spectra of complex networks we propose a general approach to the description of spectra of complex networks for the spectra of networks with uncorrelated vertices and a local treelike structure exact equations are derived these equations are generalized to the case of networks with correlations between neighboring vertices the tail of the density of eigenvalues ensuremath rho ensuremath lambda at large ensuremath lambda is related to the behavior of the vertex degree distribution p k at large k in particular as p k ensuremath sim k ensuremath ensuremath gamma ensuremath rho ensuremath lambda ensuremath sim ensuremath lambda 1 ensuremath 2 ensuremath gamma we propose a simple approximation which enables us to calculate spectra of various graphs analytically we analyze spectra of various complex networks and discuss the role of vertices of low degree we show that spectra of locally treelike random graphs may serve as a starting point in the analysis of spectral properties of real world networks e g of the internet

new solution method of linear static output feedback design problem for linear control systems abstract a new method of construction of the linear static output feedback for linear control systems is offered the essence of this method consists in construction of an initial approximation of the feedback matrix such that a numerical sequence which generated by arbitrary iterative method and beginning from this initial approximation converges to an exact solution of the feedback design problem examples are given

delocalization transition for the google matrix we study the localization properties of eigenvectors of the google matrix generated both from the world wide web and from the albert barabasi model of networks we establish the emergence of a delocalization phase for the pagerank vector when network parameters are changed for networks with localized pagerank eigenvalues of the matrix in the complex plane with a modulus above a certain threshold correspond to localized eigenfunctions while eigenvalues below this threshold are associated with delocalized relaxation modes we argue that for networks with delocalized pagerank the efficiency of information retrieval by google type search is strongly affected since the pagerank values have no clear hierarchical structure in this case

eigenvalue distribution of second order dynamic equations on time scales considered as fractals abstract let t a b be a time scale with a b t in this paper we study the asymptotic distribution of eigenvalues of the following linear problem u u with mixed boundary conditions u a u a 0 u b u b it is known that there exists a sequence of simple eigenvalues k k we consider the spectral counting function n t k k and we seek for its asymptotic expansion as a power of let d be the minkowski or box dimension of t which gives the order of growth of the number k t e of intervals of length e needed to cover t namely k t e e d we prove an upper bound of n which involves the minkowski dimension n t c d 2 where c is a positive constant depending only on the minkowski content of t roughly speaking its d volume although the minkowski content is not a measure we also consider certain limiting cases d 0 infinite minkowski content and we show a family of self similar fractal sets where n t admits two side estimates

a measurement of disorder in binary sequences abstract we propose a complex quantity a l to characterize the degree of disorder of l length binary symbolic sequences as examples we respectively apply it to typical random and deterministic sequences one kind of random sequences is generated from a periodic binary sequence and the other is generated from the logistic map the deterministic sequences are the fibonacci and thue morse sequences in these analyzed sequences we find that the modulus of a l denoted by a l is a statistically equivalent quantity to the boltzmann entropy the metric entropy the conditional block entropy and or other quantities so it is a useful quantitative measure of disorder it can be as a fruitful index to discern which sequence is more disordered moreover there is one and only one value of a l for the overall disorder characteristics it needs extremely low computational costs it can be easily experimentally realized from all these mentioned we believe that the proposed measure of disorder is a valuable complement to existing ones in symbolic sequences

exact pole assignment by output feedback part 3 abstract this part concludes a series of three papers concerning controllable and observable linear time invariant multivariable control systems in which the number of outputs plus the number of inputs exceeds the number of states following from the second paper fletcher 1987 it remains to consider the assignment by real output feedback of a set of distinct eigenvalues consisting of a mixture of real numbers and complex conjugate pairs of non real numbers we examine the possibility that in the algorithm described in the first paper fletcher and magni 1987 all the real eigenvalues can be assigned before any of the complex conjugate pairs if this is possible then the argument of fletcher 1987 can be used to complete the assignment it is only from the failure of this possibility that the pathological cases detailed in the first paper can arise

