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

convergence towards an erdős renyi graph structure in network contraction processes in a highly influential paper twenty years ago barabasi and albert science 286 509 1999 scieas0036 807510 1126 science 286 5439 509 showed that networks undergoing generic growth processes with preferential attachment evolve towards scale free structures in any finite system the growth eventually stalls and is likely to be followed by a phase of network contraction due to node failures attacks or epidemics using the master equation formulation and computer simulations we analyze the structural evolution of networks subjected to contraction processes via random preferential and propagating node deletions we show that the contracting networks converge towards an erd s renyi network structure whose mean degree continues to decrease as the contraction proceeds this is manifested by the convergence of the degree distribution towards a poisson distribution and the loss of degree degree correlations

brief accuracy enhancing methods for balancing related frequency weighted model and controller reduction we consider the solution of the balancing related frequency weighted model and controller reduction problems using accuracy enhanced numerical algorithms we propose first new stability enforcing choices of the frequency weighted grammians which can guarantee the stability of reduced models for two sided frequency weights then we show that for the frequency weighted controller reduction problems with standard stability and performance enforcing frequency weights the computation of the frequency weighted grammians can be done by solving reduced order lyapunov equations for both frequency weighted model and controller reduction problems we indicate how to compute the grammians directly in terms of their cholesky factors this allows the extension of the square root and balancing free accuracy enhancing techniques to the frequency weighted case

conditions for a symmetrized decomposable tensor to be zero let x1ast xm be a decomposable symmetrized tensor corresponding to the symmetric group sm and an irreducible character a necessary and sufficient condition on the vectors x1 xm is given for x1 xm to be zero

epidemics and percolation in small world networks we study some simple models of disease transmission on small world networks in which either the probability of infection by a disease or the probability of its transmission is varied or both the resulting models display epidemic behavior when the infection or transmission probability rises above the threshold for site or bond percolation on the network and we give exact solutions for the position of this threshold in a variety of cases we confirm our analytic results by numerical simulation

interpolatory weighted h2 model reduction this paper introduces an interpolation framework for the weighted h 2 model reduction problem we obtain a new representation of the weighted h 2 norm of siso systems that provides new interpolatory first order necessary conditions for an optimal reduced order model the h 2 norm representation also provides an error expression that motivates a new weighted h 2 model reduction algorithm several numerical examples illustrate the effectiveness of the proposed approach

mixing patterns in networks we study assortative mixing in networks the tendency for vertices in networks to be connected to other vertices that are like or unlike them in some way we consider mixing according to discrete characteristics such as language or race in social networks and scalar characteristics such as age as a special example of the latter we consider mixing according to vertex degree i e according to the number of connections vertices have to other vertices do gregarious people tend to associate with other gregarious people we propose a number of measures of assortative mixing appropriate to the various mixing types and apply them to a variety of real world networks showing that assortative mixing is a pervasive phenomenon found in many networks we also propose several models of assortatively mixed networks both analytic ones based on generating function methods and numerical ones based on monte carlo graph generation techniques we use these models to probe the properties of networks as their level of assortativity is varied in the particular case of mixing by degree we find strong variation with assortativity in the connectivity of the network and in the resilience of the network to the removal of vertices

frequency weighted model reduction via optimal projection the problem of order reduction with frequency weighting is considered necessary conditions characterizing the reduced order model which minimizes a quadratic error criterion in the frequency domain with either input or output frequency weighting are given the problem is transformed to the time domain where the weighting matrix is expressed as a shaping filter that generates colored noise at the input when the weighting matrix is singular at infinity the reduced order model has a direct transmission term even if the original system is strictly proper this additional degree of freedom can be used to obtain better approximation over a prespecified frequency range

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

h2 tf optimality conditions for a finite time horizon abstract in this paper we establish the interpolatory model reduction framework for optimal approximation of mimo dynamical systems with respect to the h 2 norm over a finite time horizon denoted as the h 2 t f norm using the underlying inner product space we derive the interpolatory first order necessary optimality conditions for approximation in the h 2 t f norm then we develop an algorithm which yields a locally optimal reduced model that satisfies the established interpolation based optimality conditions we test the algorithm on various numerical examples to illustrate its performance

a new frequency weighted balanced truncation method and an error bound in this paper a new frequency weighted model reduction method with an a priori error bound is proposed the method is a generalization of enns technique 1984 and yields stable models even when both input and output weightings are included the proposed method is compared with other existing methods using numerical examples

tailored graph ensembles as proxies or null models for real networks i tools for quantifying structure we study the tailoring of structured random graph ensembles to real networks with the objective of generating precise and practical mathematical tools for quantifying and comparing network topologies macroscopically beyond the level of degree statistics our family of ensembles can produce graphs with any prescribed degree distribution and any degree degree correlation function its control parameters can be calculated fully analytically and as a result we can calculate asymptotically formulae for entropies and complexities and for information theoretic distances between networks expressed directly and explicitly in terms of their measured degree distribution and degree correlations

