the continuum limit of critical random graphs we consider the erd s renyi random graph g n p inside the critical window that is when p 1 n n 4 3 for some fixed lambda in mathbb r we prove that the sequence of connected components of g n p considered as metric spaces using the graph distance rescaled by n 1 3 converges towards a sequence of continuous compact metric spaces the result relies on a bijection between graphs and certain marked random walks and the theory of continuum random trees our result gives access to the answers to a great many questions about distances in critical random graphs in particular we deduce that the diameter of g n p rescaled by n 1 3 converges in distribution to an absolutely continuous random variable with finite mean

two critical periods in the evolution of random planar graphs let p n m be a graph chosen uniformly at random from the family of all labeled planar graphs with n vertices and m edges in the paper we study the component structure of p n m combining counting arguments with analytic techniques we show that there are two critical periods in the evolution of p n m the first one of width n 2 3 is analogous to the phase transition observed in the standard random graph models and takes place for m n 2 o n 2 3 when the largest complex component is formed then for m n o n 3 5 when the complex components covers nearly all vertices the second critical period of width n 3 5 occurs starting from that moment increasing of m mostly affects the density of the complex components not its size

the average distances in random graphs with given expected degrees abstract random graph theory is used to examine the small world phenomenon any two strangers are connected through a short chain of mutual acquaintances we will show that for certain families of random graphs with given expected degrees the average distance is almost surely of order log n log d where d is the weighted average of the sum of squares of the expected degrees of particular interest are power law random graphs in which the number of vertices of degree k is proportional to 1 k for some fixed exponent for the case of 3 we prove that the average distance of the power law graphs is almost surely of order log n log d however many internet social and citation networks are power law graphs with exponents in the range 2 3 for which the power law random graphs have average distance almost surely of order log log n but have diameter of order log n provided having some mild constraints for the average distance and maximum degree in particular these graphs contain a dense subgraph which we call the core having nc log log n vertices almost all vertices are within distance log log n of the core although there are vertices at distance log n from the core

threshold for monotone symmetric properties through a logarithmic sobolev inequality threshold phenomena are investigated using a general approach following talagrand ann probab 22 1994 1576 1587 and friedgut and kalai proc amer math soc 12 1999 1017 10541 the general upper bound for the threshold width of symmetric monotone properties is improved this follows from a new lower bound on the maximal influence of a variable on a boolean function the method of proof is based on a well known logarithmic sobolev inequality on 0 1 n this new bound is shown to be asymptotically optimal

the potts model and the tutte polynomial this is an invited survey on the relation between the partition function of the potts model and the tutte polynomial on the assumption that the potts model is more familiar we have concentrated on the latter and its interpretations in particular we highlight the connections with abelian sandpiles counting problems on random graphs error correcting codes and the ehrhart polynomial of a zonotope where possible we use the mean field and square lattice as illustrations we also discuss in some detail the complexity issues involved

assessing intrarater interrater and test retest reliability of continuous measurements in this paper we review the problem of defining and estimating intrarater interrater and test retest reliability of continuous measurements we argue that the usual notion of product moment correlation is well adapted in a test retest situation whereas the concept of intraclass correlation should be used for intrarater and interrater reliability the key difference between these two approaches is the treatment of systematic error which is often due to a learning effect for test retest data we also consider the reliability of a sum and a difference of variables and illustrate the effects on components further we compare these approaches of reliability with the concept of limits of agreement proposed by bland and altman for evaluating the agreement between two methods of clinical measurements and show how product moment correlation is related to it we then propose new kinds of limits of agreement which are related to intraclass correlation a test battery to study the development of neuro motor functions in children and adolescents illustrates our purpose throughout the paper

the tight constant in the dvoretzky kiefer wolfowitz inequality let hat f n denote the empirical distribution function for a sample of n i i d random variables with distribution function f in 1956 dvoretzky kiefer and wolfowitz proved that p big sqrt n sup x hat f n x f x lambda big leq c exp 2 lambda 2 where c is some unspecified constant we show that c can be taken as 1 as conjectured by birnbaum and mccarty in 1958 provided that exp 2 lambda 2 leq frac 1 2 in particular the two sided inequality p big sqrt n sup x hat f n x f x lambda big leq 2 exp 2 lambda 2 holds without any restriction on lambda in the one sided as well as in the two sided case the constants cannot be further improved