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

localizations on complex networks we study the structural characteristics of complex networks using the representative eigenvectors of the adjacent matrix the probability distribution function of the components of the representative eigenvectors are proposed to describe the localization on networks where the euclidean distance is invalid several quantities are used to describe the localization properties of the representative states such as the participation ratio the structural entropy and the probability distribution function of the nearest neighbor level spacings for spectra of complex networks whole cell networks in the real world and the watts strogatz small world and barabasi albert scale free networks are considered the networks have nontrivial localization properties due to the nontrivial topological structures it is found that the ascending order ranked series of the occurrence probabilities at the nodes behave generally multifractally this characteristic can be used as a structural measure of complex networks

pole assignment by output feedback using cylindrical algebraic decomposition recently a new polynomial method for pole assignment by static output feedback has been developed in this paper we show how this method can be generalized to partial pole clustering some poles are exactly assigned the others being clustered in a specified region of the complex plane a constructive and systematic algorithm for the numerical resolution of both problems the collins cylindrical algebraic decomposition algorithm is presented and applied to an illustrative example

eigenvalue spectra of complex networks we examine the eigenvalue spectrum of the adjacency matrix of a random scale free network with an average of p edges per vertex using the replica method we show how in the dense limit when p one can obtain two relatively simple coupled equations whose solution yields for an arbitrary complex network for scale free graphs with degree distribution exponent we obtain an exact expression for the eigenvalue spectrum when 3 and show that 1 2 1 for large in the limit we recover known results for the erdos renyi random graph

universality of level correlation function of sparse random matrices the statistical properties of sparse random matrices ensembles are investigated by means of a supersymmetric approach with the use of a functional generalization of the hubbard stratonovich transformation when used to calculate the density of states the method is shown to be absolutely equivalent to the replica trick the model turns out to bear a close resemblance to the anderson model on the bethe lattice it possesses a delocalization transition that occurs with an increase in the mean connectivity parameter in the delocalized phase the level level correlation function proves to have a universal dyson form with the full density of states replaced by the contribution from the infinite cluster

quantum graphs a simple model for chaotic scattering we connect quantum graphs with infinite leads and turn them into scattering systems we show that they display all the features which characterize quantum scattering systems with an underlying classical chaotic dynamics typical poles delay time and conductance distributions ericson fluctuations and when considered statistically the ensemble of scattering matrices reproduces quite well the predictions of the appropriately defined random matrix ensembles the underlying classical dynamics can be defined and it provides important parameters which are needed for the quantum theory in particular we derive exact expressions for the scattering matrix and an exact trace formula for the density of resonances in terms of classical orbits analogous to the semiclassical theory of chaotic scattering we use this in order to investigate the origin of the connection between random matrix theory and the underlying classical chaotic dynamics being an exact theory and due to its relative simplicity it offers new insights into this problem which is at the forefront of the research in chaotic scattering and related fields

random matrix analysis of complex networks we study complex networks under random matrix theory rmt framework using nearest neighbor and next nearest neighbor spacing distributions we analyze the eigenvalues of the adjacency matrix of various model networks namely random scale free and small world networks these distributions follow the gaussian orthogonal ensemble statistic of rmt to probe long range correlations in the eigenvalues we study spectral rigidity via the ensuremath delta 3 statistic of rmt as well it follows rmt prediction of linear behavior in semilogarithmic scale with the slope being ensuremath sim 1 ensuremath pi 2 random and scale free networks follow rmt prediction for very large scale a small world network follows it for sufficiently large scale but much less than the random and scale free networks

a numerical study of sparse random matrices a numerical study is presented for the eigensolution statistics of largen n real and symmetric sparse random matrices as a function of the mean numberp of nonzero elements per row the model shows classical percolation and quantum localization transitions atp c 1 andp q 1 respectively in the rigid limitp n we demonstrate that the averaged density of states follows the wigner semicircle law and the corresponding nearest energy level spacing distribution functionp s obeys the wigner surmise in the very sparse matrix limitp n withp p q a singularity e 1 e is found as e 0 and exponential tails develop in the high e regions but thep s distribution remains consistent with level repulsion the localization properties of the model are examined by studying both the eigenvector amplitude and the density fluctuations the valuep q 1 4 is roughly estimated in agreement with previous studies of the anderson transition in dilute bethe lattices

