distance dissimilarity index and network community structure we address the question of finding the community structure of a complex network in an earlier effort h zhou phys rev e 67 041908 2003 the concept of network random walking is introduced and a distance measure defined here we calculate based on this distance measure the dissimilarity index between nearest neighboring vertices of a network and design an algorithm to partition these vertices into communities that are hierarchically organized each community is characterized by an upper and a lower dissimilarity threshold the algorithm is applied to several artificial and real world networks and excellent results are obtained in the case of artificially generated random modular networks this method outperforms the algorithm based on the concept of edge betweenness centrality for yeast s protein protein interaction network we are able to identify many clusters that have well defined biological functions

scaling properties of random walks on small world networks using both numerical simulations and scaling arguments we study the behavior of a random walker on a one dimensional small world network for the properties we study we find that the random walk obeys a characteristic scaling form these properties include the average number of distinct sites visited by the random walker the mean square displacement of the walker and the distribution of first return times the scaling form has three characteristic time regimes at short times the walker does not see the small world shortcuts and effectively probes an ordinary euclidean network in d dimensions at intermediate times the properties of the walker shows scaling behavior characteristic of an infinite small world network finally at long times the finite size of the network becomes important and many of the properties of the walker saturate we propose general analytical forms for the scaling properties in all three regimes and show that these analytical forms are consistent with our numerical simulations

pointwise rates of convergence for the oliker prussner method for the monge ampere equation we study the oliker prussner method exploiting its geometric nature we derive discrete stability and continuous dependence estimates in the max norm by using a discrete alexandroff estimate and the brunn minkowski inequality we show that the method is exact for all convex quadratic polynomials provided the underlying set of nodes is translation invariant within the domain nodes still conform to the domain boundary this gives a suitable notion of operator consistency which combined with stability leads to pointwise rates of convergence for classical and non classical solutions of the monge ampere equation

random spread on the family of small world networks we present analytical and numerical results of a random walk on the family of small world graphs the average access time shows a crossover from regular to random behavior with increasing distance from the starting point of the random walk we introduce an independent step approximation which enables us to obtain analytic results for the average access time we observe a scaling relation for the average access time in the degree of the nodes the behavior of the average access time as a function of p shows striking similarity with that of the characteristic length of the graph this observation may have important applications in routing and switching in networks with a large number of nodes

random walks in modular scale free networks with multiple traps extensive empirical investigation has shown that a plethora of real networks synchronously exhibit scale free and modular structure and it is thus of great importance to uncover the effects of these two striking properties on various dynamical processes occurring on such networks in this paper we examine two cases of random walks performed on a class of modular scale free networks with multiple traps located at several given nodes we first derive a formula of the mean first passage time mfpt for a general network which is the mean of the expected time to absorption originating from a specific node averaged over all nontrap starting nodes although the computation is complex the expression of the formula is exact moreover the computational approach and procedure are independent of the number and position of the traps we then determine analytically the mfpt for the two random walks being considered the obtained analytical results are in complete agreement with the numerical ones our results show that the number and location of traps play an important role in the behavior of the mfpt since for both cases the mfpt grows as a power law function of the number of nodes but their exponents are quite different we demonstrate that the root of the difference in the behavior of mfpt is attributed to the modular and scale free topologies of the networks this work can deepen the understanding of diffusion on networks with modular and scale free architecture and motivate relevant studies for random walks running on complex random networks with multiple traps

a rate of convergence for monotone finite difference approximations to fully nonlinear uniformly elliptic pdes we obtain an algebraic rate of convergence for monotone and consistent finite difference approximations to lipschitz continuous viscosity solutions of uniformly elliptic partial differential equations 2007 wiley periodicals inc