subspace identification methods for hammerstein systems rank constraint and dimension problem this article studies the subspace identification methods sims for hammerstein systems with major focus on a rank constraint and the related dimension problem we analyse the effects of the rank constraint on the three steps of a unifying framework for sims the rank constraint has no effect on the first two steps but does so on the third step if the rank constraint is ignored as in the existing over parametrised method opm for hammerstein system identification the optimality of the resulting estimate can still be established even so the opm may suffer from the dimension problem resulting in a low numerical efficiency to resolve the dimension problem we propose a new subspace based method named as the least parametrised method lpm for identification of hammerstein systems with non coupling input nonlinearities simulation results are provided to demonstrate the effectiveness of the lpm and show the necessity of considering the rank constraint to improve the numerical efficiency

higher moment approaches to approximate interval estimation for a certain intraclass correlation coefficient we consider the problem of constructing a confidence interval for the intraclass correlation coefficient in an interrater reliability study when the raters are assumed to be randomly selected from a population of raters a monte carlo simulation study is conducted to investigate the true coverage probabilities of the commonly used intervals proposed by fleiss and shrout which rely on satterthwaite s two moment approximation these intervals are shown to be substantially anticonservative in certain cases we propose intervals based on higher moment approximations obtained using the pearson system of distributions the modified intervals are more conservative and generally more satisfactory than those obtained by the two moment approximation the competing methods are illustrated using data from a study of the natural history of facioscapulohumeral muscular dystrophy

the exploration process of inhomogeneous continuum random trees and an extension of jeulin s local time identity we study the inhomogeneous continuum random trees icrt that arise as weak limits of birthday trees we give a description of the exploration process a function defined on 0 1 that encodes the structure of an icrt and also of its width process determining the size of layers in order of height these processes turn out to be transformations of bridges with exchangeable increments which have already appeared in other icrt related topics such as stochastic additive coalescence the results rely on two different constructions of birthday trees from processes with exchangeable increments on weak convergence arguments and on general theory on continuum random trees

entropy for translation invariant random cluster measures we study translation invariant random cluster measures with techniques from large deviation theory and convex analysis in particular we prove a large deviation principle with rate function given by a specific entropy and a dobrushinlanfordruelle variational principle that characterizes translation invariant random cluster measures as the solutions of the variational equation for free energy consequences of these theorems include inequalities for edge and cluster densities of translationinvariant random cluster measures 1 introduction the random cluster model was introduced around 1972 by fortuin and kasteleyn 18 as a family of models that contains the bernoulli bond percolation model the ising model and the q state potts models as special cases it has developed into an important and very successful tool for studying the ising and potts models our paper focuses on the properties of the random cluster measures themselves our starting point is grimmett s 23 development of elements of a thermodynamic formalism for this model beginning with a definition of infinite volume random cluster measures in terms of a dlr dobrushinlanfordruelle equation just as for gibbsian systems natural next questions concern the validity of large deviation principles and variational characterizations of infinite volume measures these questions we answer in the present paper and derive some consequences here is a brief overview of the dlr approach of 23 the random cluster measures of interest live on the graph formed by putting edges between nearest neighbor vertices of the d dimensional hypercubic lattice d the edge that connects two vertices x and y that are l 1 distance one apart is

