continuum percolation in two dimensions monte carlo tests of scaling and universality for non interacting discs detailed results are reported for the connectivity properties of a system of discs of unit radius free to be situated anywhere within a square of area 2l2 ordinary lattke percolation would correspond to the discs being situated on the vertices of a j2l x j2l lattice computer simulations are carried out for a sequence of increasing system sizes ranging from l 20 to l 1000 for each value of l a large number of realisations are generated for 25 values of the disc concentration x we calculate a variety of estimates for the threshold parameter x as well as the critical exponents p y t and v our exponent estimates are in close agreement with accepted values for ordinary lattice percolation therefore this continuum system appears to be in the same universality class as lattice percolation

influence of memory in deterministic walks in random media analytical calculation within a mean field approximation consider a random medium consisting of n points randomly distributed so that there is no correlation among the distances separating them this is the random link model which is the high dimensionality limit mean field approximation for the euclidean random point structure in the random link model at discrete time steps a walker moves to the nearest point which has not been visited in the last ensuremath mu steps memory producing a deterministic partially self avoiding walk the tourist walk we have analytically obtained the distribution of the number n of points explored by the walker with memory ensuremath mu 2 as well as the transient and period joint distribution this result enables us to explain the abrupt change in the exploratory behavior between the cases ensuremath mu 1 memoryless walker driven by extreme value statistics and ensuremath mu 2 walker with memory driven by combinatorial statistics in the ensuremath mu 1 case the mean newly visited points in the thermodynamic limit n 1 is just n e 2 72 dots while in the ensuremath mu 2 case the mean number n of visited points grows proportionally to n 1 2 also this result allows us to establish an equivalence between the random link model with ensuremath mu 2 and random map uncorrelated back and forth distances with ensuremath mu 0 and the abrupt change between the probabilities for null transient time and subsequent ones

detection of signals by information theoretic criteria a new approach is presented to the problem of detecting the number of signals in a multichannel time series based on the application of the information theoretic criteria for model selection introduced by akaike aic and by schwartz and rissanen mdl unlike the conventional hypothesis testing based approach the new approach does not requite any subjective threshold settings the number of signals is obtained merely by minimizing the aic or the mdl criteria simulation results that illustrate the performance of the new method for the detection of the number of signals received by a sensor array are presented

exploratory behavior trap models and glass transitions a random walk is performed on a disordered landscape composed of n sites randomly and uniformly distributed inside a d dimensional hypercube the walker hops from one site to another with probability proportional to exp betae d where beta 1 t is the inverse of a formal temperature and e d is an arbitrary cost function which depends on the hop distance d analytic results indicate that if e d d d and n infinity there exists a glass transition at beta d pi d 2 d 2 gamma d 2 below t d the average trapping time diverges and the system falls into an out of equilibrium regime with aging phenomena a levy flight scenario and applications of exploratory behavior are considered

deterministic walks in random networks an application to thesaurus graphs abstract in a landscape composed of n randomly distributed sites in euclidean space a walker tourist goes to the nearest one that has not been visited in the last steps this procedure leads to trajectories composed of a transient part and a final cyclic attractor of period p the tourist walk presents a simple scaling with respect to and can be performed in a wide range of networks that can be viewed as ordinal neighborhood graphs as an example we show that graphs defined by thesaurus dictionaries share some of the statistical properties of low dimensional d 2 euclidean graphs and are easily distinguished from random link networks which correspond to the d limit this approach furnishes complementary information to the usual clustering coefficient and mean minimum separation length

escaping from cycles through a glass transition a random walk is performed over a disordered media composed of n sites random and uniformly distributed inside a d dimensional hypercube the walker cannot remain in the same site and hops to one of its n neighboring sites with a transition probability that depends on the distance d between sites according to a cost function e d the stochasticity level is parametrized by a formal temperature t in the case t 0 the walk is deterministic and ergodicity is broken the phase space is divided in a o n number of attractor basins of two cycles that trap the walker for d 1 analytic results indicate the existence of a glass transition at t 1 1 2 as n infinity below t1 the average trapping time in two cycles diverges and an out of equilibrium behavior appears similar glass transitions occur in higher dimensions when the right cost function is chosen we also present some results for the statistics of distances for poisson spatial point processes

an algebraic equation for linear operators let t be a linear operator on a vector space v possibly of infinite dimension over a general field k we solve the functional equation p t f where p k x and f an algebraic operator on v are given for nilpotent f we give an explicit linear system which determines the solutions by their similarity classes the method is based on a canonical decomposition theorem