constrained spin dynamics description of random walks on hierarchical scale free networks we study a random walk problem on the hierarchical network which is a scale free network grown deterministically the random walk problem is mapped onto a dynamical ising spin chain system in one dimension with a nonlocal spin update rule which allows an analytic approach we show analytically that the characteristic relaxation time scale grows algebraically with the total number of nodes n as t n z from a scaling argument we also show the power law decay of the autocorrelation function c sigma t t alpha which is the probability to find the ising spins in the initial state sigma after t time steps with the state dependent nonuniversal exponent alpha it turns out that the power law scaling behavior has its origin in a quasiultrametric structure of the configuration space

a new three step fixed point iteration scheme for asymptotically nonexpansive mappings in the present paper we define and study a new three step iterative scheme inspired by suantai j math anal appl 311 2005 506 517 this scheme includes many well known iterations for examples modified mann type modified ishikawa type iterative schemes and the three step iterative scheme of xu and noor several convergence theorems of this scheme are established for asymptotically nonexpansive mappings our results extend and improve the recent ones announced by schu j math anal appl 158 1991 407 413 bull aust math soc 43 1991 153 159 xu and noor j math anal appl 267 2002 444 453 suantai j math anal appl 311 2005 506 517 and many others a misleading conclusion of theorem 2 6 in suantai s paper is also corrected

recent developments in numerical methods for fully nonlinear second order partial differential equations this article surveys the recent developments in computational methods for second order fully nonlinear partial differential equations pdes a relatively new subarea within numerical pdes due to their ever increasing importance in mathematics itself e g differential geometry and pdes and in many scientific and engineering fields e g astrophysics geostrophic fluid dynamics grid generation image processing optimal transport meteorology mathematical finance and optimal control numerical solutions to fully nonlinear second order pdes have garnered a great deal of interest from the numerical pde and scientific communities significant progress has been made for this class of problems in the past few years but many problems still remain open this article intends to introduce these current advancements and new results to the siam community and generate more interest in numerical methods for fully nonlinear pdes

random walks on fractals and stretched exponential relaxation stretched exponential relaxation exp t tau beta k is observed in a large variety of systems but has not been explained so far studying random walks on percolation clusters in curved spaces whose dimensions range from 2 to 7 we show that the relaxation is accurately a stretched exponential and is directly connected to the fractal nature of these clusters thus we find that in each dimension the decay exponent beta k is related to well known exponents of the percolation theory in the corresponding flat space we suggest that the stretched exponential behavior observed in many complex systems polymers colloids glasses is due to the fractal character of their configuration space

effect of trap position on the efficiency of trapping in treelike scale free networks the conventional wisdom is that the role and impact of nodes on dynamical processes in scale free networks are not homogenous because of the presence of highly connected nodes at the tail of their power law degree distribution in this paper we explore the influence of different nodes as traps on the trapping efficiency of the trapping problem taking place on scale free networks to this end we study in detail the trapping problem in two families of deterministically growing scale free networks with treelike structure one family is non fractal the other is fractal in the first part of this work we attack a special case of random walks on the two network families with a perfect trap located at a hub i e node with the highest degree the second study addresses the case with trap distributed uniformly over all nodes in the networks for these two cases we compute analytically the mean trapping time mtt a quantitative indicator characterizing the trapping efficiency of the trapping process we show that in the non fractal scale free networks the mtt for both cases follows different scalings with the network order number of network nodes implying that trap s position has a significant effect on the trapping efficiency in contrast it is presented that for both cases in the fractal scale free networks the two leading scalings exhibit the same dependence on the network order suggesting that the location of trap has no essential impact on the trapping efficiency we also show that for both cases of the trapping problem the trapping efficiency is more efficient in the non fractal scale free networks than in their fractal counterparts

weak and strong convergence theorems for three step iterations with errors for asymptotically nonexpansive mappings abstract in the present paper several weak and strong convergence theorems are established for the three step iterative schemes with errors for asymptotically nonexpansive mappings our results extend and improve the recent ones announced by tan and xu xu and noor and many others

relaxation properties of small world networks recently watts and strogatz introduced the so called small world networks in order to describe systems that combine simultaneously properties of regular and random lattices in this work we study diffusion processes defined on such structures by considering explicitly the probability for a random walker to be present at the origin the results are intermediate between the corresponding ones for fractals and cayley trees