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

controller reduction concepts and approaches the problem of passing from a linear time invariant high order controller designed for a linear time invariant plant of presumably high order to a low order approximation of the controller is discussed the approximation problem is often best posed as a frequency weighted l sub infinity approximation problem many different controller representations are possible giving different performances of the various reduction algorithms

network dismantling abstract we study the network dismantling problem which consists of determining a minimal set of vertices in which removal leaves the network broken into connected components of subextensive size for a large class of random graphs this problem is tightly connected to the decycling problem the removal of vertices leaving the graph acyclic exploiting this connection and recent works on epidemic spreading we present precise predictions for the minimal size of a dismantling set in a large random graph with a prescribed light tailed degree distribution building on the statistical mechanics perspective we propose a three stage min sum algorithm for efficiently dismantling networks including heavy tailed ones for which the dismantling and decycling problems are not equivalent we also provide additional insights into the dismantling problem concluding that it is an intrinsically collective problem and that optimal dismantling sets cannot be viewed as a collection of individually well performing nodes

linear system approximation via covariance equivalent realizations abstract a truncation technique for model reduction to simultaneously match a specified number of output covariance derivatives is described these covariance equivalent realizations which match q 1 covariance derivatives are called q covers in general q covers are not unique the additional freedom is used herein so that the q cover obtained also matches q markov parameters the truncation technique uses a form of the observability matrices of the full order system to determine a priori the order required of the reduced order model to match a specified number of output covariance derivatives and markov parameters the resulting realization is shown to be independent of the basis of the complete model to within a unitary transformation stability conditions for the reduced order model are also described and the relationships are established between stochastically equivalent realizations and the q covers

a note on closed loop balanced truncation closed loop balanced truncation is a model reduction method that aims at preserving the closed loop properties of the original controller we show that this method is equivalent to frequency weighted balanced truncation with certain weighting and that for observer based controllers a significant amount of computation can be saved because of the observer state feedback structure of the controller

two sided time domain moment matching for linear systems in this technical note we study the problem of computing reduced order models of order nu that match 2 nu moments of the given to be reduced system at two sets of nu interpolation points called two sided moment matching in the time domain framework we also study the time domain moment matching problem of the state space representations of the first order derivative of a transfer function based on this we show that the solution of the two sided time domain moment matching problem at non distinct nu interpolation points yields a unique nu order model that satisfies the following properties the model matches the moments of the given system at the interpolation points and the same property holds for the first order derivatives of the corresponding transfer functions

balanced performance preserving controller reduction two methods for controller reduction based on the closed loop system performance preserving criterion are proposed they are the closed loop block balanced and the closed loop structurally balanced truncating methods reduced order controllers are obtained directly by solving the closed loop system reduction problem with performance preserving weights

a survey of model reduction by balanced truncation and some new results balanced truncation is one of the most common model reduction schemes in this note we present a survey of balancing related model reduction methods and their corresponding error norms and also introduce some new results five balancing methods are studied 1 lyapunov balancing 2 stochastic balancing 3 bounded real balancing 4 positive real balancing and 5 frequency weighted balancing for positive real balancing we introduce a multiplicative type error bound moreover for a certain subclass of positive real systems a modified positive real balancing scheme with an absolute error bound is proposed we also develop a new frequency weighted balanced reduction method with a simple bound on the error system based on the frequency domain representations of the system gramians two numerical examples are illustrated to verify the efficiency of the proposed methods

near optimal frequency weighted interpolatory model reduction abstract this paper develops an interpolatory framework for weighted h 2 model reduction of mimo dynamical systems a new representation of the weighted h 2 inner products in mimo settings is introduced and used to derive associated first order necessary conditions satisfied by optimal weighted h 2 reduced order models equivalence of these new interpolatory conditions with earlier riccati based conditions given by halevi is also shown an examination of realizations for equivalent weighted h 2 systems leads then to an algorithm that remains tractable for large state space dimension several numerical examples illustrate the effectiveness of this approach and its competitiveness with frequency weighted balanced truncation and an earlier interpolatory approach the weighted iterative rational krylov algorithm

homogeneous complex networks abstract we discuss various ensembles of homogeneous complex networks and a monte carlo method of generating graphs from these ensembles the method is quite general and can be applied to simulate micro canonical canonical or grand canonical ensembles for systems with various statistical weights it can be used to construct homogeneous networks with desired properties or to construct a non trivial scoring function for problems of advanced motif searching

the covering number of the elements of a matroid and generalized matrix functions abstract let s be a nonempty finite set with cardinality m let m be a matroid on s with no loops the covering number of an element x in s is the smallest positive integer k such that x is a coloop of the union of k copies of m we investigate connections between the structure of m and the values of the covering numbers of elements of s applications to the study of the rank partition and generalized matrix functions are presented

approximative model reconstruction of cascade systems this letter considers how to approximately reconstruct a cascade system from a given unstructured system estimate many system identification methods including subspace methods provide reliable but generally unstructured black box models the problem we consider is how to find cascade systems that are close to such black box models for this we use model matching techniques and optimal weighted hankel norm approximation to obtain accurate low order cascade systems we show that it is possible to bound the reconstruction error in terms of an error tolerance parameter and weighted hankel singular values the suggested methods are illustrated on both a numerical example and a real double tank system with experimental data

a new algorithm for l 2 optimal model reduction in this paper the quadratically optimal model reduction problem for single input single output systems is considered the reduced order model is determined by minimizing the integral of the magnitude squared of the transfer function error it is shown that the numerator coefficients of the optimal approximant satisfy a weighted least squares problem and on this basis a two step iterative algorithm is developed combining a least squares solver with a gradient minimizer convergence of the proposed algorithm to stationary values of the quadratic cost function is proved the formulation is extended to handle the frequency weighted optimal model reduction problem three examples demonstrate the optimization algorithm

a note on the minimal polynomial of the kronecker sum of two linear operators abstract we give a lower bound for the degree of the minimal polynomial of the kronecker sum of two linear operators f and g in terms of the degrees of the minimal polynomials of f and g our result generalizes the known cauchy davenport theorem