spectral properties of the google matrix of the world wide web and other directed networks we study numerically the spectrum and eigenstate properties of the google matrix of various examples of directed networks such as vocabulary networks of dictionaries and university world wide web networks the spectra have gapless structure in the vicinity of the maximal eigenvalue for google damping parameter equal to unity the vocabulary networks have relatively homogeneous spectral density while university networks have pronounced spectral structures which change from one university to another reflecting specific properties of the networks we also determine specific properties of eigenstates of the google matrix including the pagerank the fidelity of the pagerank is proposed as a characterization of its stability

on eigenstructure assignment by gain output feedback in this paper the eigenstructure assignment of linear multivariable control systems is studied from a geometric point of view for the class of systems in which the number of outputs plus the number of inputs exceeds the number of states genericity properties relative to this problem are derived it is shown without any assumption on the genericity of the system that the pole assignment can be carried out by choosing some closed loop eigenvectors almost freely the crucial point is that all the expected degrees of freedom in pole assignment are so described without redundancy which fully justifies the practical interest of such techniques despite the technicality required for the derivation of intermediate results the main result which is an eigenstructure assignment algorithm is very easy to implement because it is only based on the computation of certain sums and intersections of characteristic subspaces furthermore it is shown how the basic tools developed here can be used to tackle the prob

trapping of continuous time quantum walks on erdos renyi graphs abstract we consider the coherent exciton transport modeled by continuous time quantum walks on erdos reny graphs in the presence of a random distribution of traps the role of trap concentration and of the substrate dilution is deepened showing that at long times and for intermediate degree of dilution the survival probability typically decays exponentially with a average decay rate which depends non monotonically on the graph connectivity when the degree of dilution is either very low or very high stationary states not affected by traps get more likely giving rise to a survival probability decaying to a finite value both these features constitute a qualitative difference with respect to the behavior found for classical walks

dissimilar behaviors of coherent exciton transport on scale free networks with identical degree sequence conventional wisdom is that the power law degree sequence has a profound impact on diverse dynamical processes taking place on scale free networks here we show that power law degree distribution alone is not sufficient to determine the behaviors of coherent exciton transport on scale free networks for this purpose we study coherent exciton transport on a class of scale free networks controlled by a tunable parameter q which have the same degree sequence independent of q but are very different in other structural properties determined by q such as average distance and fractality first through numerical determination of the time evolutions of return probabilities we compare in detail the dissimilarities in oscillation periods and oscillation amplitudes between two limiting cases of q 0 and q 1 second we demonstrate that in the network of q 0 up to the third generation the quantum transport problem between node groups partitioned by node degrees can be mapped onto a line furthermore we show that in the case of long time limit the transition probabilities in the networks of q 0 and q 1 exhibit different characteristic patterns with identical values finally we find that the degree averaged return probability decreases when the parameter q increases from 0 to 1 indicating that the stronger localization can be found in those networks with lower values of q our results suggest that it is far from enough to characterize the behaviors of coherent exciton transport on scale free networks merely by power law degree distribution

on the density of states of sparse random matrices the supersymmetric method of calculation of the density of states of a sparse random matrix is shown to be absolutely equivalent to the replica trick a functional generalization of the hubbard stratonovich transformation was used in the course of the solution

deformed gaussian orthogonal ensemble description of small world networks the study of spectral behavior of networks has gained enthusiasm over the last few years in particular random matrix theory rmt concepts have proven to be useful in discussing transition from regular behavior to fully chaotic behavior it has been found that an extrapolation formula of the brody type can be used in the present paper we analyze the regular to chaotic behavior of small world sw networks using an extension of the gaussian orthogonal ensemble this rmt ensemble coined the deformed gaussian orthogonal ensemble dgoe supplies a natural foundation of the brody formula sw networks follow goe statistics until a certain range of eigenvalue correlations depending upon the strength of random connections we show that for these regimes of sw networks where spectral correlations do not follow goe beyond a certain range dgoe statistics models the correlations very well the analysis performed in this paper proves the utility of the dgoe in network physics as much as it has been useful in other physical systems