error bounds for monotone approximation schemes for parabolic hamilton jacobi bellman equations we obtain nonsymmetric upper and lower bounds on the rate of convergence of general monotone approximation numerical schemes for parabolic hamilton jacobi bellman equations by introducing a new notion of consistency our results are robust and general they improve and extend earlier results by krylov barles and jakobsen we apply our general results to various schemes including crank nicholson type finite difference schemes splitting methods and the classical approximation by piecewise constant controls in the first two cases our results are new and in the last two cases the results are obtained by a new method which we develop here

two approximations of solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games the associated initial value problems almost never have global time classical solutions and one must deal with suitable generalized solutions the correct class of generalized solutions has only recently been established by the authors this article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates analogous results are proved by similar means for the method of vanishing viscosity author

search in power law networks many communication and social networks have power law link distributions containing a few nodes that have a very high degree and many with low degree the high connectivity nodes play the important role of hubs in communication and networking a fact that can be exploited when designing efficient search algorithms we introduce a number of local search strategies that utilize high degree nodes in power law graphs and that have costs scaling sublinearly with the size of the graph we also demonstrate the utility of these strategies on the gnutella peer to peer network

ultradiffusion the relaxation of hierarchical systems the authors show that the dynamics of relaxation in hierarchical structures leads to an anomalous decay process which they term ultradiffusion using renormalisation group techniques they explicitly calculate the behaviour of the autocorrelation function in a one dimensional hierarchical system and show its universal power law decay as a function of temperature the authors also demonstrate analytically the emergence of a hierarchy of time scales this phenomenon is relevant to systems ranging from macromolecules to computing structures

a local discontinuous galerkin method for directly solving hamilton jacobi equations in this paper we propose a new local discontinuous galerkin method to directly solve hamilton jacobi equations the scheme is a natural extension of the monotone scheme for the linear case with constant coefficients the method is equivalent to the discontinuous galerkin method for conservation laws thus stability and error analysis are obtained under the framework of conservation laws for both convex and noneconvex hamiltonian optimal k 1 th order of accuracy for smooth solutions are obtained with piecewise kth order polynomial approximations the scheme is numerically tested on a variety of one and two dimensional problems the method works well to capture sharp corners discontinuous derivatives and have the solution converges to the viscosity solution

trapping in scale free networks with hierarchical organization of modularity a wide variety of real life networks share two remarkable generic topological properties scale free behavior and modular organization and it is natural and important to study how these two features affect the dynamical processes taking place on such networks in this paper we investigate a simple stochastic process trapping problem a random walk with a perfect trap fixed at a given location performed on a family of hierarchical networks that exhibit simultaneously striking scale free and modular structure we focus on a particular case with the immobile trap positioned at the hub node having the largest degree using a method based on generating functions we determine explicitly the mean first passage time mfpt for the trapping problem which is the mean of the node to trap first passage time over the entire network the exact expression for the mfpt is calculated through the recurrence relations derived from the special construction of the hierarchical networks the obtained rigorous formula corroborated by extensive direct numerical calculations exhibits that the mfpt grows algebraically with the network order concretely the mfpt increases as a power law function of the number of nodes with the exponent much less than 1 we demonstrate that the hierarchical networks under consideration have more efficient structure for transport by diffusion in contrast with other analytically soluble media including some previously studied scale free networks we argue that the scale free and modular topologies are responsible for the high efficiency of the trapping process on the hierarchical networks

efficient sampling of complex network with modified random walk strategies abstract we present two novel random walk strategies choosing seed node csn random walk and no retracing nr random walk different from the classical random walk sampling the csn and nr strategies focus on the influences of the seed node choice and path overlap respectively three random walk samplings are applied in the erdos renyi er barabasi albert ba watts strogatz ws and the weighted usair networks respectively then the major properties of sampled subnets such as sampling efficiency degree distributions average degree and average clustering coefficient are studied the similar conclusions can be reached with these three random walk strategies firstly the networks with small scales and simple structures are conducive to the sampling secondly the average degree and the average clustering coefficient of the sampled subnet tend to the corresponding values of original networks with limited steps and thirdly all the degree distributions of the subnets are slightly biased to the high degree side however the nr strategy performs better for the average clustering coefficient of the subnet in the real weighted usair networks some obvious characters like the larger clustering coefficient and the fluctuation of degree distribution are reproduced well by these random walk strategies