critical percolation clusters in seven dimensions and on a complete graph we study critical bond percolation on a seven dimensional hypercubic lattice with periodic boundary conditions 7d and on the complete graph cg of finite volume number of vertices v we numerically confirm that for both cases the critical number density n s v of clusters of size s obeys a scaling form n s v s n over s v d f with identical volume fractal dimension d f 2 3 and exponent 1 1 d f 5 2 we then classify occupied bonds into bridge bonds which includes branch and junction bonds and nonbridge bonds a bridge bond is a branch bond if and only if its deletion produces at least one tree deleting branch bonds from percolation configurations produces leaf free configurations whereas deleting all bridge bonds leads to bridge free configurations composed of blobs it is shown that the fraction of nonbridge biconnected bonds vanishes n cg 0 for large cgs but converges to a finite value n 7d 0 0061931 7 for the 7d hypercube further we observe that while the bridge free dimension d bf 1 3 holds for both the cg and 7d cases the volume fractal dimensions of the leaf free clusters are different d lf 7d 0 669 9 2 3 and d lf cg 0 3337 17 1 3 on the cg and in 7d the whole leaf free and bridge free clusters all have the shortest path volume fractal dimension d min 1 3 characterizing their graph diameters we also study the behavior of the number and the size distribution of leaf free and bridge free clusters for the number of clusters we numerically find the number of leaf free and bridge free clusters on the cg scale as lnv while for 7d they scale as v for the size distribution we find the behavior on the cg is governed by a modified fisher exponent 1 while for leaf free clusters in 7d it is governed by fisher exponent 5 2 the size distribution of bridge free clusters in 7d displays two scaling behavior with exponents 4 and 1 the probability distribution p c 1 v dc 1 of the largest cluster of size c 1 for whole percolation configurations is observed to follow a single variable function p over x dx with x c 1 v d f for both cg and 7d up to a rescaling factor for the variable x the probability functions for cg and 7d collapse on top of each other within the entire range of x the analytical expressions in the x 0 and x limits are further confirmed our work demonstrates that the geometric structure of high dimensional percolation clusters cannot be fully accounted for by their complete graph counterparts

brief paper making parametric hammerstein system identification a linear problem in this paper we study the identification of parametric hammerstein systems with fir linear parts by a proper normalization and a clever characterization it is shown that the average squared error cost function for identification can be expressed in terms of the inner product between the true but unknown parameter vector and its estimate further the cost function is concave in the inner product and linear in the inner product square therefore the identification of parametric hammerstein systems with fir linear parts is a globally convergent problem and has one and only one local and global minimum this implies that the identification of such systems is a linear problem in terms of the inner product square and any local search based identification algorithm converges globally

the structure of a random graph at the point of the phase transition consider the random graph models g n edges m and g n prob edge p with m m n 1 n 1 3 n 2 and p p n 1 an 1 3 n for 1 1 define an i component of a random graph as a component which has exactly i more edges than vertices call an i component with i 1 a complex component for both models we show that when a is constant the expected number of complex components is bounded almost surely a s each of these components if any exist has size of order n2 3 and the maximum value of i is bounded in probability we prove that a s the largest suspended tree in each complex component has size of order n2 3 and deletion of all suspended trees results in a smoothed graph of size of order n 3 with the maximum vertex degree 3 the total number of branching vertices i e of degree 3 is bounded in probability thus each complex component is almost surely topologically equivalent to a 3 regular multigraph of a uniformly bounded size lengths of the shortest cycle and of the shortest path between two branching vertices of a smoothed graph are each of order n 3 we find a relatively simple integral formula for the limit distribution of the numbers of complex components which implies in particular that all values of the complexity spectrum have positive limiting probabilities we also answer questions raised by erdos and renyi back in 1960 it is proven that there exists p a the limiting planarity probability with 0 p a 1 p oo 1 p oo 0 in particular g n m g n p resp is almost surely nonplanar iff m n 2 n 213 00 np i n 1 3 oo resp

critical random graphs diameter and mixing time let mathcal c 1 denote the largest connected component of the critical erd s renyi random graph g n frac 1 n we show that typically the diameter of mathcal c 1 is of order n1 3 and the mixing time of the lazy simple random walk on mathcal c 1 is of order n the latter answers a question of benjamini kozma and wormald these results extend to clusters of size n2 3 of p bond percolation on any d regular n vertex graph where such clusters exist provided that p d 1 1 o n 1 3

identification of mimo hammerstein models using least squares support vector machines this paper studies a method for the identification of hammerstein models based on least squares support vector machines ls svms the technique allows for the determination of the memoryless static nonlinearity as well as the estimation of the model parameters of the dynamic arx part this is done by applying the equivalent of bai s overparameterization method for identification of hammerstein systems in an ls svm context the siso as well as the mimo identification cases are elaborated the technique can lead to significant improvements with respect to classical overparameterization methods as illustrated in a number of examples another important advantage is that no stringent assumptions on the nature of the nonlinearity need to be imposed except for a certain degree of smoothness

