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

fake projective planes a fake projective plane is a complex surface different from but has the same betti numbers as the complex projective plane it is a complex hyperbolic space form and has the smallest euler poincare characteristic among smooth surfaces of general type the first example was constructed by mumford later on two more examples were found by ishida and kato a fourth possible one was recently proposed by keum in a recent joint work with gopal prasad we showed that there are seventeen non empty classes of fake projective planes and there can at most be four more specific classes higher dimensional analogues and examples were also obtained the main purpose of the talk is to explain the joint work with prasad and other related results such as arithmeticity of the lattices involved obtained earlier by klingler and yeung independently

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

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

moving switching surfaces for robust control of second order variable structure systems abstract most of the switching surfaces proposed so far for a variable structure control system vscs have been determined independently of initial conditions the vscs with these typical switching surfaces may be sensitive to parameter variations and extraneous disturbances during the reaching phase to overcome this drawback we propose a new switching surface which is initially designed to pass arbitrary initial conditions and subsequently move towards a predetermined switching surface by rotating or and shifting we call it a moving switching surface mss using the mss a low sensitivity system is obtained through shortening the reaching phase furthermore the system robustness is guaranteed during whole intervals of control action by eliminating the reaching phase to illustrate the advantages of the proposed method a simple second order linear system subjected to external disturbance is considered as a preliminary example followed by a two link manipulator

a new family of surfaces of general type with k 2 7 and p_g 0 we construct a new family of smooth minimal surfaces of general type with k 2 7 and p g 0 we show that a surface in this family has ample canonical divisor and birational bicanonical morphism we also prove that these surfaces satisfy bloch s conjecture

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

finding community structure in networks using the eigenvectors of matrices we consider the problem of detecting communities or modules in networks groups of vertices with a higher than average density of edges connecting them previous work indicates that a robust approach to this problem is the maximization of the benefit function known as modularity over possible divisions of a network here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix which plays a role in community detection similar to that played by the graph laplacian in graph partitioning calculations this result leads us to a number of possible algorithms for detecting community structure as well as several other results including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong the algorithms and measures proposed are illustrated with applications to a variety of real world complex networks

multistep greedy algorithm identifies community structure in real world and computer generated networks we have recently introduced a multistep extension of the greedy algorithm for modularity optimization the extension is based on the idea that merging l pairs of communities l 1 at each iteration prevents premature condensation into few large communities here an empirical formula is presented for the choice of the step width l that generates partitions with close to optimal modularity for 17 real world and 1100 computer generated networks furthermore an in depth analysis of the communities of two real world networks the metabolic network of the bacterium e coli and the graph of coappearing words in the titles of papers coauthored by martin karplus provides evidence that the partition obtained by the multistep greedy algorithm is superior to the one generated by the original greedy algorithm not only with respect to modularity but also according to objective criteria in other words the multistep extension of the greedy algorithm reduces the danger of getting trapped in local optima of modularity and generates more reasonable partitions

analysis of weighted networks the connections in many networks are not merely binary entities either present or not but have associated weights that record their strengths relative to one another recent studies of networks have by and large steered clear of such weighted networks which are often perceived as being harder to analyze than their unweighted counterparts here we point out that weighted networks can in many cases be analyzed using a simple mapping from a weighted network to an unweighted multigraph allowing us to apply standard techniques for unweighted graphs to weighted ones as well we give a number of examples of the method including an algorithm for detecting community structure in weighted networks and a simple proof of the maximum flow minimum cut theorem

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

self affine fractals embedded in spectra of complex networks the scaling properties of spectra of real world complex networks are studied by using the wavelet transform it is found that the spectra of networks are multifractal according to the values of the long range correlation exponent the hust exponent h the networks can be classified into three types namely h 0 5 h 0 5 and h 0 5 all real world networks considered belong to the class of h 0 5 which may be explained by the hierarchical properties

quotients of products of curves new surfaces with pg 0 and their fundamental groups we construct many new surfaces of general type with q pg 0 whose canonical model is the quotient of the product of two curves by the action of a finite group g constructing in this way many new interesting fundamental groups which distinguish connected components of the moduli space of surfaces of general type we indeed classify all such surfaces whose canonical model is singular the smooth case was classified in an earlier work as an important tool we prove a structure theorem giving a precise description of the fundamental group of quotients of products of curves by the action of a finite group g