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

laplacian spectra of and random walks on complex networks are scale free architectures really important we study the laplacian operator of an uncorrelated random network and as an application consider hopping processes diffusion random walks signal propagation etc on networks we develop a strict approach to these problems we derive an exact closed set of integral equations which provide the averages of the laplacian operator s resolvent this enables us to describe the propagation of a signal and random walks on the network we show that the determining parameter in this problem is the minimum degree q m of vertices in the network and that the high degree part of the degree distribution is not that essential the position of the lower edge of the laplacian spectrum c appears to be the same as in the regular bethe lattice with the coordination number q m namely c 0 if q m 2 and c 0 if q m 2 in both of these cases the density of eigenvalues p 0 as c 0 but the limiting behaviors near c are very different in terms of a distance from a starting vertex the hopping propagator is a steady moving gaussian broadening with time this picture qualitatively coincides with that for a regular bethe lattice our analytical results include the spectral density p near c and the long time asymptotics of the autocorrelator and the propagator

convergent semi lagrangian methods for the monge ampere equation on unstructured grids this paper is concerned with developing and analyzing convergent semi lagrangian methods for the fully nonlinear elliptic monge ampere equation on general triangular grids this is done by establishing an equivalent in the viscosity sense hamilton jacobi bellman formulation of the monge ampere equation a significant benefit of the reformulation is the removal of the convexity constraint from the admissible space as convexity becomes a built in property of the new formulation moreover this new approach allows one to tap the wealthy numerical methods such as semi lagrangian schemes for hamilton jacobi bellman equations to solve monge ampere type equations it is proved that the considered numerical methods are monotone pointwise consistent and uniformly stable consequently its solutions converge uniformly to the unique convex viscosity solution of the monge ampere dirichlet problem a superlinearly convergent howard s algorithm which is a newton type method is utilized as the nonlinear

explicit determination of mean first passage time for random walks on deterministic uniform recursive trees the determination of mean first passage time mfpt for random walks in networks is a theoretical challenge and is a topic of considerable recent interest within the physics community in this paper according to the known connections between mfpt effective resistance and the eigenvalues of graph laplacian we first study analytically the mfpt between all node pairs of a class of growing treelike networks which we term deterministic uniform recursive trees durts since one of its particular cases is a deterministic version of the famous uniform recursive tree the interesting quantity is determined exactly through the recursive relation of the laplacian spectra obtained from the special construction of durts the analytical result shows that the mfpt between all couples of nodes in durts varies as n text text ln text n for large networks with node number n second we study trapping on a particular network of durts focusing on a special case with the immobile trap positioned at a node having largest degree we determine exactly the average trapping time att that is defined as the average of fpt from all nodes to the trap in contrast to the scaling of the mfpt the leading behavior of att is a linear function of n interestingly we show that the behavior for att of the trapping problem is related to the trapping location which is in comparison with the phenomenon of trapping on fractal t graph although both networks exhibit tree structure finally we believe that the methods could open the way to exactly calculate the mfpt and att in a wide range of deterministic media

