eigenvalues and degree deviation in graphs abstract let g be a graph with n vertices and m edges and let g 1 g n g be the eigenvalues of its adjacency matrix set s g u v g d u 2 m n we prove that s 2 g 2 n 2 2 m g 2 m n s g in addition we derive similar inequalities for bipartite g we also prove that the inequality k g n k 2 g 1 2 2 s g holds for every k 2 n finally we prove that for every graph g of order n n g n g 1 s 2 g 2 n 3 we show that these inequalities are tight up to a constant factor

world wide web scaling exponent from simon s 1955 model the statistical properties of the world wide web have attracted considerable attention recently since self similar regimes were first observed in the scaling of its link structure one characteristic quantity is the number of in links k that point to a particular web page its probability distribution p k shows a pronounced power law scaling p k approximately k gamma that is not readily explained by standard random graph theory here we recall a simple and elegant model for scaling phenomena in general copy and growth processes as proposed by simon in 1955 when combined with an experimental measurement of network growth in the world wide web this classical model is able to model the in link dynamics and predicts the scaling exponent gamma 2 1 in accordance with observation

a bound on the spectral radius of graphs with e edges the spectral radius p a of the adjacency matrix a of a graph g with e edges satisfiesp a i l equalityoccnrsifandonlyif e i andgisa disjoint union of the complete graph k and isolated vertices 0

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

weighted estimators for proportional hazards regression with missing covariates missing covariate data are common in epidemiologic studies and disease prevention trials in this article regression parameter estimation in the cox proportional hazards model is considered when certain covariates are observed for all study subjects and other covariate data are collected only for a subset the article presents both simple weighted and kernel assisted fully augmented weighted estimators that use the partially incomplete data nonparametrically we use nonparametric methods to estimate selection probabilities in the simple weighted estimating functions we also use nonparametric kernel smoothing techniques to estimate certain conditional expectations in fully augmented weighted estimating functions the proposed methods are nonparametric in the sense that they require neither a model for the missing data mechanism nor specification of the conditional distribution of missing covariates given observed covariates these estimators allow the missing data mechanism to depend on outcome variables

optimal symmetric networks in terms of minimizing average shortest path length and their sub optimal growth model abstract homogeneous entangled networks characterized by small world large girths and no community structure have attracted much attention due to some of their favorable performances however the optimization algorithm proposed by donetti et al is very time consuming and will lose its efficiency when the size of the target network becomes large in this paper an alternative optimization algorithm is provided to get optimal symmetric networks by minimizing the average shortest path length it is shown that the synchronizability of a symmetric network is enhanced when the average shortest path length of the network is shortened as the optimization proceeds which suggests that the optimal symmetric networks in terms of minimizing average shortest path length will be very close to those entangled networks in order to overcome the time consuming obstacle of the optimization algorithms proposed by us and donetti et al a growth model is proposed to get large scale sub optimal symmetric networks numerical simulations show that the symmetric networks derived by our growth model will have small world property and besides these networks will have many other similar favorable performances as entangled networks e g robustness against errors and attacks very good load balancing ability and strong synchronizability

analytical relationships between metric and centrality measures of a network and its dual the centrality and efficiency measures of a network g are strongly related to the respective measures on the dual g and the bipartite b g associated networks we show some relationships between the bonacich centralities c g c g and c b g and between the efficiencies e g and e g and we compute the behavior of these parameters in some examples

proportional hazards regression with missing covariates abstract nonparametric maximum likelihood npml is used to estimate regression parameters in a proportional hazards regression model with missing covariates the npml estimator is shown to be consistent and asymptotically normally distributed under some conditions em type algorithms are applied to solve the maximization problem variance estimates of the regression parameters are obtained by a profile likelihood approach that uses em aided numerical differentiation simulation results indicate that the npml estimates of the regression parameters are more efficient than the approximate partial likelihood estimates and estimates from complete case analysis when missing covariates are missing completely at random and that the proposed method corrects for bias when the missing covariates are missing at random

likelihood based methods for missing covariates in the cox proportional hazards model problems associated with missing covariate data are well known but often ignored we present a method for estimating the parameters in the cox proportional hazards model when the missing data are missing at random mar and censoring is noninformative due to the computational burden of this method we introduce an approximation that allows us to use a weighted expectation maximization em algorithm to estimate the parameters more easily when the missing covariates are continuous rather than categorical we implement a monte carlo version of the em algorithm along with the gibbs sampler to obtain parameter estimates we also give the asymptotic distribution of these estimates the primary advantage of this method over complete case analysis is that it produces more efficient parameter estimates and corrects for bias in the mar setting to motivate the methodology we present an analysis of a phase iii melanoma clinical trial conducted by the eastern cooperative oncology group