spectra of modular random graphs we compute spectra of symmetric random matrices defined on graphs exhibiting a modular structure modules are initially introduced as fully connected sub units of a graph by contrast inter module connectivity is taken to be incomplete two different types of inter module connectivities are considered one where the number of intermodule connections per node diverges and one where this number remains finite in the infinite module size limit in the first case the results can be understood as a perturbation of a superposition of semicircular spectral densities one would obtain for uncoupled modules in the second case matters can be more involved and depend in detail on inter module connectivities for suitable parameters we even find near triangular shaped spectral densities similar to those observed in certain scale free networks in a system consisting of just two coupled modules analytic results are presented for the infinite module size limit they are well corroborated by numerical simulations

green s functions and eigenvalue comparisons for a focal problem on time scales abstract the study of dynamic systems on time scales unifies continuous and discrete processes in this paper we shall examine green s function for an n th order focal boundary value problem with this we can then consider eigenvalue comparisons for higher order focal boundary value problems on time scales using the theory of operators on a banach space

normal mode analysis of spectra of random networks abstract several spectral fluctuation measures of random matrix theory rmt have been applied in the study of spectral properties of networks however the calculation of those statistics requires performing an unfolding procedure which may not be an easy task in this work network spectra are interpreted as time series and we show how their short and long range correlations can be characterized without implementing any previous unfolding in particular we consider three different representations of erd s renyi er random networks standard er networks er networks with random weighted self edges and fully random weighted er networks in each case we apply singular value decomposition svd such that the spectra are decomposed in trend and fluctuation normal modes we obtain that the fluctuation modes exhibit a clear crossover between the poisson and the gaussian orthogonal ensemble statistics when the average degree of er networks changes moreover by using the trend modes we perform a data adaptive unfolding to calculate for comparison purposes traditional fluctuation measures such as the nearest neighbor spacing distribution number variance 2 as well as 3 and n statistics the thorough comparison of rmt short and long range correlation measures make us identify the svd method as a robust tool for characterizing random network spectra

a geometric framework for pole assignment algorithms the problem of pole assignment by gain output feedback or by low order dynamical compensator is considered from a geometric point of view this makes it possible to unify in a general framework most of the existing pole assignment methods formulated in a state space context such as the minimal order observers the brasch pearson compensator and the methods proposed by h kimura and to simplify their presentation moreover new pole assignment algorithms may be derived from this general formulation

random matrix analysis of localization properties of gene coexpression network we analyze gene coexpression network under the random matrix theory framework the nearest neighbor spacing distribution of the adjacency matrix of this network follows gaussian orthogonal statistics of random matrix theory rmt spectral rigidity test follows random matrix prediction for a certain range and deviates afterwards eigenvector analysis of the network using inverse participation ratio suggests that the statistics of bulk of the eigenvalues of network is consistent with those of the real symmetric random matrix whereas few eigenvalues are localized based on these ipr calculations we can divide eigenvalues in three sets a the nondegenerate part that follows rmt b the nondegenerate part at both ends and at intermediate eigenvalues which deviates from rmt and expected to contain information about important nodes in the network c the degenerate part with zero eigenvalue which fluctuates around rmt predicted value we identify nodes corresponding to the dominant modes of the corresponding eigenvectors and analyze their structural properties

density of states of sparse random matrices a supersymmetric formalism is used to derive a set of equations giving the density of states of any real symmetric sparse random matrix as a function of the distribution of non zero elements and the mean number of non zero elements per row p in the matrix where the non zero elements take the values or 1 with equal probability the equations are solved as p to infinity recovering results obtained previously with the replica method as e to 0 the density of states rho e behaves as 1 e ln e 2 the more general case where or 1 occur with unequal probabilities is also considered

