measures of centrality based on the spectrum of the laplacian we introduce a family of new centralities the k spectral centralities k spectral centrality is a measurement of importance with respect to the deformation of the graph laplacian associated with the graph due to this connection k spectral centralities have various interpretations in terms of spectrally determined information we explore this centrality in the context of several examples while for sparse unweighted networks 1 spectral centrality behaves similarly to other standard centralities for dense weighted networks they show different properties in summary the k spectral centralities provide a novel and useful measurement of relevance for single network elements as well as whole subnetworks distinct from other known measures

some rules on resistance distance with applications in this work some rules for resistance distances of a graph g are established let s be a set of vertices of g such that all vertices in s have the same neighborhood n in g s if s 2 3 4 simple formulae are derived to compute resistance distances between vertices in s in terms of the cardinality of n these show that resistance distances between vertices in s depend only on the cardinality of n and the induced subgraph g s one question arises naturally does this property hold for s with arbitrarily many vertices we answer this question by the following reduction principle resistance distances between vertices in s can be computed as in the subgraph obtained from g s n by deleting all the edges between vertices in n

dense output for extrapolation methods this paper is concerned with dense output formulas for extrapolation methods for ordinary differential equations in particular the extrapolated explicit euler method the gbs method for non stiff equations and the extrapolated linearly implicit euler method for stiff and differential algebraic equations are considered existence and uniqueness questions for dense output formulas are discussed and an algorithmic description for their construction is given several numerical experiments illustrate the theoretical results

robustness of heterogeneous complex networks abstract in this paper we study the robustness of heterogeneous preferential attachment networks the robustness of a network measures its structural tolerance to the random removal of nodes and links we numerically analyze the influence of the affinity parameters on a set of ensemble averaged robustness metrics we show that the presence of heterogeneity does not fundamentally alter the smooth nature of the fragmentation process of the models we also show that a moderate level of locality translates into slight improvements in the robustness metrics which prompts us to conjecture an evolutionary argument for the existence of real networks with power law scaling in their connectivity and clustering distributions

on smoothing and order reduction effects for implicit runge kutta formulae abstract it is well known that many important classes of runge kutta methods suffer an order reduction phenomenon when applied to certain classes of stiff problems in particular the s stage gauss methods with stage order s and order of consistency 2s behave like methods of order s when applied to the class of singularly perturbed problems in this paper we will show that the process of smoothing can ameliorate this effect when dealing with initial value problems by first studying the effect of smoothing on the standard prothero robinson problem and then by extending the analysis to the general class of singularly perturbed problems

modern convergence theory for stiff initial value problems abstract in this paper we give a brief review of available theoretical results about convergence and error structures for discretizations of stiff initial value problems we point out limitations of the various approaches and discuss some recent developments

approximating frustration scores in complex networks via perturbed laplacian spectra systems of many interacting components as found in physics biology infrastructure and the social sciences are often modeled by simple networks of nodes and edges the real world systems frequently confront outside intervention or internal damage whose impact must be predicted or minimized and such perturbations are then mimicked in the models by altering nodes or edges this leads to the broad issue of how to best quantify changes in a model network after some type of perturbation in the case of node removal there are many centrality metrics which associate a scalar quantity with the removed node but it can be difficult to associate the quantities with some intuitive aspect of physical behavior in the network this presents a serious hurdle to the application of network theory real world utility networks are rarely altered according to theoretic principles unless the kinetic impact on the network s users are fully appreciated beforehand in pursuit of a kinetically interpretable centrality score we discuss the f score or frustration score each f score quantifies whether a selected node accelerates or inhibits global mean first passage times to a second independently selected target node we show that this is a natural way of revealing the dynamical importance of a node in some networks after discussing merits of the f score metric we combine spectral and laplacian matrix theory in order to quickly approximate the exact f score values which can otherwise be expensive to compute following tests on both synthetic and real medium sized networks we report f score runtime improvements over exact brute force approaches in the range of 0 to 400 with low error 3