percolation on finite graphs and isoperimetric inequalities consider a uniform expanders family gn with a uniform bound on the degrees it is shown that for any p and c 0 a random subgraph of gn obtained by retaining each edge randomly and independently with probability p will have at most one cluster of size at least c gn with probability going to one uniformly in p the method from ajtai komlos and szemeredi combinatorica 2 1982 1 7 is applied to obtain some new results about the critical probability for the emergence of a giant component in random subgraphs of finite regular expanding graphs of high girth as well as a simple proof of a result of kesten about the critical probability for bond percolation in high dimensions several problems and conjectures regarding percolation on finite transitive graphs are presented

the multiplicative coalescent inhomogeneous continuum random trees and new universality classes for critical random graphs one major open conjecture in the area of critical random graphs formulated by statistical physicists and supported by a large amount of numerical evidence over the last decade braunstein et al in phys rev lett 91 16 168701 2003 wu et al in phys rev lett 96 14 148702 2006 braunstein et al int j bifurc chaos 17 07 2215 2255 2007 chen et al in phys rev lett 96 6 068702 2006 is as follows for a wide array of random graph models with degree exponent tau in 3 4 distances between typical points both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like n tau 3 tau 1 in this paper we study the metric space structure of maximal components of the multiplicative coalescent in the regime where the sizes converge to excursions of levy processes without replacement aldous and limic electron in j probab 3 3 59 1998 yielding a completely new class of limiting random metric spaces a by product of the analysis yields the continuum scaling limit of one fundamental class of random graph models with degree exponent tau in 3 4 where edges are rescaled by n tau 3 tau 1 yielding the first rigorous proof of the above conjecture the limits in this case are compact tree like random fractals with a dense collection of hubs infinite degree vertices a finite number of which are identified with leaves to form shortcuts in a special case we show that the minkowski dimension of the limiting spaces equal tau 2 tau 3 a s in stark contrast to the erd s renyi scaling limit whose minkowski dimension is 2 a s it is generally believed that dynamic versions of a number of fundamental random graph models as one moves from the barely subcritical to the critical regime can be approximated by the multiplicative coalescent in work in progress the general theory developed in this paper is used to prove analogous limit results for other random graph models with degree exponent tau in 3 4 our proof makes crucial use of inhomogeneous continuum random trees icrt which have previously arisen in the study of the entrance boundary of the additive coalescent we show that tilted versions of the same objects using the associated mass measure describe connectivity properties of the multiplicative coalescent since convergence of height processes of corresponding approximating mathbf p trees is not known we use general methodology in athreya et al 2014 and develop novel techniques relying on first showing convergence in the gromov weak topology and then extending this to gromov hausdorff prokhorov convergence by proving a global lower mass bound

a modified large sample approach to approximate interval estimation for a particular intraclass correlation coefficient we consider the problem of constructing a confidence interval for the intraclass correlation coefficient in an interrater reliability study when both raters and subjects are random effects in a two way random effects model a simulation study is conducted to investigate and compare the interval coverage per cents of three methods of approximation i satterthwaite s two moments the standard approach ii modified three moments a newer approach and iii modified large sample the newest approach for this particular problem one sided lower and two sided bounds are examined for the two moment approach coverage of confidence intervals can be understated for not only one sided bounds but also two sided bounds coverage of confidence intervals for the modified large sample approach is either correct or conservative and provides narrower two sided widths than the modified three moment approach which can understate coverage on one sided and two sided bounds for two or three raters the competing methods are illustrated with data from a reliability study of four raters evaluating the number of decayed missing and filled surfaces in the permanent teeth of ten subjects overall the modified large sample approach provides the most satisfactory performance of the three methods