random geometric graphs we analyze graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present the critical connectivity is found numerically by examining the size of the largest cluster we derive an analytical expression for the cluster coefficient which shows that the graphs are distinctly different from standard random graphs even for infinite dimensionality insights relevant for graph bipartitioning are included

percolation like scaling exponents for minimal paths and trees in the stochastic mean field model author s aldous dj abstract in the mean field or random link model there are n points and inter point distances are independent random variables for 0 l l l and in the n limit let l 1 n times the maximum number of steps in a path whose average step length is l the function l is analogous to the percolation function in percolation theory there is a critical value l e 1 at which becomes non zero and presumably a scaling exponent in the sense l equivalent to sign l l recently developed probabilistic methodology in some sense a rephrasing of the cavity method developed in the 1980s by mezard and parisi provides a simple albeit non rigorous way of writing down such functions in terms of solutions of fixed point equations for probability distributions solving numerically gives convincing evidence that 3 a parallel study with trees and connected edge sets in place of paths gives scaling exponent 2 while the analogue for classical percolation has scaling exponent 1 the new exponents coincide with those recently found in a different context comparing optimal and near optimal solutions of the mean field travelling salesman problem tsp and the minimum spanning tree mst problem and reinforce the suggestion that scaling exponents determine universality classes for optimization problems on random points 2005 the royal society

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

random walks on fractals and stretched exponential relaxation stretched exponential relaxation exp t tau beta k is observed in a large variety of systems but has not been explained so far studying random walks on percolation clusters in curved spaces whose dimensions range from 2 to 7 we show that the relaxation is accurately a stretched exponential and is directly connected to the fractal nature of these clusters thus we find that in each dimension the decay exponent beta k is related to well known exponents of the percolation theory in the corresponding flat space we suggest that the stretched exponential behavior observed in many complex systems polymers colloids glasses is due to the fractal character of their configuration space

analytical results for the statistical distribution related to a memoryless deterministic walk dimensionality effect and mean field models consider a medium characterized by n points whose coordinates are randomly generated by a uniform distribution along the edges of a unitary d dimensional hypercube a walker leaves from each point of this disordered medium and moves according to the deterministic rule to go to the nearest point which has not been visited in the preceding ensuremath mu steps deterministic tourist walk each trajectory generated by this dynamics has an initial nonperiodic part of t steps transient and a final periodic part of p steps attractor the neighborhood rank probabilities are parametrized by the normalized incomplete beta function i d i 1 4 1 2 d 1 2 the joint distribution s ensuremath mu d n t p is relevant and the marginal distributions previously studied are particular cases we show that for the memory less deterministic tourist walk in the euclidean space this distribution is s 1 d ensuremath infty t p ensuremath gamma 1 i d phantom rule 0 1em 0ex ensuremath 1 t i d phantom rule 0 1em 0ex ensuremath 1 ensuremath gamma t p i d phantom rule 0 1em 0ex ensuremath 1 ensuremath delta p 2 where t 0 1 2 dots ensuremath infty ensuremath gamma z is the gamma function and ensuremath delta i j is the kronecker delta the mean field models are the random link models which correspond to d ensuremath rightarrow ensuremath infty and the random map model which even for ensuremath mu 0 presents nontrivial cycle distribution s 0 rm n p ensuremath propto p ensuremath 1 s 0 rm n t p ensuremath gamma n ensuremath gamma n 1 ensuremath t p n phantom rule 0 1em 0ex t p the fundamental quantities are the number of explored points n e t p and i d although the obtained distributions are simple they do not follow straightforwardly and they have been validated by numerical experiments

classification of invariant image representations using a neural network a neural network nn based approach for classification of images represented by translation scale and rotation invariant features is presented the utilized network is a multilayer perceptron mlp classifier with one hidden layer back propagation learning is used for its training two types of features are used moment invariants derived from geometrical moments of the image and features based on zernlike moments which are the mapping of the image onto a set of complex orthogonal polynomials the performance of the mlp is compared to the bayes nearest neighbor and minimum mean distance statistical classifiers through extensive experimentation with noiseless as well as noisy binary images of all english characters 26 classes the following conclusions are reached 1 the mlp outperforms the other three classifiers especially when noise is present 2 the nearest neighbor classifier performs about the same as the nn for the noiseless case 3 the nn can do well even with a very small number of training samples 4 the nn has a good degree of fault tolerance and 5 the zernlike moment based features possess strong class separability power and are more powerful than moment invariants