geometry of complex networks and topological centrality abstract we explore the geometry of complex networks in terms of an n dimensional euclidean embedding represented by the moore penrose pseudo inverse of the graph laplacian l the squared distance of a node i to the origin in this n dimensional space l i i yields a topological centrality index defined as c i 1 l i i in turn the sum of reciprocals of individual node centralities i 1 c i i l i i or the trace of l yields the well known kirchhoff index k an overall structural descriptor for the network to put into context this geometric definition of centrality we provide alternative interpretations of the proposed indices that connect them to meaningful topological characteristics first as forced detour overheads and frequency of recurrences in random walks that has an interesting analogy to voltage distributions in the equivalent electrical network and then as the average connectedness of i in all the bi partitions of the graph these interpretations respectively help establish the topological centrality c i of node i as a measure of its overall position as well as its overall connectedness in the network thus reflecting the robustness of i to random multiple edge failures through empirical evaluations using synthetic and real world networks we demonstrate how the topological centrality is better able to distinguish nodes in terms of their structural roles in the network and along with kirchhoff index is appropriately sensitive to perturbations re wirings in the network

an interactive multiple objective linear programming method for a class of underlying nonlinear utility functions this paper develops a method for interactive multiple objective linear programming assuming an unknown pseudo concave utility function satisfying certain general properties the method is an extension of our earlier method published in this journal zionts s wallenius j 1976 an interactive programming method for solving the multiple criteria problem management sci 22 6 652 663 various technical problems present in predecessor versions have been resolved in addition to presenting the supporting theory and algorithm we discuss certain options in implementation and summarize our practical experience with several versions of the method

communicability in complex networks we propose a new measure of the communicability of a complex network which is a broad generalization of the concept of the shortest path according to the new measure most of the real world networks display the largest communicability between the most connected popular nodes of the network assortative communicability there are also several networks with the disassortative communicability where the most popular nodes communicate very poorly to each other using this information we classify a diverse set of real world complex systems into a small number of universality classes based on their structure dynamic correlation in addition the new communicability measure is able to distinguish finer structures of networks such as communities into which a network is divided a community is unambiguously defined here as a set of nodes displaying larger communicability among them than to the rest of the nodes in the network

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

implicit explicit one leg methods for nonlinear stiff neutral equations abstract in this paper by adapting the underlying implicit explicit imex one leg methods cf 1 2 a class of extended imex one leg eieol methods are suggested for solving nonlinear stiff neutral equations snes it is proven under some suitable conditions that eieol methods are d convergent of order 2 and stable for nonlinear snes several numerical examples are given to testify the obtained theoretical results and the computational effectiveness of eieol methods moreover a comparison with the fully implicit one leg methods is presented which shows that eieol methods have the higher computational efficiency

operator splitting for chemical reaction systems with fast chemistry abstract splitting methods are frequently used for large scale chemical reaction systems the main reason is the reduced computational cost for solving the subproblems in comparison to the time integration of the full problem however a splitting error is introduced the most popular splitting schemes are the first order lie trotter splitting and the second order strang splitting however in case of stiff differential equations the strang splitting suffers from order reduction and both schemes have order one for stiff differential equations therefore the step size restrictions due to the low order can result in a prohibitive small step size hence splitting schemes with order larger than one are necessary for stiff differential equations the richardson extrapolation of the lie trotter splitting is a second order scheme in this paper we examine differential equations with a fast chemical source term and a slow transport term thus all stiffness of the system is related to the chemical source term we show that the richardson extrapolation does not suffer from order reduction in this case furthermore we perform a stability analysis of the extrapolated splitting scheme for a linear test problem thereby the operators of the test problem do not commute