generating simple random graphs with prescribed degree distribution let f be a probability distribution with support on the non negative integers four methods for generating a simple undirected graph with approximate degree distribution f are described and compared two methods are based on the so called configuration model with modifications ensuring a simple graph one method is an extension of the classical erdos renyi graph where the edge probabilities are random variables and the last method starts with a directed random graph which is then modified to a simple undirected graph all methods are shown to give the correct distribution in the limit of large graph size but under different assumptions on the degree distribution f and also using different order of operations

an optimal two stage identification algorithm for hammerstein wiener nonlinear systems abstract an optimal two stage identification algorithm is presented for hammerstein wiener systems where two static nonlinear elements surround a linear block the proposed algorithm consists of two steps the first one is the recursive least squares and the second one is the singular value decomposition of two matrices whose dimensions are fixed and do not increase as the number of the data point increases moreover the algorithm is shown to be convergent in the absence of noise and convergent with probability one in the presence of white noise

random graph asymptotics on high dimensional tori we investigate the scaling of the largest critical percolation cluster on a large d dimensional torus for nearest neighbor percolation in sufficiently high dimensions or when d 6 for sufficiently spread out percolation we use a relatively simple coupling argument to show that this largest critical cluster is with high probability bounded above by a large constant times v 2 3 and below by a small constant times v 2 3 log v 4 3 where v is the volume of the torus we also give a simple criterion in terms of the subcritical percolation two point function on mathbb z d under which the lower bound can be improved to small constant times v 2 3 i e we prove random graph asymptotics for the largest critical cluster on the high dimensional torus this establishes a conjecture by 1 apart from logarithmic corrections we discuss implications of these results on the dependence on boundary conditions for high dimensional percolation

the phase transition in a random hypergraph we show that in the evolution of the random d uniform hypergraph gd n m the phase transition occurs when m n d d 1 o n2 3 we also prove local limit theorems for the distribution of the size of the largest component of gd n m in the subcritical and in the early supercritical phase

brief an estimator of the inverse covariance matrix and its application to ml parameter estimation in dynamical systems an exact formula of the inverse covariance matrix of an autoregressive stochastic process is obtained using the gohberg semencul explicit inverse of the toeplitz matrix this formula is used to build an estimator of the inverse covariance matrix of a stochastic process based on a single realization in this paper we show that this estimator can be conveniently applied to maximum likelihood parameter estimation in nonlinear dynamical system with correlated measurement noise the efficiency of the estimation scheme is illustrated via monte carlo simulations it is shown that the statistical properties of the estimated parameters are largely improved using the proposed inverse covariance matrix estimator in comparison to the classical variance estimator

rigidity percolation in a field rigidity percolation with g degrees of freedom per site is analyzed on randomly diluted erdos renyi graphs with average connectivity gamma in the presence of a field h in the gamma h plane the rigid and flexible phases are separated by a line of first order transitions whose location is determined exactly this line ends at a critical point with classical critical exponents analytic expressions are given for the densities n f of uncanceled degrees of freedom and gamma r of redundant bonds upon crossing the coexistence line gamma r and n f are continuous although their first derivatives are discontinuous we extend for the case of nonzero field a recently proposed hypothesis namely that the density of uncanceled degrees of freedom is a free energy for rigidity percolation analytic expressions are obtained for the energy entropy and specific heat some analogies with a liquid vapor transition are discussed particularizing to zero field we find that the existence of a g 1 core is a necessary condition for rigidity percolation with g degrees of freedom at the transition point gamma c maxwell counting of degrees of freedom is exact on the rigid cluster and on the g 1 rigid core i e the average coordination of these subgraphs is exactly 2g although gamma c the average coordination of the whole system is smaller than 2g gamma c is found to converge to 2g for large g i e in this limit maxwell counting is exact globally as well

finite model order accuracy in hammerstein model estimation hammerstein models is one of the most commonly used model classes used for identifying nonlinear systems a static input nonlinearity followed by a linear dynamical part is an adequate way to model many real life systems this paper investigates the asymptotic in terms of sample size variance of hammerstein model estimates the work extends earlier results by ninness and gibson 2002 in the following ways not only frequency function estimation but estimation of general quantities is considered the expressions are not restricted to be valid asymptotically in the model order in addition the results cover model structures having noise models and allow for data generated under feedback the increase in variance due to the estimation of the input nonlinearity is characterized in particular under open loop operation white additive noise and the assumption of a separable process it is shown that the variance increase is exactly a term that was observed in ninness and gibson 2002 to result in good agreement with simulations this term vanishes in the formal asymptotic in model order analysis in ninness and gibson 2002