reweighting estimators for cox regression with missing covariates missingness in covariates is a common problem in survival data in this article we propose a reweighting method for estimating the regression parameters in the cox model with missing covariates we also consider the augmented reweighting method by subtracting the projection term onto the nuisance tangent space the proposed method provides consistent and asymptotically normally distributed estimators when the missing data mechanism depends on the outcome variables as well as on the observed covariates with either monotone or arbitrary nonmonotone missingness patterns simulation results indicate that in most situations the proposed reweighting estimators are more efficient than the inverse probability weighting estimators for the regression coefficients of the missing covariates and are as efficient as or more efficient than the inverse probability weighting estimators for the regression coefficients of the completely observed covariates the simple reweighting estimators can be easily implemented in st

vulnerability of labeled networks abstract we propose a metric for vulnerability of labeled graphs that has the following two properties 1 when the labeled graph is considered as an unlabeled one the metric reduces to the corresponding metric for an unlabeled graph and 2 the metric has the same value for differently labeled fully connected graphs reflecting the notion that any arbitrarily labeled fully connected topology is equally vulnerable as any other a vulnerability analysis of two real world networks the power grid of the european union and an autonomous system network has been performed the networks have been treated as graphs with node labels the analysis consists of calculating characteristic path lengths between labels of nodes and determining largest connected cluster size under two node and edge attack strategies results obtained are more informative of the networks vulnerability compared to the case when the networks are modeled with unlabeled graphs

double semiparametric method for missing covariates in cox regression models the problem of nuisance covariate model specification is considered in cox regression where the maximum semiparametric likelihood method is used to handle the missing covariates a component of the covariates is modeled nonparametrically to achieve robustness against covariate model misspecification and to reduce the number of possibly intractable integrations involved in the parametric modeling of the covariates the statistical properties of the proposed method are examined it is found that in some important situations the maximum semiparametric likelihood can be applied without making any additional parametric model assumptions on covariates the proposed method can yield a more efficient estimator than the nonparametric imputation methods and does not require specification of the missingness mechanism when compared with the inverse probability weighting method a real data example is analyzed to demonstrate use of the proposed method

preferential attachment with information filtering node degree probability distribution properties abstract a network growth mechanism based on a two step preferential rule is investigated as a model of network growth in which no global knowledge of the network is required in the first filtering step a subset of fixed size m of existing nodes is randomly chosen in the second step the preferential rule of attachment is applied to the chosen subset the characteristics of thus formed networks are explored using two approaches computer simulations of network growth and a theoretical description based on a master equation the results of the two approaches are in excellent agreement special emphasis is put on the investigation of the node degree probability distribution it is found that the tail of the distribution has the exponential form given by exp k m implications of the node degree distribution with such tail characteristics are briefly discussed

degree and component size distributions in the generalized uniform recursive tree we propose a generalized model for the famous uniform recursive tree urt by introducing an imperfect growth process which may generate disconnected components clusters the model undergoes an interesting phase transition from a singly connected network to a graph consisting of fully isolated nodes we investigate the distributions of degree and component sizes by both theoretical predictions and numerical simulations for the nontrivial cases we show that the network has an exponential degree distribution while its component size distribution follows a power law both of which are related to the imperfect growth process we also predict the growth dynamics of the individual components all analytical solutions are successfully contrasted with computer simulations

communicability in temporal networks a first principles approach to quantify the communicability between pairs of nodes in temporal networks is proposed it corresponds to the imaginary time propagator of a quantum random walk in the temporal network which accounts for unique structural and temporal characteristics of both streaming and nonstreaming temporal networks the influence of the system s temperature on the perdurability of information and how the communicability identifies patterns of communication hidden in the temporal and topological structure of the networks are also studied for synthetic and real world systems

interlacing eigenvalues and graphs abstract we give several old and some new applications of eigenvalue interlacing to matrices associated to graphs bounds are obtained for characteristic numbers of graphs such as the size of a maximal co clique the chromatic number the diameter and the bandwidth in terms of the eigenvalues of the standard adjacency matrix or the laplacian matrix we also deal with inequalities and regularity results concerning the structure of graphs and block designs