convergent finite difference methods for one dimensional fully nonlinear second order partial differential equations this paper develops a new framework for designing and analyzing convergent finite difference methods for approximating both classical and viscosity solutions of second order fully nonlinear partial differential equations pdes in 1 d the goal of the paper is to extend the successful framework of monotone consistent and stable finite difference methods for first order fully nonlinear hamilton jacobi equations to second order fully nonlinear pdes such as monge ampere and bellman type equations new concepts of consistency generalized monotonicity and stability are introduced among them the generalized monotonicity and consistency which are easier to verify in practice are natural extensions of the corresponding notions of finite difference methods for first order fully nonlinear hamilton jacobi equations the main component of the proposed framework is the concept of a numerical operator and the main idea used to design consistent generalized monotone and stable finite difference methods is the concept of a numerical moment these two new concepts play the same roles the numerical hamiltonian and the numerical viscosity play in the finite difference framework for first order fully nonlinear hamilton jacobi equations in the paper two classes of consistent and monotone finite difference methods are proposed for second order fully nonlinear pdes the first class contains lax friedrichs like methods which also are proved to be stable and the second class contains godunov like methods numerical results are also presented to gauge the performance of the proposed finite difference methods and to validate the theoretical results of the paper

finite element approximations of general fully nonlinear second order elliptic partial differential equations based on the vanishing moment method the vanishing moment method was introduced by the authors in feng and neilan 2009 as a reliable methodology for computing viscosity solutions of fully nonlinear second order partial differential equations pdes it is based on the simple idea of approximating a hard to handle fully nonlinear second order pde by a family parameterized by a small parameter e of easy to handle quasilinear fourth order pdes the primary objective of this article is to present a comprehensive finite element analysis for the vanishing moment approximation of general fully nonlinear second order elliptic pdes which fulfill some structure conditions abstract methodological and convergence analysis frameworks of conforming finite element methods are first developed for fully nonlinear second order pdes in a general setting the abstract framework is then applied to three prototypical nonlinear equations namely the monge ampere equation the equation of prescribed gauss curvature and the infinity laplacian equation numerical experiments are presented for each problem to validate the theoretical error estimate results and to gauge the efficiency of the proposed numerical methods and the vanishing moment methodology

numerical methods for fully nonlinear elliptic equations of the monge ampere type abstract in this article we discuss the numerical solution of the dirichlet problem for the monge ampere equation in two dimensions the solution of closely related problems is also discussed these include a family of pucci s equations the equation prescribing the harmonic mean of the eigenvalues of the hessian of a smooth function of two variables and a minimization problem from nonlinear elasticity where the cost functional involves the determinant of the gradient of vector valued functions to solve the monge ampere equation we consider two methods the first one reduces the monge ampere equation to a saddle point problem for a well chosen augmented lagrangian to solve this saddle point problem we advocate an uzawa douglas rachford algorithm the second method combines nonlinear least squares and operator splitting this second method being simpler to implements we apply variants of it to the solution of the other problems for the space discretization we use mixed finite element approximations closely related to methods already used for the solution of linear and nonlinear bi harmonic problems through these approximations the solution of the above problems is essentially reduced to the solution of discrete poisson problems the methods discussed in this article are validated by the results of numerical experiments

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

mean first passage times in confined media from markovian to non markovian processes we review recent theoretical works that enable the accurate evaluation of the mean first passage time mfpt of a random walker to a target in confinement for markovian memory less and non markovian walkers for the markovian problem we present a general theory which allows one to accurately evaluate the mfpt and its extensions to related first passage observables such as splitting probabilities and occupation times we show that this analytical approach provides a universal scaling dependence of the mfpt on both the volume of the confining domain and the source target distance in the case of general scale invariant processes this analysis is applicable to a broad range of stochastic processes characterized by length scale invariant properties and reveals the key role that can be played by the starting position of the random walker we then present an extension to non markovian walks by taking the specific example of a tagged monomer of a polymer chain looking for a target in confinement we show that the mfpt can be calculated accurately by computing the distribution of the positions of all the monomers in the chain at the instant of reaction such a theory can be used to derive asymptotic relations that generalize the scaling dependence with the volume and the initial distance to the target derived for markovian walks finally we present an application of this theory to the problem of the first contact time between the two ends of a polymer chain and review the various theoretical approaches of this nonmarkovian problem

