growing network models we compare the properties of growing network models with the present real internet data analysis

deterministic self similar models of complex networks based on very symmetric graphs abstract using very symmetric graphs we generalize several deterministic self similar models of complex networks and we calculate the main network parameters of our generalization more specifically we calculate the order size and the degree distribution and we give an upper bound for the diameter and a lower bound for the clustering coefficient these results yield conditions under which the network is a self similar and scale free small world network we remark that all these conditions are posed on a small base graph which is used in the construction as a consequence we can construct complex networks having prescribed properties we demonstrate this fact on the clustering coefficient we propose eight new infinite classes of complex networks one of these new classes is so rich that it is parametrized by three independent parameters

analytical solution of average path length for apollonian networks with the help of recursion relations derived from the self similar structure we obtain the solution of average path length overline d t for apollonian networks in contrast to the well known numerical result overline d t ensuremath propto text ln text n t 3 4 j s andrade jr et al phys rev lett 94 018702 2005 our rigorous solution shows that the average path length grows logarithmically as overline d t ensuremath propto text ln text n t in the infinite limit of network size n t the extensive numerical calculations completely agree with our closed form solution

structural and spectral properties of a family of deterministic recursive trees rigorous solutions as one of the most significant models the uniform recursive tree urt has found many applications in a variety of fields in this paper we study rigorously the structural features and spectral properties of the adjacency matrix for a family of deterministic uniform recursive trees durts that are deterministic versions of urt firstly from the perspective of complex networks we investigate analytically the main structural characteristics of durts and obtain the accurate solutions for these properties which include degree distribution average path length distribution of node betweenness and degree correlations then we determine the complete eigenvalues and their corresponding eigenvectors of the adjacency matrix for durts our research may shed light on a better understanding of the features for urt also the analytical methods used here are capable of being extended to many other deterministic networks making the precise computation of their properties especially the full spectrum characteristics possible

error analysis of piecewise constant pressure approximations of darcy s law abstract in this paper we present a short discussion on some finite element formulations for linear elliptic problems for the sake of simplicity we consider the poisson equation p f taking the notation from darcy s law among the zillions of such methods we shall concentrate our attention on fem leading to a final system of linear algebraic equations mp f where each unknown pi represents the constant value of the approximated pressure ph in a single element it is indeed well known that for some applications there is a certain demand for these types of schemes

a general geometric growth model for pseudofractal scale free web abstract we propose a general geometric growth model for pseudofractal scale free web psw which is controlled by two tunable parameters we derive exactly the main characteristics of the networks degree distribution second moment of degree distribution degree correlations distribution of clustering coefficient as well as the diameter which are partially determined by the parameters analytical results show that the resulting networks are disassortative and follow power law degree distributions with a more general degree exponent tuned from 2 to 1 ln 3 ln 2 the clustering coefficient of each individual node is inversely proportional to its degree and the average clustering coefficient of all nodes approaches to a large nonzero value in the infinite network order the diameter grows logarithmically with the number of network nodes all these reveal that the networks described by our model have small world effect and scale free topology

epidemic dynamics and endemic states in complex networks we study by analytical methods and large scale simulations a dynamical model for the spreading of epidemics in complex networks in networks with exponentially bounded connectivity we recover the usual epidemic behavior with a threshold defining a critical point below that the infection prevalence is null on the contrary on a wide range of scale free networks we observe the absence of an epidemic threshold and its associated critical behavior this implies that scale free networks are prone to the spreading and the persistence of infections whatever spreading rate the epidemic agents might possess these results can help understanding computer virus epidemics and other spreading phenomena on communication and social networks

from regular to growing small world networks abstract we propose a growing model which interpolates between one dimensional regular lattice and small world networks the model undergoes an interesting phase transition from large to small worlds we investigate the structural properties by both theoretical predictions and numerical simulations our growing model is a complementarity for the important static watts strogatz network model

on the spectrum of the normalized laplacian of iterated triangulations of graphs the eigenvalues of the normalized laplacian of a graph provide information on its topological and structural characteristics and also on some relevant dynamical aspects specifically in relation to random walks in this paper we determine the spectra of the normalized laplacian of iterated triangulations of a generic simple connected graph as an application we also find closed forms for their multiplicative degree kirchhoff index kemeny s constant and number of spanning trees

geometric fractal growth model for scale free networks we introduce a deterministic model for scale free networks whose degree distribution follows a power law with the exponent ensuremath gamma at each time step each vertex generates its offspring whose number is proportional to the degree of that vertex with proportionality constant m ensuremath 1 mg1 we consider the two cases first each offspring is connected to its parent vertex only forming a tree structure second it is connected to both its parent and grandparent vertices forming a loop structure we find that both models exhibit power law behaviors in their degree distributions with the exponent ensuremath gamma 1 mathrm ln 2m ensuremath 1 mathrm ln m thus by tuning m the degree exponent can be adjusted in the range 2l ensuremath gamma l3 we also solve analytically a mean shortest path distance d between two vertices for the tree structure showing the small world behavior that is d ensuremath sim mathrm ln n mathrm ln overline k where n is system size and overline k is the mean degree finally we consider the case that the number of offspring is the same for all vertices and find that the degree distribution exhibits an exponential decay behavior