efficiency of scale free networks error and attack tolerance the concept of network efficiency recently proposed to characterize the properties of small world networks is here used to study the effects of errors and attacks on scale free networks two different kinds of scale free networks i e networks with power law p k are considered 1 scale free networks with no local clustering produced by the barabasi albert model and 2 scale free networks with high clustering properties as in the model by klemm and eguiluz and their properties are compared to the properties of random graphs exponential graphs by using as mathematical measures the global and the local efficiency we investigate the effects of errors and attacks both on the global and the local properties of the network we show that the global efficiency is a better measure than the characteristic path length to describe the response of complex networks to external factors we find that at variance with random graphs scale free networks display both on a global and on a local scale a high degree of error tolerance and an extreme vulnerability to attacks in fact the global and the local efficiency are unaffected by the failure of some randomly chosen nodes though they are extremely sensitive to the removal of the few nodes which play a crucial role in maintaining the network s connectivity

highly clustered scale free networks we propose a model for growing networks based on a finite memory of the nodes the model shows stylized features of real world networks power law distribution of degree linear preferential attachment of new links and a negative correlation between the age of a node and its link attachment rate notably the degree distribution is conserved even though only the most recently grown part of the network is considered as the network grows the clustering reaches an asymptotic value larger than that for regular lattices of the same average connectivity and similar to the one observed in the networks of movie actors coauthorship in science and word synonyms these highly clustered scale free networks indicate that memory effects are crucial for a correct description of the dynamics of growing networks

implicitly restarted arnoldi with purification for the shift invert transformation the need to determine a few eigenvalues of a large sparse generalised eigenvalue problem az bx with positive semidefinite b arises in many physical situations for example in a stability analysis of the discretised navier stokes equation a common technique is to apply arnoldi s method to the shift invert transformation but this can suffer from numerical instabilities as is illustrated by a numerical example in this paper a new method that avoids instabilities is presented which is based on applying the implicitly restarted arnoldi method with the b semi inner product and a purification step the paper contains a rounding error analysis and ends with brief comments on some extensions

synchronizability of dynamical scale free networks subject to random errors in this paper the robustness of network synchronizability against random deletion of nodes i e errors in dynamical scale free networks was studied to this end two measures of network synchronizability namely the eigenratio of the laplacian and the order parameter quantifying the degree of phase synchrony were adopted and the synchronizability robustness on preferential attachment scale free graphs was investigated the findings revealed that as the network size decreases the robustness of its synchronizability against random removal of nodes declines i e the more the number of randomly removed nodes from the network the worse its synchronizability we also showed that this dependence of the synchronizability on the network size is different with that in the growing scale free networks the profile of a number of network properties such as clustering coefficient efficiency assortativity and eccentricity as a function of the network size was investigated in these two cases growing scale free networks and those with randomly removed nodes the results showed that these processes are also different in terms of these metrics

reweighting estimators for cox regression with missing covariate data analysis of insulin resistance and risk of stroke in the northern manhattan study incomplete covariates often obscure analysis results from a cox regression in an analysis of the northern manhattan study nomas to determine the influence of insulin resistance on the incidence of stroke in non diabetic individuals insulin level is unknown for 34 1 of the subjects the available data suggest that the missingness mechanism depends on outcome variables which may generate biases in estimating the parameters of interest if only using the complete observations this article aimed to introduce practical strategies to analyze the nomas data and present sensitivity analyses by using the reweighting method in standard statistical packages when the data set structure is in counting process style the reweighting estimates can be obtained by built in procedures with variance estimated by the jackknife method simulation results indicate that the jackknife variance estimate provides reasonable coverage probability in moderate sample sizes we subsequently conducted sensitivity analyses for the nomas data showing that the risk estimates are robust to a variety of missingness mechanisms at the end of this article we present the core sas and r programs used in the analysis

on the α nonbacktracking centrality for complex networks existence and limit cases abstract nonbactracking centrality was introduced as an attempt to correct some deficiencies of eigenvector centrality in this work the nonbacktracking centrality is introduced as an extension that interpolates between the nonbacktracking centrality of the edges of a directed network and the eigenvector centrality of the corresponding directed line graph the existence of this new nonbacktracking centrality is proved in terms of the connectivity of the original network we prove that the limit of the nonbacktracking centrality when decreases to zero exists and is well defined moreover it coincides with the nonbacktracking centrality when this measure is defined with the same techniques we also prove the convergence of pagerank vectors to the eigenvector centrality vector when the damping factor tends to 1