netons vibrations of complex networks we consider atoms interacting with each other through the topological structure of ac omplex network and investigate lattice vibrations of the system the quanta of which we call netons for convenience the density of neton levels obtained numerically reveals that unlike a local regular lattice the system develops a gap of finite width manifesting extreme rigidity of the network structure at low energies two different network models the small world network and the scale free network are compared the characteristic structure of the former is described by an additional peak in the level density whereas a power law tail is observed in the latter indicating excitability of netons at arbitrarily high energies the gap width is also found to vanish in the small world network when the connection range r 1

on generalized trade off directions in nonconvex multiobjective optimization trade off information related to pareto optimal solutions is important in multiobjective optimization problems with conflicting objectives recently the concept of trade off directions has been introduced for convex problems these trade offs are characterized with the help of tangent cones generalized trade off directions for nonconvex problems can be defined by replacing convex tangent cones with nonconvex contingent cones here we study how the convex concepts and results can be generalized into a nonconvex case giving up convexity naturally means that we need local instead of global analysis

a vibrational approach to node centrality and vulnerability in complex networks we propose a new measure of vulnerability of a node in a complex network the measure is based on the analogy in which the nodes of the network are represented by balls and the links are identified with springs we define the measure as the node displacement or the amplitude of vibration of each node under fluctuation due to the thermal bath in which the network is supposed to be submerged we prove exact relations among the thus defined node displacement the information centrality and the kirchhoff index the relation between the first two suggests that the node displacement has a better resolution of the vulnerability than the information centrality because the latter is the sum of the local node displacement and the node displacement averaged over the entire network

asymptotic error expansions for stiff equations an analysis for the implicit midpoint and trapezoidal rules in the strongly stiff case the structure of the global discretization error is studied for the implicit midpoint and trapezoidal rules applied to nonlinearstiff initial value problems the point is that in general the global error contains nonsmooth oscillating terms at the dominanth 2 level however it is shown in the present paper that for special classes of stiff problems these nonsmooth terms contain an additional factor where 1 is the magnitude of the stiff eigenvalues in these cases a full asymptotic error expansion exists in thestrongly stiff case sufficiently small compared to the stepsizeh the general case where the oscillating error components areo h 2 and noto h 2 and applications of our results extrapolation and defect correction algorithims will be studied in separate papers

average resistance of toroidal graphs abstract the average effective resistance of a graph is a relevant performance index in many applications including distributed estimation and control of network systems in this paper we study how the average resistance depends on the graph topology and specifically on the dimension of the graph we concentrate on d dimensional toroidal grids and we exploit the connection between resistance and laplacian eigenvalues our analysis provides tight estimates of the average resistance which are key to study its asymptotic behavior when the number of nodes grows to infinity in dimension two the average resistance diverges in this case we are able to capture its rate of growth when the sides of the grid grow at different rates in higher dimensions the average resistance is bounded uniformly in the number of nodes in this case we conjecture that its value is of order 1 d for large d we prove this fact for hypercubes and when the side lengths go to infinity

trade off rates in the hyperplane method for multiobjective optimization problems abstract recently as a generalization of the well known scalarizing methods for characterizing pareto optimal solutions of multiobjective optimization problems a fully integrated scalarizing method called the hyperplane method has been introduced by the authors in this paper a meaningful formula which relates the trade off rates within the pareto optimal solution set to the lagrange multipliers of the hyperplane problems is derived by realizing that trade off information is significantly useful in almost all the interactive multiobjective decision making methods moreover it is shown that such a formula can be regarded as a generalized version of the trade off rate formulae for all of the well known scalarizing problems finally various aspects of the results developed in this paper are illustrated by a numerical example

asymptotic error expansions for stiff equations the implicit euler scheme this paper discusses the existence of an asymptotic expansion for the global error of the implicit euler scheme applied to stiff nonlinear systems of ordinary differential equations it is shown that in strongly stiff situations a full asymptotic expansion exists at all gridpoints for the mildly stiff case it is shown that the full order of the remainder term which inevitably breaks down at the first gridpoints after a stepsize change reappears at the subsequent gridpoints our analysis is based on singular perturbation techniques