correlations in random apollonian network abstract in this paper by both simulations and theoretical predictions we study two and three node or degree correlations in random apollonian network ran which have small world and scale free topologies using the rate equation approach under the assumption of continuous degree we first give the analytical solution for two node correlations expressed by average nearest neighbor degree annd then we revisit the degree distribution of ran using rate equation method and get the exact connection distribution based on which we derive a more accurate result for mean clustering coefficient as an average quantity of three degree correlations than the one previously reported analytical results reveal that annd has no correlations with respect to degree while clustering coefficient is dependent on degree showing a power law behavior as c k k 1 the obtained expressions are successfully contrasted with extensive numerical simulations

stability and error analysis of mixed finite volume methods for advection dominated problems abstract we consider a convection diffusion reaction problem and we analyze a stabilized mixed finite volume scheme introduced in 1 the scheme is presented in the format of discontinuous galerkin methods and error bounds are given proving o h convergence in the l 2 norm for the scalar variable which is approximated with piecewise constant elements

pseudofractal scale free web we find that scale free random networks are excellently modeled by simple deterministic graphs our graph has a discrete degree distribution degree is the number of connections of a vertex which is characterized by a power law with exponent gamma 1 ln 3 ln 2 properties of this compact structure are surprisingly close to those of growing random scale free networks with gamma in the most interesting region between 2 and 3 we succeed to find exactly and numerically with high precision all main characteristics of the graph in particular we obtain the exact shortest path length distribution for a large network ln n 1 the distribution tends to a gaussian of width approximately sqrt ln n centered at l approximately ln n we show that the eigenvalue spectrum of the adjacency matrix of the graph has a power law tail with exponent 2 gamma

a nonconforming finite element method for the biot s consolidation model in poroelasticity a stable finite element scheme that avoids pressure oscillations for a three field biot s model in poroelasticity is considered the involved variables are the displacements fluid flux darcy velocity and the pore pressure and they are discretized by using the lowest possible approximation order crouzeix raviart finite elements for the displacements lowest order raviart thomas nedelec elements for the darcy velocity and piecewise constant approximation for the pressure mass lumping technique is introduced for the raviart thomas nedelec elements in order to eliminate the darcy velocity and therefore reduce the computational cost we show convergence of the discrete scheme which is implicit in time and use these types of elements in space with and without mass lumping finally numerical experiments illustrate the convergence of the method and show its effectiveness to avoid spurious pressure oscillations when mass lumping for the raviart thomas nedelec elements is used

deterministic weighted scale free small world networks abstract we propose a deterministic weighted scale free small world model for considering pseudofractal web with the co evolution of topology and weight considering the fluctuations in traffic flow constitute a main reason for congestion of packet delivery and poor performance of communication networks we suggest a recursive algorithm to generate the network which restricts the traffic fluctuations on it effectively during the evolutionary process we provide a relatively complete view of topological structure and weight dynamics characteristics of the networks such as weight and strength distribution degree correlations average clustering coefficient and degree cluster correlations as well as the diameter

inverted berezinskii kosterlitz thouless singularity and high temperature algebraic order in an ising model on a scale free hierarchical lattice small world network we have obtained exact results for the ising model on a hierarchical lattice incorporating three key features characterizing many real world networks a scale free degree distribution a high clustering coefficient and the small world effect by varying the probability p of long range bonds the entire spectrum from an unclustered non small world network to a highly clustered small world system is studied using the self similar structure of the network we obtain analytic expressions for the degree distribution p k and clustering coefficient c for all p as well as the average path length ensuremath ell for p 0 and 1 the ferromagnetic ising model on this network is studied through an exact renormalization group transformation of the quenched bond probability distribution using up to 562 phantom rule 0 2em 0ex 500 renormalized probability bins to represent the distribution for pl0 494 we find power law critical behavior of the magnetization and susceptibility with critical exponents continuously varying with p and exponential decay of correlations away from t c for p ensuremath ge 0 494 in fact where the network exhibits small world character the critical behavior radically changes we find a highly unusual phase transition namely an inverted berezinskii kosterlitz thouless singularity between a low temperature phase with nonzero magnetization and finite correlation length and a high temperature phase with zero magnetization and infinite correlation length with power law decay of correlations throughout the phase approaching t c from below the magnetization and the susceptibility respectively exhibit the singularities of mathrm exp ensuremath c sqrt t c ensuremath t and mathrm exp d sqrt t c ensuremath t with c and d positive constants with long range bond strengths decaying with distance we see a phase transition with power law critical singularities for all p and evaluate an unusually narrow critical region and important corrections to power law behavior that depend on the exponent characterizing the decay of long range interactions