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

discontinuous galerkin finite element differential calculus and applications to numerical solutions of linear and nonlinear partial differential equations this paper develops a discontinuous galerkin dg finite element differential calculus theory for approximating weak derivatives of sobolev functions and piecewise sobolev functions by introducing numerical one sided derivatives as building blocks various first and second order numerical operators such as the gradient divergence hessian and laplacian operator are defined and their corresponding calculus rules are established among the calculus rules are product and chain rules integration by parts formulas and the divergence theorem approximation properties and the relationship between the proposed dg finite element numerical derivatives and some well known finite difference numerical derivative formulas on cartesian grids are also established besides independent interest in numerical differentiation the primary motivation and goal of developing the dg finite element differential calculus is to solve partial differential equations it is shown that several existing finite element finite difference and discontinuous galerkin methods can be rewritten compactly using the proposed dg finite element differential calculus framework moreover new discontinuous galerkin methods for linear and nonlinear pdes are also obtained from the framework

degree dependent intervertex separation in complex networks we study the mean length l k of the shortest paths between a vertex of degree k and other vertices in growing networks where correlations are essential in a number of deterministic scale free networks we observe a power law correction to a logarithmic dependence l k a ln n k gamma 1 2 ck gamma 1 n in a wide range of network sizes here n is the number of vertices in the network gamma is the degree distribution exponent and the coefficients a and c depend on a network we compare this law with a corresponding l k dependence obtained for random scale free networks growing through the preferential attachment mechanism in stochastic and deterministic growing trees with an exponential degree distribution we observe a linear dependence on degree l k approximately a ln n ck we compare our findings for growing networks with those for uncorrelated graphs

mixed finite element methods for the fully nonlinear monge ampere equation based on the vanishing moment method this paper studies mixed finite element approximations of the viscosity solution to the dirichlet problem for the fully nonlinear monge ampere equation det d 2u 0 f 0 based on the vanishing moment method which was proposed recently by the authors in x feng and m neilan j scient comp doi 10 1007 s10915 008 9221 9 2008 in this approach the second order fully nonlinear monge ampere equation is approximated by the fourth order quasilinear equation varepsilon delta 2 u varepsilon det d 2u varepsilon f it was proved in x feng trans ams submitted that the solution u varepsilon converges to the unique convex viscosity solution u 0 of the dirichlet problem for the monge ampere equation this result then opens a door for constructing convergent finite element methods for the fully nonlinear second order equations a task which has been impracticable before the goal of this paper is threefold first we develop a family of hermann miyoshi type mixed finite element methods for approximating the solution u varepsilon of the regularized fourth order problem which computes simultaneously u varepsilon and the moment tensor sigma varepsilon d 2u varepsilon second we derive error estimates which track explicitly the dependence of the error constants on the parameter varepsilon for the errors u varepsilon u varepsilon h and sigma 0 sigma h varepsilon finally we present a detailed numerical study on the rates of convergence in terms of powers of varepsilon for the error u 0 u h varepsilon and sigma varepsilon sigma h varepsilon and numerically examine what is the best mesh size h in relation to varepsilon in order to achieve these rates due to the strong nonlinearity of the underlying equation the standard perturbation argument for error analysis of finite element approximations of nonlinear problems does not work for the problem to overcome the difficulty we employ a fixed point technique which strongly relies on the stability of the linearized problem and its mixed finite element approximations

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

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

scaling of random spreading in small world networks in this study we have carried out computer simulations of random walks on watts strogatz type small world networks and measured the mean number of visited sites and the return probabilities these quantities were found to obey scaling behavior with intuitively reasoned exponents as long as the probability p of having a long range bond was sufficiently low

discrete methods for fully nonlinear elliptic equations this paper exhibits and proves the stability of discrete approximations for the solution of the dirichlet problem for fully nonlinear uniformly elliptic partial differential equations in situations where classical solutions are not known to exist the resulting solutions are characterized in the viscosity sense of crandall and lions traps amer math soc 177 1983 pp 1 42 and our stability analysis depends on estimates for nonlinear difference equations of positive type which are consequences of earlier work on linear difference equations with random coefficients