experimental and numerical investigation of the reflection coefficient and the distributions of wigner s reaction matrix for irregular graphs with absorption we present the results of an experimental and numerical study of the distribution of the reflection coefficient p r and the distributions of the imaginary p v and the real p u parts of the wigner reaction k matrix for irregular fully connected hexagon networks graphs in the presence of strong absorption in the experiment we used microwave networks which were built of coaxial cables and attenuators connected by joints in the numerical calculations experimental networks were described by quantum fully connected hexagon graphs the presence of absorption introduced by attenuators was modeled by optical potentials the distribution of the reflection coefficient p r and the distributions of the reaction k matrix were obtained from measurements and numerical calculations of the scattering matrix s of the networks and graphs respectively we show that the experimental and numerical results are in good agreement with the exact analytic ones obtained within the framework of random matrix theory

random networks eigenvalue spectra abstract we analyze the spectra of eigenvalues for random graphs with a local tree like structure the exact equations to the spectra of networks with a local tree like structure are presented we propose a simple approximation and in the framework of effective medium approximation calculate spectra of various graphs analytically we show that spectra of locally tree like random graphs gives a good description of the spectral properties of real life networks like the internet

localization transition on complex networks via spectral statistics the spectral statistics of complex networks are numerically studied the features of the anderson metal insulator transition are found to be similar for a wide range of different networks a metal insulator transition as a function of the disorder can be observed for different classes of complex networks for which the average connectivity is small the critical index of the transition corresponds to the mean field expectation when the connectivity is higher the amount of disorder needed to reach a certain degree of localization is proportional to the average connectivity though a precise transition cannot be identified the absence of a clear transition at high connectivity is probably due to the very compact structure of the highly connected networks resulting in a small diameter even for a large number of sites

spectra of modular and small world matrices we compute spectra of symmetric random matrices describing graphs with general modular structure and arbitrary inter and intra module degree distributions subject only to the constraint of finite mean connectivities we also evaluate spectra of a certain class of small world matrices generated from random graphs by introducing shortcuts via additional random connectivity components both adjacency matrices and the associated graph laplacians are investigated for the laplacians we find lifshitz type singular behaviour of the spectral density in a localized region of small values in the case of modular networks we can identify contributions of local densities of state from individual modules for small world networks we find that the introduction of short cuts can lead to the creation of satellite bands outside the central band of extended states exhibiting only localized states in the band gaps results for the ensemble in the thermodynamic limit are in excellent agreement with those obtained via a cavity approach for large finite single instances and with direct diagonalization results

scattering and transport properties of tight binding random networks we study numerically scattering and transport statistical properties of tight binding random networks characterized by the number of nodes n and the average connectivity we use a scattering approach to electronic transport and concentrate on the case of a small number of single channel attached leads we observe a smooth crossover from insulating to metallic behavior in the average scattering matrix elements the conductance probability distribution w t the average conductance the shot noise power p and the elastic enhancement factor f by varying from small 0 to large 1 values we also show that all these quantities are invariant for fixed n moreover we proposes a heuristic and universal relation between and p and the disorder parameter

spectral properties of the apollonian network abstract a first characterization of the spectral properties of the apollonian network is presented results are based on the numerically evaluated eigenvalue spectra of adjacency matrices for the first nine generations of the network the main features refer to the presence of several gaps on the eigenvalue axis and a very large number of degenerated eigenvalues the increase in the degeneracy of a family of eigenvalues is found to obey a very simple empirical relation

anderson model on a cayley tree the density of states the authors describe a method of calculating the electronic density of states on an arbitrary tree they use it for the problem of anderson localization on a regular tree with random site energies and obtain the first term of a weak disorder expansion of the density of states

random matrix analysis of network laplacians abstract we analyse the eigenvalue fluctuations of the laplacian of various networks under the random matrix theory framework analyses of random networks scale free networks and small world networks show that the nearest neighbor spacing distribution of the laplacian of these networks follow gaussian orthogonal ensemble statistics of the random matrix theory furthermore we study the nearest neighbor spacing distribution as a function of the random connections and find that the transition to the gaussian orthogonal ensemble statistics occurs at the small world transition