generating random networks that consist of a single connected component with a given degree distribution we present a method for the construction of ensembles of random networks that consist of a single connected component with a given degree distribution this approach extends the construction toolbox of random networks beyond the configuration model framework in which one controls the degree distribution but not the number of components and their sizes unlike configuration model networks which are completely uncorrelated the resulting single component networks exhibit degree degree correlations moreover they are found to be disassortative namely high degree nodes tend to connect to low degree nodes and vice versa we demonstrate the method for single component networks with ternary exponential and power law degree distributions

cluster tails for critical power law inhomogeneous random graphs recently the scaling limit of cluster sizes for critical inhomogeneous random graphs of rank 1 type having finite variance but infinite third moment degrees was obtained in bhamidi et al ann probab 40 2299 2361 2012 it was proved that when the degrees obey a power law with exponent tau in 3 4 the sequence of clusters ordered in decreasing size and multiplied through by n tau 2 tau 1 converges as n rightarrow infty to a sequence of decreasing non degenerate random variables here we study the tails of the limit of the rescaled largest cluster i e the probability that the scaling limit of the largest cluster takes a large value u as a function of u this extends a related result of pittel j combin theory ser b 82 2 237 269 2001 for the erd s renyi random graph to the setting of rank 1 inhomogeneous random graphs with infinite third moment degrees we make use of delicate large deviations and weak convergence arguments

the random cluster model on a homogeneous tree the random cluster model on a homogeneous tree is defined and studied it is shown that for 1 q 2 the percolation probability in the maximal random cluster measure is continuous inp while forq 2 it has a discontinuity at the critical valuep p c q it is also shown that forq 2 there is nonuniqueness of random cluster measures for an entire interval of values ofp the latter result is in sharp contrast to what happens on the integer lattice z d

the random cluster model on the complete graph the random cluster model of fortuin and kasteleyn contains as special cases the percolation ising and potts models of statistical physics when the underlying graph is the complete graph onn vertices then the associated processes are called mean field in this study of the mean field random cluster model with parametersp n andq we show that its properties for any value ofq 0 may be derived from those of an erd s renyi random graph in this way we calculate the critical point c q of the model and show that the associated phase transition is continuous if and only ifq 2 exact formulae are given for c q the density of the largest component the density of edges of the model and the free energy this work generalizes earlier results valid for the potts model whereq is an integer satisfyingq 2 equivalent results are obtained for a fixed edge number random cluster model as a consequence of the results of this paper one obtains large deviation theorems for the number of components in the classical random graph models whereq 1

continuum limit of critical inhomogeneous random graphs the last few years have witnessed tremendous interest in understanding the structure as well as the behavior of dynamics for inhomogeneous random graph models to gain insight into real world systems in this study we analyze the maximal components at criticality of one famous class of such models the rank one inhomogeneous random graph model norros and reittu adv appl probab 38 1 59 75 2006 bollobas et al random struct algorithms 31 1 3 122 2007 section 16 4 viewing these components as measured random metric spaces under finite moment assumptions for the weight distribution we show that the components in the critical scaling window with distances scaled by n 1 3 converge in the gromov haussdorf prokhorov metric to rescaled versions of the limit objects identified for the erd s renyi random graph components at criticality in addario berry et al probab theory related fields 152 3 4 367 406 2012 a key step is the construction of connected components of the random graph through an appropriate tilt of a fundamental class of random trees called mathbf p trees camarri and pitman electron j probab 5 2 1 18 2000 aldous et al probab theory related fields 129 2 182 218 2004 this is the first step in rigorously understanding the scaling limits of objects such as the minimal spanning tree and other strong disorder models from statistical physics braunstein et al phys rev lett 91 16 168701 2003 for such graph models by asymptotic equivalence janson random struct algorithms 36 1 26 45 2010 the same results are true for the chung lu model chung and lu proc natl acad sci 99 25 15879 15882 2002 chung and lu ann combin 6 2 125 145 2002 chung and lu complex graphs and networks 2006 and the britton deijfen martin lof model britton et al j stat phys 124 6 1377 1397 2006 a crucial ingredient of the proof of independent interest are tail bounds for the height of mathbf p trees the techniques developed in this paper form the main technical bedrock for the general program developed in bhamidi et al scaling limits of random graph models at criticality universality and the basin of attraction of the erd s renyi random graph arxiv preprint 2014 for proving universality of the continuum scaling limits in the critical regime for a wide array of other random graph models including the configuration model and inhomogeneous random graphs with general kernels bollobas et al random struct algorithms 31 1 3 122 2007