extremal optimization for graph partitioning extremal optimization is a new general purpose method for approximating solutions to hard optimization problems we study the method in detail by way of the computationally hard np hard graph partitioning problem we discuss the scaling behavior of extremal optimization focusing on the convergence of the average run as a function of run time and system size the method has a single free parameter which we determine numerically and justify using a simple argument on random graphs our numerical results demonstrate that extremal optimization maintains consistent accuracy for increasing system sizes with an approximation error decreasing over run time roughly as a power law t 20 4 on geometrically structured graphs the scaling of results from the average run suggests that these are far from optimal with large fluctuations between individual trials but when only the best runs are considered results consistent with theoretical arguments are recovered

deterministic and random partially self avoiding walks in random media abstract consider a set of n cities randomly distributed in the bulk of a hypercube with d dimensions a walker with memory begins his route from a given city of this map and moves at each discrete time step to the nearest point which has not been visited in the preceding steps after reviewing the more interesting general results we consider one dimensional disordered media and show that the walker needs not to have full memory of its trajectory to explore the whole system it suffices to have memory of order ln n ln 2

an analytical calculation of neighbourhood order probabilities for high dimensional poissonian processes and mean field models consider that the coordinates of n points are randomly generated along the edges of a d dimensional hypercube random point problem the probability p d n m n that an arbitrary point is the mth nearest neighbour to its own nth nearest neighbour cox probabilities plays an important role in spatial statistics also it has been useful in the description of physical processes in disordered media here we propose a simpler derivation of cox probabilities where we stress the role played by the system dimensionality d in the limit d the distances between pair of points become independent random link model and closed analytical forms for the neighbourhood probabilities are obtained both for the thermodynamic limit and finite size system breaking the distance symmetry constraint drives us to the random map model for which the cox probabilities are obtained for two cases whether a point is its own nearest neighbour or not

contragredient equivalence a canonical form and some applications abstract let a and c be m by n complex matrices and let b and d be n by m complex matrices the pair a b is contragrediently equivalent to the pair c d if there are square nonsingular complex matrices x and y such that xay 1 c and ybx 1 d contragredient equivalence is a common generalization of four basic equivalence relations similarity consimilarity complex orthogonal equivalence and unitary equivalence we develop a complete set of invariants and an explicit canonical form for contragredient equivalence and show that a a t is contragrediently equivalent to c c t if and only if there are complex orthogonal matrices p and q such that c paq using this result we show that the following are equivalent for a given n by n complex matrix a 1 1 a qs for some complex orthogonal q and some complex symmetric s 2 2 a t a is similar to aa t 3 3 a a t is contragrediently equivalent to a t a 4 4 a q 1 a t q 2 for some complex orthogonal q 1 q 2 5 5 a pa t p for some complex orthogonal p we then consider a linear operator on n by n complex matrices that shares the following properties with transpose operators for every pair of n by n complex matrices a and b a preserves the spectrum of a b a a and c ab b a we show that a a is contragrediently similar to b b if and only if a x 1 bx 2 for some nonsingular x 1 x 2 that satisfy x 1 1 x 1 and x 1 2 x 2 we also consider a factorization of the form a xy where x 1 x and y y we use the canonical form for the contragredient equivalence relation to give a new proof of a theorem of flanders concerning the relative sizes of the nilpotent jordan blocks of ab and ba we present a sufficient condition for the existence of square roots of ab and ba and close with a canonical form for complex orthogonal equivalence

the stochastic traveling salesman problem finite size scaling and the cavity prediction we study the random link traveling salesman problem where lengths lij between city i and city j are taken to be independent identically distributed random variables we discuss a theoretical approach the cavity method that has been proposed for finding the optimum tour length over this random ensemble given the assumption of replica symmetry using finite size scaling and a renormalized model we test the cavity predictions against the results of simulations and find excellent agreement over a range of distributions we thus provide numerical evidence that the replica symmetric solution to this problem is the correct one finally we note a surprising result concerning the distribution of k th nearest neighbor links in optimal tours and invite a theoretical understanding of this phenomenon

local equivalence of rank 2 quantum mixed states we investigate the equivalence of quantum mixed states under local unitary transformations for a class of rank 2 mixed states a sufficient and necessary condition of local equivalence is obtained by giving a complete set of invariants under local unitary transformations such that two states in this class are locally equivalent if and only if all these invariants have equal values for them

