spectra and eigenvectors of scale free networks we study the spectra and eigenvectors of the adjacency matrices of scale free networks when bidirectional interaction is allowed so that the adjacency matrix is real and symmetric the spectral density shows an exponential decay around the center followed by power law long tails at both spectrum edges the largest eigenvalue ensuremath lambda 1 depends on system size n as ensuremath lambda 1 ensuremath sim n 1 4 for large n and the corresponding eigenfunction is strongly localized at the hub the vertex with largest degree the component of the normalized eigenfunction at the hub is of order unity we also find that the mass gap scales as n ensuremath 0 68

pinning dynamic systems of networks with markovian switching couplings and controller node set in this paper we study pinning control problem of coupled dynamical systems with stochastically switching couplings and stochastically selected controller node set here the coupling matrices and the controller node sets change with time induced by a continuous time markov chain by constructing lyapunov functions we establish tractable sufficient conditions for exponential stability of the coupled system two scenarios are considered here first we prove that if each subsystem in the switching system i e with the fixed coupling can be stabilized by the fixed pinning controller node set and in addition the markovian switching is sufficiently slow then the time varying dynamical system is stabilized second we conclude that if the system with the average coupling and pinning gains can be stabilized and the switching is sufficiently fast the time varying system is stabilized two numerical examples are provided to demonstrate the validity of these theoretical results including a switching dynamical system between several stable subsystems and a dynamical system with mobile nodes and spatial pinning control towards the nodes when these nodes are being in a pre designed region

inequalities for eigenvalues of fourth order elliptic operators in divergence form on riemannian manifolds abstract in this paper we study eigenvalue of linear fourth order elliptic operators in divergence form with dirichlet boundary condition on a bounded domain in a compact riemannian manifolds with boundary possibly empty and provide a general inequality for them as an application by using this inequality we study eigenvalues of this operator on compact domains of complete submanifolds in a euclidean space

pinning synchronization of fractional order delayed complex networks with non delayed and delayed couplings abstractthis paper considers the global synchronization of fractional order delayed complex networks with non delayed and delayed couplings based on pinning control with the help of comparison theorems for fractional order linear systems with multiple time delays some novel and useful synchronisation criteria are presented by adding linear error feedback controllers to certain selected nodes here the coupling matrices need not be symmetric or irreducible what is more it is shown that the model and results in the paper are more general than those in the literature a numerical example is given to demonstrate the effectiveness and feasibility of the obtained results

on trace identities and universal eigenvalue estimates for some partial differential operators in this article we prove and exploit a trace identity for the spectra of schrodinger operators and similar operators this identity leads to universal bounds on the spectra which apply to low lying eigenvalues eigenvalue asymptotics and to partition functions traces of heat operators in many cases they are sharp in the sense that there are specific examples for which the inequalities are saturated special cases corresponding to known inequalities include those of hile and protter and of yang introduction in this article we prove and exploit an identity for the spectra of self adjoint operators h modeled on the dirichlet laplacian or more generally on schrodinger operators of the form p a x 2 v x 1 where p 1i is the usual momentum operator in convenient units and a x is the magnetic vector potential we recover and extend several known inequalities involving sums differences and ratios of eigenvalues let j j 1 denote the ordered eigenvalues of the dirichlet laplacian on a bounded d dimensional domain with zero dirichlet boundary conditions and recall that hile and protter hipr80 proved that d 4 1 n n j 1 j n 1 j 2 thereby extending an earlier inequality of payne polya and weinberger papowe56 in the last few years it has become clear that these and many similar relationships can be realized as special cases of abstract variational bounds involving the interplay among commutators of 2 a cartesian coordinate xj and the corresponding derivative xj for this analysis and various extensions of 2 see ha88 ho90 ha93 hami95 while inequalities of this type have been fairly sharp for low lying eigenvalues they have been mostly disappointing for higher eigenvalues yang ya91 however received by the editors september 28 1995 1991 mathematics subject classification primary 35j10 35j25 58g25