spectra of random graphs with given expected degrees in the study of the spectra of power law graphs there are basically two competing approaches one is to prove analogues of wigner s semicircle law whereas the other predicts that the eigenvalues follow a power law distribution although the semicircle law and the power law have nothing in common we will show that both approaches are essentially correct if one considers the appropriate matrices we will prove that under certain mild conditions the eigenvalues of the normalized laplacian of a random power law graph follow the semicircle law whereas the spectrum of the adjacency matrix of a power law graph obeys the power law our results are based on the analysis of random graphs with given expected degrees and their relations to several key invariants of interest are a number of new values for the exponent where phase transitions for eigenvalue distributions occur the spectrum distributions have direct implications to numerous graph algorithms such as for example randomized algorithms that involve rapidly mixing markov chains

laplacian spectra as a diagnostic tool for network structure and dynamics we examine numerically the three way relationships among structure laplacian spectra and frequency synchronization dynamics on complex networks we study the effects of clustering degree distribution and a particular type of coupling asymmetry input normalization all of which are known to have effects on the synchronizability of oscillator networks we find that these topological factors produce marked signatures in the laplacian eigenvalue distribution and in the localization properties of individual eigenvectors using a set of coordinates based on the laplacian eigenvectors as a diagnostic tool for synchronization dynamics we find that the process of frequency synchronization can be visualized as a series of quasi independent transitions involving different normal modes particular features of the partially synchronized state can be understood in terms of the behavior of particular modes or groups of modes for example there are important partially synchronized states in which a set of low lying modes remain unlocked while those in the main spectral peak are locked we find therefore that spectra are correlated with dynamics in ways that go beyond results relating a single threshold to a single extremal eigenvalue

fibred surfaces varieties isogenous to a product and related moduli spaces a fibration of an algebraic surface s over a curve b with fibres of genus at least 2 has constant moduli iff it is birational to the quotient of a product of curves by the action of a finite group g a variety isogenous to a higher product is the quotient of a product of curves of genus at least 2 by the free action of a finite group theorem b gives a characterization of surfaces isogenous to a higher product in terms of the fundamental group and of the euler number theorem c classifies the groups thus occurring and shows that after fixing the group and the euler number one obtains an irreducible moduli space the result of theorem b is extended to higher dimension in theorem g thus generalizing cf also theorem h results of jost yau and mok concerning varieties whose universal cover is a polydisk theorem a shows that fibrations where the fibre genus and the genus of the base b are at least 2 are invariants of the oriented differentiable structure the main theorems d and e characterize surfaces carrying constant moduli fibrations as surfaces having a zariski open set satisfying certain topological conditions e g having the right euler number the right fundamental group and the right fundamental group at infinity 0 introduction the study of fibrations of a smooth algebraic surface s over a smooth algebraic curve b lies at the heart of the classification theory and of the geometry of algebraic surfaces our main results in this paper theo rems d e concern a topological characterization of surfaces admitting a constant moduli fibration these results are intimately related to a series of variations on the same theme also in higher dimension and to a circle of allied techniques which we present in the course of the paper the first new result we have in this direction and which is presented in section 2 after we recall several known results some classical some quite recent is the following theorem on the dif ferentiable invariance of the genus of fibres cf theorem 2 9 for a more precise statement theorem a let f s b be a fibration of the surface s over a curve b of genus b 2 and with fibre f of genus g 1 assume that s is another smooth compact kahler surface which is orientedly diffeomorphic to s then the subspace of h 1 s corresponding to f h 1 b determines a fibration f s b with

random matrix analysis of human eeg data we use random matrix theory to demonstrate the existence of generic and subject independent features of the ensemble of correlation matrices extracted from human eeg data in particular the spectral density as well as the level spacings was analyzed and shown to be generic and subject independent we also investigate number variance distributions in this case we show that when the measured subject is visually stimulated the number variance displays deviations from the random matrix prediction

laplacian spectra of and random walks on complex networks are scale free architectures really important we study the laplacian operator of an uncorrelated random network and as an application consider hopping processes diffusion random walks signal propagation etc on networks we develop a strict approach to these problems we derive an exact closed set of integral equations which provide the averages of the laplacian operator s resolvent this enables us to describe the propagation of a signal and random walks on the network we show that the determining parameter in this problem is the minimum degree q m of vertices in the network and that the high degree part of the degree distribution is not that essential the position of the lower edge of the laplacian spectrum c appears to be the same as in the regular bethe lattice with the coordination number q m namely c 0 if q m 2 and c 0 if q m 2 in both of these cases the density of eigenvalues p 0 as c 0 but the limiting behaviors near c are very different in terms of a distance from a starting vertex the hopping propagator is a steady moving gaussian broadening with time this picture qualitatively coincides with that for a regular bethe lattice our analytical results include the spectral density p near c and the long time asymptotics of the autocorrelator and the propagator

spectra of real world graphs beyond the semicircle law results on the spectra of adjacency matrices corresponding to models of real world graphs we find that when the number of links grows as the number of nodes the spectral density of uncorrelated random matrices does not converge to the semicircle law furthermore the spectra of real world graphs have specific features depending on the details of the corresponding models in particular scale free graphs develop a trianglelike spectral density with a power law tail while small world graphs have a complex spectral density consisting of several sharp peaks these and further results indicate that the spectra of correlated graphs represent a practical tool for graph classification and can provide useful insight into the relevant structural properties of real networks