exact pole assignment by output feedback part 1 abstract we shall prove that exact assignment of distinct poles using output feedback is possible in any controllable and observable linear time invariant multivariable system in which the number of inputs plus the number of outputs exceeds the number of states except in clearly identified circumstances furthermore we give a construction for the entire class of pathological cases the main part of our argument is concerned with the assigning of complex conjugate eigenvalues by means of a real feedback matrix indeed we begin by giving a simple algorithm which is effective over any field although this problem has been discussed by numerous authors in the past they have usually presented only generic results we examine the complete class of systems satisfying the hypotheses mentioned

disturbance decoupling by measurement feedback with stability or pole placement in this paper we solve the disturbance decoupling problem by measurement feedback and requiring stability or pole placement on the closed loop system the problem is attacked using the geometric approach through the concepts of a bmod mathcal b invariant and controllability subspaces and their duals a mid k invariant and complementary observability subspaces the solution of this problem has an interesting structure consisting of a feedback processor which decomposes into i a disturbance decoupling loop ii a disturbance input stabilization or pole placement loop and iii a controlled output stabilization or pole placement loop

universality in complex networks random matrix analysis we apply random matrix theory to complex networks we show that nearest neighbor spacing distribution of the eigenvalues of the adjacency matrices of various model networks namely scale free small world and random networks follow universal gaussian orthogonal ensemble statistics of random matrix theory second we show an analogy between the onset of small world behavior quantified by the structural properties of networks and the transition from poisson to gaussian orthogonal ensemble statistics quantified by brody parameter characterizing a spectral property we also present our analysis for a protein protein interaction network in budding yeast

universal spectral statistics in quantum graphs we prove that the spectrum of an individual chaotic quantum graph shows universal spectral correlations as predicted by random matrix theory the stability of these correlations with regard to nonuniversal corrections is analyzed in terms of the linear operator governing the classical dynamics on the graph

spectral analysis of deformed random networks we study spectral behavior of sparsely connected random networks under the random matrix framework subnetworks without any connection among them form a network having perfect community structure as connections among the subnetworks are introduced the spacing distribution shows a transition from the poisson statistics to the gaussian orthogonal ensemble statistics of random matrix theory the eigenvalue density distribution shows a transition to the wigner s semicircular behavior for a completely deformed network the range for which spectral rigidity measured by the dyson mehta ensuremath delta 3 statistics follows the gaussian orthogonal ensemble statistics depends upon the deformation of the network from the perfect community structure the spacing distribution is particularly useful to track very slight deformations of the network from a perfect community structure whereas the density distribution and the ensuremath delta 3 statistics remain identical to the undeformed network on the other hand the ensuremath delta 3 statistics is useful for the larger deformation strengths finally we analyze the spectrum of a protein protein interaction network for helicobacter and compare the spectral behavior with those of the model networks

experimental simulation of quantum graphs by microwave networks we present the results of experimental and theoretical study of irregular tetrahedral microwave networks consisting of coaxial cables annular waveguides connected by t joints the spectra of the networks were measured in the frequency range 0 0001 16 ghz in order to obtain their statistical properties such as the integrated nearest neighbor spacing distribution and the spectral rigidity d3sld the comparison of our experimental and theoretical results shows that microwave networks can simulate quantum graphs with time reversal symmetry in particular we use the spectra of the microwave networks to study the periodic orbits of the simulated quantum graphs we also present experimental study of directional microwave networks consisting of coaxial cables and faraday isolators for which the time reversal symmetry is broken in this case our experimental results indicate that spectral statistics of directional microwave networks deviate from predictions of gaussian orthogonal ensembles in random matrix theory approaching especially for small eigenfrequency spacing s results for gaussian unitary ensembles experimental results are supported by the theoretical analysis of directional graphs

spectral density of complex networks with a finite mean degree in order to clarify the statistical features of complex networks the spectral density of adjacency matrices has often been investigated adopting a static model introduced by goh kahng and kim we analyse the spectral density of complex scale free networks for this purpose we utilize the replica method and effective medium approximation ema in statistical mechanics as a result we identify a new integral equation which determines the asymptotic spectral density of scale free networks with a finite mean degree p in the limit p known asymptotic formulae are rederived moreover the 1 p corrections to known results are analytically calculated by a perturbative method