random walks on the random graph we study random walks on the giant component of the erd s renyi random graph g n p g n p where p np n for 1 1 fixed the mixing time from a worst starting point was shown by fountoulakis and reed and independently by benjamini kozma and wormald to have order log2nlog2 n we prove that starting from a uniform vertex equivalently from a fixed vertex conditioned to belong to the giant both accelerates mixing to o logn o log n and concentrates it the cutoff phenomenon occurs the typical mixing is at d 1logn logn 1 2 o 1 d 1log n log n 1 2 o 1 where and dd are the speed of random walk and dimension of harmonic measure on a poisson poisson galton watson tree analogous results are given for graphs with prescribed degree sequences where cutoff is shown both for the simple and for the nonbacktracking random walk

a phase transition in the random transposition random walk our work is motivated by bourque pevzner s simulation study of the effectiveness of the parsimony method in studying genome rearrangement and leads to a surprising result about the random transposition walk in continuous time on the group of permutations on n elements starting from the identity let d t be the minimum number of transpositions needed to go back to the identity element from the location at time t d t undergoes a phase transition for 0 1 d cn 2 u c n where u is an explicit function satisfying u x x 2 moreover we describe the fluctuations of d cn 2 about its mean at each of the three stages subcritical critical and supercritical the techniques used involve viewing the cycles in the random permutation as a coagulation fragmentation process and relating the behavior to the erd s renyi random graph model

revealing the microstructure of the giant component in random graph ensembles the microstructure of the giant component of the erd s renyi network and other configuration model networks is analyzed using generating function methods while configuration model networks are uncorrelated the giant component exhibits a degree distribution which is different from the overall degree distribution of the network and includes degree degree correlations of all orders we present exact analytical results for the degree distributions as well as higher order degree degree correlations on the giant components of configuration model networks we show that the degree degree correlations are essential for the integrity of the giant component in the sense that the degree distribution alone cannot guarantee that it will consist of a single connected component to demonstrate the importance and broad applicability of these results we apply them to the study of the distribution of shortest path lengths on the giant component percolation on the giant component and spectra of sparse matrices defined on the giant component we show that by using the degree distribution on the giant component one obtains high quality results for these properties which can be further improved by taking the degree degree correlations into account this suggests that many existing methods currently used for the analysis of the whole network can be adapted in a straightforward fashion to yield results conditioned on the giant component

approximating the partition function of the ferromagnetic potts model we provide evidence that it is computationally difficult to approximate the partition function of the ferromagnetic q state potts model when q 2 specifically we show that the partition function is hard for the complexity class rhpi under approximation preserving reducibility thus it is as hard to approximate the partition function as it is to find approximate solutions to a wide range of counting problems including that of determining the number of independent sets in a bipartite graph our proof exploits the first order phase transition of the random cluster model which is a probability distribution on graphs that is closely related to the q state potts model