fragmentation of random trees we study fragmentation of a random recursive tree into a forest by repeated removal of nodes the initial tree consists of n nodes and it is generated by sequential addition of nodes with each new node attaching to a randomly selected existing node as nodes are removed from the tree one at a time the tree dissolves into an ensemble of separate trees namely a forest we study statistical properties of trees and nodes in this heterogeneous forest and find that the fraction of remaining nodes m characterizes the system in the limit we obtain analytically the size density of trees of size s the size density has power law tail with exponent therefore the tail becomes steeper as further nodes are removed and the fragmentation process is unusual in that exponent increases continuously with time we also extend our analysis to the case where nodes are added as well as removed and obtain the asymptotic size density for growing trees

estimates on eigenvalues of laplacian in this paper we study eigenvalues of laplacian on either a bounded connected domain in an n dimensional unit sphere sn 1 or a compact homogeneous riemannian manifold or an n dimensional compact minimal submanifold in an n dimensional unit sphere sn 1 we estimate the k 1 th eigenvalue by the first k eigenvalues as a corollary we obtain an estimate of difference between consecutive eigenvlaues our results are sharper than ones of p c yang and yau 25 leung 19 li 20 and harrel ii and stubbe 12 respectively from weyl s asymptotical formula we know that our estimates are optimal in the sense of the order of k for eigenvalues of laplacian on a bounded connected domain in an n dimensional unit sphere sn 1

transmission of packets on a hierarchical network statistics and explosive percolation we analyze an idealized model for the transmission or flow of particles or discrete packets of information in a weight bearing branching hierarchical two dimensional network and its variants the capacities add hierarchically down the clusters each node can accommodate a limited number of packets depending on its capacity and the packets hop from node to node following the links between the nodes the statistical properties of this system are given by the maxwell boltzmann distribution we obtain analytical expressions for the mean occupation numbers as functions of capacity for different network topologies the analytical results are shown to be in agreement with the numerical simulations the traffic flow in these models can be represented by the site percolation problem it is seen that the percolation transitions in the 2d model and in its variant lattices are continuous transitions whereas the transition is found to be explosive discontinuous for the v lattice the critical case of the 2d lattice the scaling behavior of the second order percolation case is studied in detail we discuss the implications of our analysis

complete synchronisation of linear oscillator networks oscillators with individual natural frequencies are not synchronisable by diffusive couplings which are usually used for solving consensus and synchronisation problems this article proves that however such oscillators can be synchronised by dynamical couplings if and only if the interconnection network includes local dynamical units with integrator dynamics a necessary and sufficient synchronisation condition shows that oscillators with arbitrary positive parameters are synchronised by any coupling network that possesses a spanning tree and has sufficiently large coupling strength simulation results illustrate the results of this article

what is the real size of a sampled network the case of the internet internet mapping projects generally consist in sampling the network from a limited set of sources by usingtraceroute probes this methodology akin to the merging of spanning trees from the different sources to a set of destinations leads necessarily to a partial incomplete map of the internet accordingly determination of internet topology characteristics from such sampled maps is in part a problem of statistical inference our contribution begins with the observation that the inference of many of the most basic topological quantities including network size and degree characteristics fromtraceroute measurements is in fact a version of the so called species problem in statistics this obse rvation has important implications since species problems are often quite challenging we focus here on the most fundamental example of atraceroute internet species the number of nodes in a network specifica lly

estimates for the eigenvalues of the bi drifting laplacian on complete metric measure spaces as we know it is very difficult to prove an eigenvalue inequality of ashbaugh cheng ichikawa mametsuka type for the bi drifting laplacian on the bounded domain of the complete metric measure spaces even assumed that the differential operator is bi laplacian on the n dimensional unit euclidean sphere this problem still remains open however a general inequality with respect to the bi drifting laplacian is successfully established under certain conditions in this paper applying the general inequality we prove some eigenvalue inequalities of ashbaugh cheng ichikawa mametsuka type on the gaussian shrinking ricci soliton and the n dimensional cigar metric measure spaces cmms for short in particular we obtain two interesting eigenvalue inequalities of ashbaugh cheng ichikawa mametsuka type for the cmms with lower dimensional topology