a radar target discrimination scheme using the discrete wavelet transform for reduced data storage a correlative radar target discrimination scheme using the transient scattered field response is proposed this scheme uses a one dimensional discrete wavelet transform on the temporal response to reduce the amount of data that must be stored for each anticipated aspect angle experimental results show that a reduction in stored data of sixteen to one still allows accurate discrimination in adverse noise situations with signal to noise ratios as low as 5 db

texture analysis using graphs generated by deterministic partially self avoiding walks texture is one of the most important visual attributes for image analysis it has been widely used in image analysis and pattern recognition a partially self avoiding deterministic walk has recently been proposed as an approach for texture analysis with promising results this approach uses walkers called tourists to exploit the gray scale image contexts in several levels here we present an approach to generate graphs out of the trajectories produced by the tourist walks the generated graphs embody important characteristics related to tourist transitivity in the image computed from these graphs the statistical position degree mean and dispersion entropy of two vertices with the same degree measures are used as texture descriptors a comparison with traditional texture analysis methods is performed to illustrate the high performance of this novel approach

recognition of blurred images by the method of moments the article is devoted to the feature based recognition of blurred images acquired by a linear shift invariant imaging system against an image database the proposed approach consists of describing images by features that are invariant with respect to blur and recognizing images in the feature space the psf identification and image restoration are not required a set of symmetric blur invariants based on image moments is introduced a numerical experiment is presented to illustrate the utilization of the invariants for blurred image recognition robustness of the features is also briefly discussed

detection of the number of coherent signals by the mdl principle an approach is presented to the problem of detecting the number of sources impinging on a passive sensor array that is based on j rissanen s 1983 minimum description length mdl principle the approach is applicable to any type of sources including the case of sources which are fully correlated referred to as the coherent signals case two slightly different detection criteria are derived both requiring the estimation of the locations of the sources the first is tailored to the detection problem per se whereas the second is tailored to the combined detection estimation problem consistency of the two criteria is proved and their performance is demonstrated by computer simulations

scaling and universality in continuous length combinatorial optimization we consider combinatorial optimization problems defined over random ensembles and study how solution cost increases when the optimal solution undergoes a small perturbation delta for the minimum spanning tree the increase in cost scales as delta2 for the minimum matching and traveling salesman problems in dimension d 2 the increase scales as delta3 this is observed in monte carlo simulations in d 2 3 4 and in theoretical analysis of a mean field model we speculate that the scaling exponent could serve to classify combinatorial optimization problems of this general kind into a small number of distinct categories similar to universality classes in statistical physics

nature s way of optimizing we propose a general purpose method for finding high quality solutions to hard optimization problems inspired by self organizing processes often found in nature the method called extremal optimization successively eliminates extremely undesirable components of sub optimal solutions drawing upon models used to simulate far from equilibrium dynamics it complements approximation methods inspired by equilibrium statistical physics such as simulated annealing with only one adjustable parameter its performance proves competitive with and often superior to more elaborate stochastic optimization procedures we demonstrate it here on two classic hard optimization problems graph partitioning and the traveling salesman problem 2000 elsevier science b v all rights reserved

efficient radar target recognition using the music algorithm and invariant features an efficient technique is developed to recognize target type using one dimensional range profiles the proposed technique utilizes the multiple signal classification algorithm to generate superresolved range profiles their central moments are calculated to provide translation invariant and level invariant feature vectors next the computed central moments are mapped into values between zero and unity followed by a principal component analysis to eliminate the redundancy of feature vectors the obtained features are classified based on the bayes classifier which is one of the statistical classifiers recognition results using five different aircraft models measured at compact range are presented to assess the effectiveness of the proposed technique and they are compared with those of the conventional range profiles obtained by inverse fast fourier transform

computation of canonical matrices for chains and cycles of linear mappings van dooren linear algebra appl 27 1979 103 constructed an algorithm for the computation of all irregular summands in kronecker s canonical form of a matrix pencil the algorithm is numerically stable since it uses only unitary transformations we construct a unitary algorithm for computation of the canonical form of the matrices of a chain of linear mappings v1 v2 vt andextend van dooren s algorithm to the matrices of a cycle of linear mappings whereall vi are complex vector spaces and each line denotes or