spectral centrality measures in complex networks complex networks are characterized by heterogeneous distributions of the degree of nodes which produce a large diversification of the roles of the nodes within the network several centrality measures have been introduced to rank nodes based on their topological importance within a graph here we review and compare centrality measures based on spectral properties of graph matrices we shall focus on pagerank pr eigenvector centrality ev and the hub and authority scores of the hits algorithm we derive simple relations between the measures and the in degree of the nodes in some limits we also compare the rankings obtained with different centrality measures

effect of edge removal on topological and functional robustness of complex networks abstract we study the robustness of several network models subject to edge removal the robustness is measured by the statistics of network breakdowns where a breakdown is defined as the destroying of the total connectedness of a network rather than the disappearance of the giant component we introduce a simple traffic dynamics as the function of a network topology and the total connectedness can be destroyed in the sense of either the topology or the function the overall effect of the topological breakdown and the functional breakdown as well as the relative importance of the topological robustness and the functional robustness are studied under two edge removal strategies

asymptotic estimates for efficiency vulnerability and cost for random networks an investigation both analytical and numerical of the values of efficiency vulnerability and cost functions is provided several comparisons among these functions are made following a pairwise strategy and taking into account different efficiency vulnerability and cost functions it is shown in all cases that these magnitudes display strong correlations which are typically piecewise linear in log space as well as universal i e independent of the number of nodes n to a great extent the analytical part of the work develops explicit bounds linking all pairs of such parameters these results are numerically compared with the result of a random testing for erdos renyi networks

a key heterogeneous structure of fractal networks based on inverse renormalization scheme abstract self similarity property of complex networks was found by the application of renormalization group theory based on this theory network topologies can be classified into universality classes in the space of configurations in return through inverse renormalization scheme a given primitive structure can grow into a pure fractal network then adding different types of shortcuts it exhibits different characteristics of complex networks however the effect of primitive structure on networks structural property has received less attention in this paper we introduce a degree variance index to measure the dispersion of nodes degree in the primitive structure and investigate the effect of the primitive structure on network structural property quantified by network efficiency numerical simulations and theoretical analysis show a primitive structure is a key heterogeneous structure of generated networks based on inverse renormalization scheme whether or not adding shortcuts and the network efficiency is positively correlated with degree variance of the primitive structure

new results on computable efficiency and its stability for complex networks the concept of efficiency in a complex dynamic network plays the role of measuring its ability for the exchange of information and its response for the spread of perturbations in diverse applications this concept is strongly related to the topological properties of the network a new framework for the definition of network efficiency is discussed and some of its consequences are established we also consider how the efficiency changes as the network structure evolves by introducing the concept of stability associated to the efficiency of a network as an accurate tool to measure the evolution of a dynamic complex network

the dynamic correlation between degree and betweenness of complex network under attack abstract complex networks are often subjected to failure and attack recent work has addressed the resilience of complex networks to either random or intentional deletion of nodes or links here we simulate the breakdown of the small world network and the scale free network under node failure or attacks we analyze and discuss the dynamic correlation between degree and betweenness in the process of attack the simulation results show that the correlation for scale free network obeys a power law distribution until the network collapses while it represents irregularly for small world network

harmony in the small world the small world phenomenon popularly known as six degrees of separation has been mathematically formalized by watts and strogatz in a study of the topological properties of a network small world networks are defined in terms of two quantities they have a high clustering coefficient c like regular lattices and a short characteristic path length l typical of random networks physical distances are of fundamental importance in applications to real cases nevertheless this basic ingredient is missing in the original formulation here we introduce a new concept the connectivity length d that gives harmony to the whole theory d can be evaluated on a global and on a local scale and plays in turn the role of l and 1 c moreover it can be computed for any metrical network and not only for the topological cases d has a precise meaning in terms of information propagation and describes in a unified way both the structural and the dynamical aspects of a network small worlds are defined by a small global and local d i e by a high efficiency in propagating information both on a local and global scale the neural system of the nematode c elegans the collaboration graph of film actors and the oldest us subway system can now be studied also as metrical networks and are shown to be small worlds