synchronisation of complex dynamical networks with different dynamics of nodes via decentralised dynamical compensation controllers this paper investigates the synchronisation of nonlinear coupled complex dynamical networks with different nonlinear nodes and different orders by using decentralised dynamical compensation controllers we propose a dynamical network mathematical model with similar nonlinear nodes whose dimensions are different for this kind of network model the decentralised dynamical compensation controllers are designed for the state synchronisation of the coupled nodes in addition the synchronisation manifold is defined as an invariant manifold which is regarded as the generalised case of dynamical networks with the same nodes dynamics furthermore some stability criteria for the synchronisation are derived by means of rigorous theoretical analysis finally numerical examples are presented to verify the effectiveness of the obtained theoretical results

pinning control for cluster synchronisation of complex dynamical networks with semi markovian jump topology in this paper a pinning controller is designed for cluster synchronisation of complex dynamical networks with semi markovian jump topology which covers the concept of the conventional markovian jump as a special case to decide on the pinned node for the cluster synchronisation a novel procedure is proposed based on a topological structure in addition criteria for the synchronisation are separately derived for the controller with the mode independent and mode dependent pinned node respectively a practical example is given to illustrate the effectiveness of the proposed methods

cluster synchronization for nonlinearly time varying delayed coupling complex networks with stochastic perturbation via periodically intermittent pinning control this paper investigate the mean square exponential cluster synchronization of a class complex networks with nonlinear hybrid time varying delays and stochastic noises via periodically intermittent pinning control to be more practical the network topology can be directed and the time delay is assumed to be time varying based on the lyapunov stability theory stochastic analysis theory and linear matrix inequalities lmi the sufficient conditions are derived to guarantee the mean square exponential cluster synchronization furthermore our results can cover and extend the related ones in existing literatures some numerical simulations are provided to show the effectiveness of our theoretical results

phase synchronization control of complex networks of lagrangian systems on adaptive digraphs this paper presents a formation control and synchronization method that utilizes adaptive network topologies for a class of complex dynamical networks comprised of a large number of highly nonlinear euler lagrange el systems a time varying and switching network topology constructed by the adaptive graph laplacian matrix relaxes the standard requirement of consensus stability even permitting exponential synchronization on an unbalanced digraph or a weakly connected digraph that can sporadically lose connectivity the time varying graph laplacian matrix is adapted by an adaptive control scheme based on relative positions and errors of synchronization and tracking the adaptive graph laplacian is integrated with a phase synchronization controller that synchronizes the relative motions of el systems moving in elliptical orbits thereby yielding a smaller synchronization error than an uncoupled tracking control law in the presence of bounded disturbances and modeling errors an example of reconfiguring hundreds of spacecraft in low earth orbit shows the effectiveness of the proposed phase synchronization controller for a large number of complex el systems moving in periodic elliptical orbits

reduced order filtering for networks with markovian jumping parameters and missing measurements abstractthe problem of reduced order h filters design for markovian jumping complex networks with polytopic time varying transition probability matrices is first addressed in this paper where th

universal bounds for eigenvalues of the polyharmonic operators we study eigenvalues of polyharmonic operators on compact riemannian manifolds with boundary possibly empty in particular we prove a universal inequality for the eigenvalues of the polyharmonic operators on compact domains in a euclidean space this inequality controls the kth eigenvalue by the lower eigenvalues independently of the particular geometry of the domain our inequality is sharper than the known payne polya weinberg type inequality and also covers the important yang inequality on eigenvalues of the dirichlet laplacian we also prove universal inequalities for the lower order eigenvalues of the polyharmonic operator on compact domains in a euclidean space which in the case of the biharmonic operator and the buckling problem strengthen the estimates obtained by ashbaugh finally we prove universal inequalities for eigenvalues of polyharmonic operators of any order on compact domains in the sphere