the centralizer of an endomorphism over an arbitrary field abstract the centralizer of an endomorphism of a finite dimensional vector space is known when the endomorphism is nonderogatory or when its minimal polynomial splits over the field it is also known for the real jordan canonical form in this paper we characterize the centralizer of an endomorphism over an arbitrary field and compute its dimension the result is obtained via generalized jordan canonical forms for separable and nonseparable minimal polynomials in addition we also obtain the corresponding generalized weyr canonical forms and the structure of its centralizers which in turn allows us to compute the determinant of its elements

new heuristic percolation criterion for continuum systems a heuristic criterion is introduced and applied for derivation of percolation thresholds in continuum systems the criterion is comprehensive in the sense that it applies to systems with diverse bonding criteria as well as to systems with a correlated distribution of sites its usefulness is manifested by the excellent quantitative agreement between the values obtained and available monte carlo data

patterns of a spatial exploration under time evolution of the attractiveness persistent nodes degree distribution and spectral properties abstract this work explores the features of a graph generated by agents that hop from one node to another node where the nodes have evolutionary attractiveness the jumps are governed by boltzmann like transition probabilities that depend both on the euclidean distance between the nodes and on the ratio of the attractiveness between them it is shown that persistent nodes i e nodes that never been reached by this special random walk are possible in the stationary limit differently from the case where the attractiveness is fixed and equal to one for all nodes 1 simultaneously one also investigates the spectral properties and statistics related to the attractiveness and degree distribution of the evolutionary network finally a study of the crossover between persistent phase and no persistent phase was performed and it was also observed the existence of a special type of transition probability which leads to a power law behaviour for the time evolution of the persistence

precise determination of the critical threshold and exponents in a three dimensional continuum percolation model we present a large scale computer simulation of the prototypical three dimensional continuum percolation model consisting of a distribution of overlapping spatially uncorrelated spheres by using simulations of up to particles and studying the finite size scaling of various effective percolation thresholds we obtain a value of this value is significantly smaller than the values obtained for simulations that have been carried out using smaller systems employing this value of and systems of size l 160 relative to a sphere of unit radius we also obtain estimates of the critical exponents and for the continuum system and show that the values are different than those obtained using previous values of

radar hrrp statistical recognition parametric model and model selection statistical modeling for radar high resolution range profile hrrp is a challenging task in radar hrrp statistical recognition theoretical analysis and experimental results show that elements in an hrrp sample are statistically correlated and non gaussian distributed first this paper introduces three joint gaussian models i e subspace approximation model probability principal components analysis ppca model and factor analysis fa model into radar hrrp statistical recognition due to the experimental results we can have the conclusion that the jointly non gaussian distributed hrrp samples approximately follow the joint gaussian distribution described by fa model therefore we can apply fa model to radar hrrp statistical recognition rather than a joint gaussian mixture model e g ppca mixture model or fa mixture model which is a more accurate choice for modeling non gaussian distributed correlations in multidimensional data but with high learning complexity and large computation burden and the difficulty in the statistical modeling for hrrp samples is largely reduced second this paper concerns model selection of fa model in radar hrrp statistical recognition in which there are two issues i e the partition of target aspect frames and the determination of the number of factors in each frame based on the akaike information criterion aic and the bayes information criterion bic an iterated algorithm for model selection is proposed in this paper which can automatically give the optimal aspect frame boundaries and determine the optimal number of factors in each aspect frame the recognition experiments based on measured data show that the proposed adaptive partition approach can further improve the recognition performance with higher recognition efficiency

graph bipartitioning and spin glasses on a random network of fixed finite valence the authors study the problem of bipartitioning a random graph of fixed finite valence using a mean field replica symmetric theory of an ising ferromagnet with zero magnetisation constraint the thermodynamics is determined by the probability distribution of an auxiliary field the expression for the ground state energy agrees with that proposed by mezard and parisi 1986 using a cavity field method but their expression for the fraction of crazy spins is reinterpreted