spectral partitioning works planar graphs and finite element meshes abstract spectral partitioning methods use the fiedler vector the eigenvector of the second smallest eigenvalue of the laplacian matrix to find a small separator of a graph these methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extremely well in this paper we show that spectral partitioning methods work well on bounded degree planar graphs and finite element meshes the classes of graphs to which they are usually applied while naive spectral bisection does not necessarily work we prove that spectral partitioning techniques can be used to produce separators whose ratio of vertices removed to edges cut is o n for bounded degree planar graphs and two dimensional meshes and o n1 d for well shaped d dimensional meshes the heart of our analysis is an upper bound on the second smallest eigenvalues of the laplacian matrices of these graphs we prove a bound of o 1 n for bounded degree planar graphs and o 1 n2 d for well shaped d dimensional meshes

extrapolation at stiff differential equations asymptotic expansions of the global error of numerical methods are well understood if the differential equation is non stiff this paper is concerned with such expansions for the implicit euler method the linearly implicit euler method and the linearly implicit mid point rule when they are applied tostiff differential equations in this case perturbation terms are present whose dominant one is given explicitly this permits us to better understand the behaviour ofextrapolation methods at stiff differential equations numerical examples supporting the theoretical results are included

communicability graph and community structures in complex networks we use the concept of the network communicability e estrada n hatano communicability in complex networks phys rev e 77 2008 036111 to define communities in a complex network the communities are defined as the cliques of a communicability graph which has the same set of nodes as the complex network and links determined by the communicability function then the problem of finding the network communities is transformed to an all clique problem of the communicability graph we discuss the efficiency of this algorithm of community detection in addition we extend here the concept of the communicability to account for the strength of the interactions between the nodes by using the concept of inverse temperature of the network finally we develop an algorithm to manage the different degrees of overlapping between the communities in a complex network we then analyze the usa airport network for which we successfully detect two big communities of the eastern airports and of the western central airports as well as two bridging central communities in striking contrast a well known algorithm groups all but two of the continental airports into one community

the concept of b convergence a useful concept of convergence for nonlinear stiff ode s will be developed which permits the derivation of uniform global error bounds independent of the stiffness of the considered problem this concept will be discussed for three simple methods for stiff systems the implicit euler scheme the implicit midpoint rule and the implicit trapezoidal rule

tradeoff directions in multiobjective optimization problems multiobjective optimization problems mop typically have conflicting objectives wherein a gain in one objective is at the expense of another tradeoff directions which measure the change in some objectives relative to changes in others provide important information as to the best direction of improvement from the current solution in this paper we present a general definition of tradeoffs as a cone of directions and provide a general method of calculating tradeoffs at every pareto optimal point of a convex mop this extends current definitions of tradeoffs which assume certain conditions on the feasible set and the objective functions two comprehensive numerical examples are provided to illustrate the tradeoff directions and the methods used to calculate them

extrapolated implicit explicit time stepping this paper constructs extrapolated implicit explicit time stepping methods that allow one to efficiently solve problems with both stiff and nonstiff components the proposed methods are based on euler steps and can provide very high order discretizations of odes index 1 daes and pdes in the method of lines framework implicit explicit schemes based on extrapolation are simple to construct easy to implement and straightforward to parallelize this work establishes the existence of perturbed asymptotic expansions of global errors explains the convergence orders of these methods and studies their linear stability properties numerical results with stiff ode dae and pde test problems confirm the theoretical findings and illustrate the potential of these methods to solve multiphysics multiscale problems

a semi implicit mid point rule for stiff systems of ordinary differential equations the paper introduces a new semi implicit extrapolation method especially designed for the numerical solution of stiff systems of ordinary differential equations the existence of a quadratic asymptotic expansion in terms of the stepsize is shown moreover the new discretization is analyzed in the light of well known stability models the efficiency of the new integrator is clearly demonstrated by solving a series of challenging test problems including real life examples