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

on images of mori dream spaces the purpose of this paper is to study the geometry of images of morphisms from mori dream spaces first we prove that a variety which admits a surjective morphism from a mori dream space is again a mori dream space secondly we introduce a natural fan structure on the effective cone of mori dream spaces we show that it encodes the information of zariski decompositions which in turn is equivalent to the information of the variation of git quotients of their cox rings finally we show that under a surjective morphism between mori dream spaces the fan of the target space coincides with the restriction of the fan of the source

description of chaos order transition with random matrices within the maximum entropy principle the deformed gaussian orthogonal ensemble introduced earlier is developed here for large dimensional matrices both the spacing and eigenvector distributions are studied and compared to other ones suggested for the chaos order transition problem the concept of a universal lower entropy with respect to the gaussian orthogonal ensemble entropy is proved very useful

cox rings of surfaces and the anticanonical iitaka dimension abstract in this paper we investigate the relation between the finite generation of the cox ring r x of a smooth projective surface x and its anticanonical iitaka dimension k x

negative curves on algebraic surfaces we study curves of negative self intersection on algebraic surfaces our main result shows there exist smooth complex projective surfaces x related to hilbert modular surfaces such that x contains reduced irreducible curves c of arbitrarily negative self intersection c 2 previously the only known examples of surfaces for which c 2 was not bounded below were in positive characteristic and the general expectation was that no examples could arise over the complex numbers indeed we show that the idea underlying the examples in positive characteristic cannot produce examples over the complex number field and thus our complex examples require a different approach

on the spectrum of the normalized graph laplacian we investigate how the spectrum of the normalized geometric graph laplacian is affected by operations like motif dougling graph splitting or joining the multiplicity of the eigenvalue 1 or equivalently the dimension of the kernel of the adjacency matrix of the graph is of particular interest this multiplicity can be increased for instance by motif doubling

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

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

differential equations as a tool for community identification we consider the task of identification of a cluster structure in random networks the results of two methods are presented i the newman algorithm m e j newman and m girvan phys rev e 69 026113 2004 and ii our method based on differential equations a series of computer experiments is performed to check if in applying these methods we are able to determine the structure of the network the trial networks consist initially of well defined clusters and are disturbed by introducing noise into their connectivity matrices further we show that an improvement of the previous version of our method is possible by an appropriate choice of the threshold parameter beta with this change the results obtained by the two methods above are similar and our method works better for all the computer experiments we have done

community detection as an inference problem we express community detection as an inference problem of determining the most likely arrangement of communities we then apply belief propagation and mean field theory to this problem and show that this leads to fast accurate algorithms for community detection

examples of mori dream surfaces of general type with pg 0 abstract in this paper we study effective nef and semiample cones of minimal surfaces of general type with p g 0 we provide examples of minimal surfaces of general type with p g 0 and 2 k 2 9 which are mori dream spaces on these examples we also give explicit description of their effective cones with all negative curves we also present non minimal surfaces of general type with p g 0 that are not mori dream surfaces

statistics of atmospheric correlations for a large class of quantum systems the statistical properties of their spectrum show remarkable agreement with random matrix predictions recent advances show that the scope of random matrix theory is much wider in this work we show that the random matrix approach can be beneficially applied to a completely different classical domain namely to the empirical correlation matrices obtained from the analysis of the basic atmospheric parameters that characterize the state of atmosphere we show that the spectrum of atmospheric correlation matrices satisfy the random matrix prescription in particular the eigenmodes of the atmospheric empirical correlation matrices that have physical significance are marked by deviations from the eigenvector distribution

derived categories of burniat surfaces and exceptional collections we construct an exceptional collection varupsilon of maximal possible length 6 on any of the burniat surfaces with k x 2 6 a 4 dimensional family of surfaces of general type with p g q 0 we also calculate the dg algebra of endomorphisms of this collection and show that the subcategory generated by this collection is the same for all burniat surfaces the semiorthogonal complement mathcal a of varupsilon is an almost phantom category it has trivial hochschild homology and k 0 mathcal a mathbb z 2 6