rational krylov sequence methods for eigenvalue computation abstract algorithms to solve large sparse eigenvalue problems are considered a new class of algorithms which is based on rational functions of the matrix is described the lanczos method the arnoldi method the spectral transformation lanczos method and rayleigh quotient iteration all are special cases but there are also new algorithms which correspond to rational functions with several poles in the simplest case a basis of a rational krylov subspace is found in which the matrix eigenvalue problem is formulated as a linear matrix pencil with a pair of hessenberg matrices

a note on the irregularity of graphs abstract denote by 1 g the largest eigenvalue of a real 0 1 adjacency matrix of a graph g and by d g the mean degree of g collatz and sinogowitz proposed 1 g d g as a measure of irregularity of g a second such measure is the variance of the vertex degrees of g the most irregular graphs according to these measures are determined for certain classes of graphs and the two measures are shown to be incompatible for some pairs of graphs

topology and correlations in structured scale free networks we study a recently introduced class of scale free networks showing a high clustering coefficient and nontrivial connectivity correlations we find that the connectivity probability distribution strongly depends on the fine details of the model we solve exactly the case of low average connectivity providing also exact expressions for the clustering and degree correlation functions the model also exhibits a lack of small world properties in the whole parameter range we discuss the physical properties of these networks in the light of the present detailed analysis

restrictions on implicit filtering techniques for orthogonal projection methods abstract we consider the class of the orthogonal projection methods opm to solve iteratively large eigenvalue problems an opm is a method that projects a large eigenvalue problem on a smaller subspace in this subspace an approximation of the eigenvalue spectrum can be computed from a small eigenvalue problem using a direct method examples of opms are the arnoldi and the davidson method we show how an opm can be restarted implicitly and explicitly this restart can be used to remove a specific subset of vectors from the approximation subspace this is called explicit filtering an implicit restart can also be combined with an implicit filtering step i e the application of a polynomial or rational function on the subspace even if inaccurate arithmetic is assumed however the condition for the implicit application of a filter is that the rank of the residual matrix must be small

augmented inverse probability weighted estimator for cox missing covariate regression summary this article investigates an augmented inverse selection probability weighted estimator for cox regression parameter estimation when covariate variables are incomplete this estimator extends the horvitz and thompson 1952 journal of the american statistical association 47 663 685 weighted estimator this estimator is doubly robust because it is consistent as long as either the selection probability model or the joint distribution of covariates is correctly specified the augmentation term of the estimating equation depends on the baseline cumulative hazard and on a conditional distribution that can be implemented by using an em type algorithm this method is compared with some previously proposed estimators via simulation studies the method is applied to a real example

remarks on the energy of regular graphs abstract the energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix this note is about the energy of regular graphs it is shown that graphs that are close to regular can be made regular with a negligible change of the energy also a k regular graph can be extended to a k regular graph of a slightly larger order with almost the same energy as an application it is shown that for every sufficiently large n there exists a regular graph g of order n whose energy g satisfies g 1 2 n 3 2 n 13 10 several infinite families of graphs with maximal or submaximal energy are given and the energy of almost all regular graphs is determined

using mapping entropy to identify node centrality in complex networks abstract the problem of finding the best strategy to attack a network or immunize a population with a minimal number of nodes has attracted much current research interest the assessment of node importance has been a fundamental issue in the research of complex networks in this paper we propose a new concept called mapping entropy me to identify the importance of a node in the complex network the concept is established according to the local information which considers the correlation among all neighbors of a node we evaluate the efficiency of the centrality by static and dynamic attacks on standard network models and real world networks the simulation result shows that the new centrality is more efficient than traditional attack strategies whether it is static or dynamic

estimation of regression coefficients when some regressors are not always observed abstract in applied problems it is common to specify a model for the conditional mean of a response given a set of regressors a subset of the regressors may be missing for some study subjects either by design or happenstance in this article we propose a new class of semiparametric estimators based on inverse probability weighted estimating equations that are consistent for parameter vector 0 of the conditional mean model when the data are missing at random in the sense of rubin and the missingness probabilities are either known or can be parametrically modeled we show that the asymptotic variance of the optimal estimator in our class attains the semiparametric variance bound for the model by first showing that our estimation problem is a special case of the general problem of parameter estimation in an arbitrary semiparametric model in which the data are missing at random and the probability of observing complete data is bounded away from 0 and then deriving a representation for the efficient score