organization of growing random networks the organizational development of growing random networks is investigated these growing networks are built by adding nodes successively and linking each to an earlier node of degree k with an attachment probability a k when a k grows more slowly than linearly with k the number of nodes with k links n k t decays faster than a power law in k while for a k growing faster than linearly in k a single node emerges which connects to nearly all other nodes when a k is asymptotically linear n k t ensuremath sim mathrm tk ensuremath ensuremath nu with ensuremath nu dependent on details of the attachment probability but in the range 2l ensuremath nu l ensuremath infty the combined age and degree distribution of nodes shows that old nodes typically have a large degree there is also a significant correlation in the degrees of neighboring nodes so that nodes of similar degree are more likely to be connected the size distributions of the in and out components of the network with respect to a given node namely its descendants and ancestors are also determined the in component exhibits a robust s ensuremath 2 power law tail where s is the component size the out component has a typical size of order mathrm ln t and it provides basic insights into the genealogy of the network

stabilization of continuous time randomly switched systems via the lmi approach this paper is concerned on the stabilization problem of continuous time switched systems with random switching signal where the dwell time in each subsystem consists of a fixed part and random part new sufficient conditions for stabilizing controllers including mode independent case are provided in terms of lmis which could be solved directly moreover the obtained results are extended to a more general case that the corresponding operation mode of a controller experiences a disordering phenomenon without designing a controller depending on new operation modes noms a kind of controller only depending on the original operation modes ooms is proposed and satisfies a minimum variance approximation whose solvable conditions are also with lmi forms finally numerical examples are used to demonstrate the effectiveness of the proposed methods

commutators spectral trace identities and universal estimates for eigenvalues using simple commutator relations we obtain several trace identities involving eigenvalues and eigenfunctions of an abstract self adjoint operator acting in a hilbert space applications involve abstract universal estimates for the eigenvalue gaps as particular examples we present simple proofs of the classical universal estimates for eigenvalues of the dirichlet laplacian as well as of some known and new results for other differential operators and systems we also suggest an extension of the methods to the case of non self adjoint operators

random networks with tunable degree distribution and clustering we present an algorithm for generating random networks with arbitrary degree distribution and clustering frequency of triadic closure we use this algorithm to generate networks with exponential power law and poisson degree distributions with variable levels of clustering such networks may be used as models of social networks and as a testable null hypothesis about network structure finally we explore the effects of clustering on the point of the phase transition where a giant component forms in a random network and on the size of the giant component some analysis of these effects is presented

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

domain independent upper bounds for eigenvalues of elliptic operators let q c ir m be a bounded open set aq its boundary and a the laplacian on rm consider the elliptic differential equation 1 au au in q u 0 on aq it is known that the eigenvalues i of 1 satisfy n m n 2 i l n 1l 1 provided that n l an in this paper we abstract the method used by hile and protter 2 to establish 2 and apply the method to a variety of second order elliptic problems in particular to all constant coefficient problems we then consider a variety of higher order problems and establish an extension of 2 for problem 1 where the laplacian is replaced by a more general operator in a hilbert space

a sharp bound for the ratio of the first two eigenvalues of dirichlet laplacians and extensions the first author s work was partially supported by fondecyt chile project 0132 88 and by a summer research fellowship provided by the research council of the university of missouri columbia he would like to thank the physics department and others at the universidad de chile for their hospitality during his visit in april 1990 when much of this research was completed the second author was supported in part by fondecyt chile projects 0132 88 and 1238 90 both authors also thank fritz gesztesy for general comments and encouragement

universal inequalities for eigenvalues of the buckling problem on spherical domains in this paper we study the eigenvalues of the buckling problem on domains in a unit sphere we obtain universal bounds on the k 1 th eigenvalue in terms of the first k eigenvalues independent of the domains

a second proof of the payne polya weinberger conjecture without using product representations or elaborate comparisons of zeros we prove the two key properties of the bessel function ratiojp 1jp 1 1x jpjp 1x that we used to prove the payne polya weinberger conjecture in these new proofs we use only differential equations and the rayleigh ritz method for estimating lowest eigenvalues the new proofs admit generalization to other related problems where our previous proofs fail