on ptak s derivation of the jordan normal form some readers of 1 might appreciate the following comments that make more explicit how ptak s beautiful insight there leads to a trivial proof of the basics of the jordan normal form the proof of theorem 1 of 1 can also be based on the observation that x being finite dimensional the sequence 0 ker a ker a2 must eventually be stationary i e ker a ker aq p for some q and all p 0 for such q let xr and xs be the range and the kernel respectively of a hence dim x dim xr dim xs further for any x xr xs x az for some z and so z ker a2q ker a hence x 0 therefore x is the direct sum of the two a invariant subspaces xs and xr and a is regular on xr since a is and nilpotent on xs in the setup and notation of theorem 2 of 1 there must be by duality some y0 in y for which x0aq 1 y0 0 hence the q order matrix x0aj 1 y0a q i i j 1 q is triangular with nonzero diagonal entries therefore invertible and this guarantees that x is the direct sum x0 x with x0 the linear span of x0aj 1 j 1 q and x the annihilator of y0a q i i 1 q both of which are ainvariant moreover it shows x0aj 1 j 1 q to be a basis for x0 and the matrix representation with respect to this basis of a restricted to x0 has the familiar form of a jordan block for the eigenvalue 0 now x being finite dimensional there are a invariant direct sum decompositions x x1 xm that are minimal in the sense that none of its summands is the direct sum of two nontrivial a invariant subspaces take any one such then the matrix representation for a with respect to any basis made up from bases for the summands xi is block diagonal with the ith block the matrix representation of the restriction ai of a to xi with respect to the chosen basis for xi assuming the underlying field to be algebraically closed the restriction ai of a to xi has some eigenvalue i and in view of the minimality of xi theorem 1 ensures

on image analysis by the methods of moments various types of moments have been used to recognize image patterns in a number of applications a number of moments are evaluated and some fundamental questions are addressed such as image representation ability noise sensitivity and information redundancy moments considered include regular moments legendre moments zernike moments pseudo zernike moments rotational moments and complex moments properties of these moments are examined in detail and the interrelationships among them are discussed both theoretical and experimental results are presented

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

scaling and percolation in the small world network model in this paper we study the small world network model of watts and strogatz which mimics some aspects of the structure of networks of social interactions we argue that there is one nontrivial length scale in the model analogous to the correlation length in other systems which is well defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero giving a normal critical point in this limit this length scale governs the crossover from large to small world behavior in the model as well as the number of vertices in a neighborhood of given radius on the network we derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex vertex distance on the network and using series expansion and pad e approximants find an approximate analytic form for the scaling function we calculate the effective dimension of small world graphs and show that this dimension varies as a function of the length scale on which it is measured in a manner reminiscent of multifractals we also study the problem of site percolation on small world networks as a simple model of disease propagation and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms in epidemiological terms the point at which an epidemic occurs the typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph all our analytic results are confirmed by extensive numerical simulations of the model

are randomly grown graphs really random we analyze a minimal model of a growing network at each time step a new vertex is added then with probability ensuremath delta two vertices are chosen uniformly at random and joined by an undirected edge this process is repeated for t time steps in the limit of large t the resulting graph displays surprisingly rich characteristics in particular a giant component emerges in an infinite order phase transition at ensuremath delta 1 8 at the transition the average component size jumps discontinuously but remains finite in contrast a static random graph with the same degree distribution exhibits a second order phase transition at ensuremath delta 1 4 and the average component size diverges there these dramatic differences between grown and static random graphs stem from a positive correlation between the degrees of connected vertices in the grown graph older vertices tend to have higher degree and to link with other high degree vertices merely by virtue of their age we conclude that grown graphs however randomly they are constructed are fundamentally different from their static random graph counterparts

efficient measurement of the percolation threshold for fully penetrable discs we study the percolation threshold for fully penetrable discs by measuring the average location of the frontier for a statistically inhomogeneous distribution of fully penetrable discs we use two different algorithms to efficiently simulate the frontier including the continuum analogue of an algorithm previously used for gradient percolation on a square lattice we find that c 0 676 339 0 000 004 thus providing an extra significant digit of accuracy to this constant some figures in this article appear in colour in the electronic version see www iop org

applications of the duality method to generalizations of the jordan canonical form abstract the jordan normal form for a matrix over an arbitrary field and the canonical form for a pair of matrices under contragredient equivalence are derived using ptak s duality method

extremal optimization of graph partitioning at the percolation threshold the benefits of a recently proposed method to approximate hard optimization problems are demonstrated on the graph partitioning problem the performance of this new method called extremal optimization eo is compared with simulated annealing sa in extensive numerical simulations while generally a complex np hard problem the optimization of the graph partitions is particularly difficult for sparse graphs with average connectivities near the percolation threshold at this threshold the relative error of sa for large graphs is found to diverge relative to eo at equalized runtime on the other hand eo based on the extremal dynamics of self organized critical systems reproduces known results about optimal partitions at this critical point quite well