localization and centrality in networks eigenvector centrality is a common measure of the importance of nodes in a network here we show that under common conditions the eigenvector centrality displays a localization transition that causes most of the weight of the centrality to concentrate on a small number of nodes in the network in this regime the measure is no longer useful for distinguishing among the remaining nodes and its efficacy as a network metric is impaired as a remedy we propose an alternative centrality measure based on the nonbacktracking matrix which gives results closely similar to the standard eigenvector centrality in dense networks where the latter is well behaved but avoids localization and gives useful results in regimes where the standard centrality fails

a jackknife estimator of variance for cox regression for correlated survival data studies in the health sciences often give rise to correlated survival data wei lin and weissfeld 1989 journal of the american statistical association 84 1065 1073 and lee wei and amato 1992 in survival analysis state of the art showed that if the marginal distributions of the correlated survival times follow a proportional hazards model then the estimates from cox s partial likelihood cox d r 1972 journal of the royal statistical society series b 24 187 220 naively treating the correlated survival times as independent give consistent estimates of the relative risk parameters however because of the correlation between survival times the inverse of the information matrix may not be a consistent estimate of the asymptotic variance wei et al 1989 and lee et al 1992 proposed a robust variance estimate that is consistent for the asymptotic variance we show that a one step jackknife estimator of variance is asymptotically equivalent to their variance estimator the jackknife variance estimator may be preferred because an investigator needs only to write a simple loop in a computer package instead of a more involved program to compute wei et al 1989 and lee et al s 1992 estimator

bounds on graph eigenvalues i abstract we prove three results about the spectral radius g of a graph g a let t r n be the r partite turan graph of order n if g is a k r 1 free graph of order n then g t r n unless g t r n b for most irregular graphs g of order n and size m g 2 m n 1 2 m 2 n c let 0 k l if g is a graph of order n with no k 2 k k 1 and no k 2 l 1 then g min g k l 1 k l 1 2 4 l n 1 2

new attack strategies for complex networks abstract the invulnerability of complex networks is an important issue in that the behavior of scale free network differs from that of exponential network according to the structural characteristics of the networks we propose two new attack strategies named idb initial degree and betweenness and rdb recalculated degree and betweenness the strategies are originated from id initial degree distribution and rd recalculated degree distribution strategies in which attacks are based on initial structural information of a network the probability of node removals depends on a new metric combining degree centrality and betweenness centrality we evaluate the efficiency of the proposed strategies on one real world network and three network models experimental results indicate that the proposed strategies are more efficient than the traditional id and rd strategies specially the ws small world network behaves more sensitive to the proposed strategies the attack efficiency of rdb strategy is improved by 20 to rd strategy and idb strategy is improved by 40 to id strategy

mean field theory for scale free random networks random networks with complex topology are common in nature describing systems as diverse as the world wide web or social and business networks recently it has been demonstrated that most large networks for which topological information is available display scale free features here we study the scaling properties of the recently introduced scale free model that can account for the observed power law distribution of the connectivities we develop a mean eld method to predict the growth dynamics of the individual vertices and use this to calculate analytically the connectivity distribution and the scaling exponents the mean eld method can be used to address the properties of two variants of the scale free model that do not display power law scaling c 1999 elsevier science b v all rights reserved

more eigenvalue problems of nordhaus gaddum type abstract let g be a graph of order n and let 1 g n g be the eigenvalues of its adjacency matrix this note studies eigenvalue problems of nordhaus gaddum type let g be the complement of a graph g it is shown that if s 2 and n 15 s 1 then s g s g n 2 s 1 1 also if s 1 and n 4 s then n s 1 g n s 1 g n 2 s 1 if s 2 k 1 for some integer k these bounds are asymptotically tight these results settle infinitely many cases of a general open problem

vulnerability of complex networks under intentional attack with incomplete information we study the vulnerability of complex networks under intentional attack with incomplete information which means that one can only preferentially attack the most important nodes among a local region of a network the known random failure and the intentional attack are two extreme cases of our study using the generating function method we derive the exact value of the critical removal fraction fc of nodes for the disintegration of networks and the size of the giant component to validate our model and method we perform simulations of intentional attack with incomplete information in scale free networks we show that the attack information has an important effect on the vulnerability of scale free networks we also demonstrate that hiding a fraction of the nodes information is a cost efficient strategy for enhancing the robustness of complex networks