eigenvalues of the buckling problem of arbitrary order and of the polyharmonic operator on ricci flat manifolds abstract in this paper we study eigenvalues of the buckling problem of arbitrary order and of the polyharmonic operator on bounded domains in ricci flat manifolds supporting a special function and obtain universal bounds on the k 1 th eigenvalue in terms of the first k eigenvalues independent of the domains

packet transport and load distribution in scale free network models we study a problem of data packet transport in scale free networks whose degree distribution follows a power law with the exponent we define load at each vertex as the accumulated total number of data packets passing through that vertex when every pair of vertices sends and receives a data packet along the shortest path connecting the pair it is found that the load distribution follows a power law with the exponent 2 2 1 insensitive to different values of in the range 2 3 and different mean degrees which is valid for both undirected and directed cases thus we conjecture that the load exponent is a universal quantity to characterize scale free networks

fluctuation driven dynamics of the internet topology we study the dynamics of the internet topology based on empirical data on the level of the autonomous systems it is found that the fluctuations occurring in the stochastic process of connecting and disconnecting edges are important features of the internet dynamics the network s overall growth can be described approximately by a single characteristic degree growth rate g eff approximately 0 016 and the fluctuation strength sigma eff approximately 0 14 together with the vertex growth rate alpha approximately 0 029 a stochastic model which incorporates these values and an adaptation rule newly introduced reproduces several features of the real internet topology such as the correlations between the degrees of different vertices

the evolutionary game of pressure or interference resistance and collaboration in this paper we extend the framework of the evolutionary inspection game put forward recently by the author and coworkers to a large class of conflict interactions to address the pressure executed by the major player or principal on the large group of small players who can resist this pressure or collaborate with the major player we prove rigorous results on the convergence of various markov decision models of interacting small agents including evolutionary growth i e pairwise in groups and by coalition formation to a deterministic evolution on the distributions of the state spaces of small players paying main attention to situations with an infinite state space of small players we supply precise rates of convergence the theoretical results of the paper are applied to the analysis of the processes of inspection corruption cyber security counter terrorism banks and firms merging strategically enhanced preferential attachment and many other

evolutionary mixed games in structured populations cooperation and the benefits of heterogeneity evolutionary games on networks traditionally involve the same game at each interaction here we depart from this assumption by considering mixed games where the game played at each interaction is drawn uniformly at random from a set of two different games while in well mixed populations the random mixture of the two games is always equivalent to the average single game in structured populations this is not always the case we show that the outcome is in fact strongly dependent on the distance of separation of the two games in the parameter space effectively this distance introduces payoff heterogeneity and the average game is returned only if the heterogeneity is small for higher levels of heterogeneity the distance to the average game grows which often involves the promotion of cooperation the presented results support preceding research that highlights the favorable role of heterogeneity regardless of its origin and they also emphasize the importance of the population structure in amplifying facilitators of cooperation

zero pearson coefficient for strongly correlated growing trees we obtained pearson s coefficient of strongly correlated recursive networks growing by preferential attachment of every new vertex by m edges we found that the pearson coefficient is exactly zero in the infinite network limit for the recursive trees m 1 if the number of connections of new vertices exceeds one mg1 then the pearson coefficient in the infinite networks equals zero only when the degree distribution exponent ensuremath gamma does not exceed 4 we calculated the pearson coefficient for finite networks and observed a slow power law like approach to an infinite network limit our findings indicate that pearson s coefficient strongly depends on size and details of networks which makes this characteristic virtually useless for quantitative comparison of different networks

universal bounds for eigenvalues of the biharmonic operator on riemannian manifolds in this paper we consider eigenvalues of the dirichlet biharmonic operator on compact riemannian manifolds with boundary possibly empty and prove a general inequality for them by using this inequality we study eigenvalues of the dirichlet biharmonic operator on compact domains in a euclidean space or a minimal submanifold of it and a unit sphere we obtain universal bounds on the k 1 th eigenvalue on such objects in terms of the first k eigenvalues independent of the domains the estimate for the k 1 th eigenvalue of bounded domains in a euclidean space improves an important inequality obtained recently by cheng and yang

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