parameter dependent spectral statistics of chaotic quantum graphs neumann versus circular orthogonal ensemble boundary conditions the parameter dependent spectral statistics of totally connected quantum graphs with n 4 30 vertices such as the parametric velocities correlation functions and the distribution of curvatures are studied the inverse participation ratio ipr an important measure of localization effects was also numerically investigated in the calculations we successfully used two different theoretical approaches the first approach was based on the graphs eigenenergies and wave functions calculations while the second one used the eigenphases and the eigenvectors of the bond scattering matrix s k we considered graphs with neumann and circular orthogonal ensemble coe boundary conditions we show that in contrast to large neumann graphs for which the departure of many parameter dependent spectral statistics from the random matrix theory rmt predictions is observed for large coe graphs the spectral statistics and ipr are in good agreement with the rmt predictions

spectral properties of directed random networks with modular structure we study spectra of directed networks with inhibitory and excitatory couplings we investigate in particular eigenvector localization properties of various model networks for different values of correlation among their entries spectra of random networks with completely uncorrelated entries show a circular distribution with delocalized eigenvectors whereas networks with correlated entries have localized eigenvectors in order to understand the origin of localization we track the spectra as a function of connection probability and directionality as connections are made directed eigenstates start occurring in complex conjugate pairs and the eigenvalue distribution combined with the localization measure shows a rich pattern moreover for a very well distinguished community structure the whole spectrum is localized except few eigenstates at the boundary of the circular distribution as the network deviates from the community structure there is a sudden change in the localization property for a very small value of deformation from the perfect community structure we search for this effect for the whole range of correlation strengths and for different community configurations furthermore we investigate spectral properties of a metabolic network of zebrafish and compare them with those of the model networks

eigenvalue distribution of large dilute random matrices we study the eigenvalue distribution of dilute n n random matrices hn that in the pure undiluted case describe the hopfield model we prove that for the fixed dilution parameter a the normalized counting function ncf of hn converges as n to a unique a we find the moments of this distribution explicitly analyze the 1 a correction and study the asymptotic properties of a for large we prove that a converges as a to the wigner semicircle distribution scd we show that the scd is the limit of the ncf of other ensembles of dilute random matrices this could be regarded as evidence of stability of the scd to dilution or more generally to random modulations of large random matrices

a new approach to eigenstructure assignment by output feedback the eigenstructure assignment problem for a linear time invariant multi input multi output system using output feedback is considered a new approach is developed which identifies the eigenspaces for the desired set of all the closed loop eigenvalues for the assignment of this desired set necessary and sufficient conditions are established these conditions contain two coupled sylvester matrix equations one of which is proven to be a reduced order square sylvester matrix equation this results in an efficient analytical procedure numerically superior to known techniques for the determination of the output feedback gain matrix

spectral correlations of individual quantum graphs we investigate the spectral properties of chaotic quantum graphs we demonstrate that the energy average over the spectrum of individual graphs can be traded for the functional average over a supersymmetric nonlinear model action this proves that spectral correlations of individual quantum graphs behave according to the predictions of wigner dyson random matrix theory we explore the stability of the universal random matrix behavior with regard to perturbations and discuss the crossover between different types of symmetries

von neumann entropy and localization delocalization transition of electron states in quantum small world networks the von neumann entropy for an electron in periodic disorder and quasiperiodic quantum small world networks qswn s is studied numerically for the disorder qswn s the derivative of the spectrum averaged von neumann entropy is maximal at a certain density of shortcut links p which can be as a signature of the localization delocalization transition of electron states the transition point p is agreement with that obtained by the level statistics method for the quasiperiodic qswn s it is found that there are two regions of the potential parameter the behaviors of electron states in different regions are similar to that of periodic and disorder qswn s respectively

higher order self adjoint boundary value problems on time scales in this study higher order self adjoint differential expressions on time scales and their associated self adjoint boundary conditions are discussed the symmetry property of the corresponding green s functions is shown together with specific formulas of green s functions for select time scales