variable selection and identification of high dimensional nonparametric additive nonlinear systems this paper considers variable selection and identification of dynamic additive nonlinear systems via kernel based nonparametric approaches where the number of variables and additive functions may be large variable selection aims to find which additive functions contribute and which do not the proposed variable selection consists of two successive steps at the first step one estimates each additive function by kernel based nonparametric identification approaches without suffering from the curse of dimensionality at the second step a nonnegative garrote estimator is applied to identify which additive functions are nonzero by utilizing the obtained nonparametric estimates of each function under weak conditions the nonparametric estimates of each additive function can achieve the same asymptotic properties as for 1d nonparametric identification based on kernel functions it is also established that the nonnegative garrote estimator turns a consistent estimate for each additive function into a consistent variable selection with probability one as the number of samples tends to infinity two simulation examples are presented to verify the effectiveness of the variable selection and identification approaches proposed in the paper

the scaling limit of the minimum spanning tree of the complete graph consider the minimum spanning tree mst of the complete graph with nn vertices when edges are assigned independent random weights endow this tree with the graph distance renormalized by n1 3n1 3 and with the uniform measure on its vertices we show that the resulting space converges in distribution as n n to a random compact measured metric space in the gromov hausdorff prokhorov topology we additionally show that the limit is a random binary rr tree and has minkowski dimension 33 almost surely in particular its law is mutually singular with that of the brownian continuum random tree or any rescaled version thereof our approach relies on a coupling between the mst problem and the erd s renyi random graph we exploit the explicit description of the scaling limit of the erd s renyi random graph in the so called critical window established in probab theory related fields 152 2012 367 406 and provide a similar description of the scaling limit for a critical minimum spanning forest contained within the mst in order to accomplish this we introduce the notion of rr graphs which generalise rr trees and are of independent interest

spectral statistics of erdős renyi graphs i local semicircle law we consider the ensemble of adjacency matrices of erd s renyi random graphs that is graphs on n vertices where every edge is chosen independently and with probability p p n we rescale the matrix so that its bulk eigenvalues are of order one we prove that as long as pn with a speed at least logarithmic in n the density of eigenvalues of the erd s renyi ensemble is given by the wigner semicircle law for spectral windows of length larger than n 1 up to logarithmic corrections as a consequence all eigenvectors are proved to be completely delocalized in the sense that the l norms of the l2 normalized eigenvectors are at most of order n 1 2 with a very high probability the estimates in this paper will be used in the companion paper spectral statistics of erd s renyi graphs ii eigenvalue spacing and the extreme eigenvalues 2011 preprint to prove the universality of eigenvalue distributions both in the bulk and at the spectral edges under the further restriction that pn n2 3

on a random graph evolving by degrees abstract we consider a random multi graph growth process g m on a vertex set n which is a special case of a more general process proposed by laci lovasz in 2002 g 0 is empty and g m 1 is obtained from g m by inserting a new edge e at random specifically the conditional probability that e joins two currently disjoint vertices i and j is proportional to d i d j where d i d j are the degrees of i j in g m and 0 is a fixed parameter the limiting case is the erd s renyi graph process we show that whp g m contains a unique giant component iff c 2 m n c 1 and the size of this giant is asymptotic to n 1 c c where c c is the root of c c 2 c c 2 a phase transition window is proved to be contained essentially in c a n 1 3 c b n 1 4 and we conjecture that 1 4 may be replaced with 1 3 for the multigraph version mg m we show that mg m is connected whp iff m m n n 1 1 we conjecture that for 1 m n is the threshold for connectedness of g m itself

robustness of onionlike correlated networks against targeted attacks recently it was found by schneider et al proc natl acad sci usa 108 3838 2011 using simulations that scale free networks with onion structure are very robust against targeted high degree attacks the onion structure is a network where nodes with almost the same degree are connected motivated by this work we propose and analyze based on analytical considerations an onionlike candidate for a nearly optimal structure against simultaneous random and targeted high degree node attacks the nearly optimal structure can be viewed as a set of hierarchically interconnected random regular graphs the degrees and populations of whose nodes are specified by the degree distribution this network structure exhibits an extremely assortative degree degree correlation and has a close relationship to the onion structure after deriving a set of exact expressions that enable us to calculate the critical percolation threshold and the giant component of a correlated network for an arbitrary type of node removal we apply the theory to the cases of random scale free networks that are highly vulnerable against targeted high degree node removal our results show that this vulnerability can be significantly reduced by implementing this onionlike type of degree degree correlation without much undermining the almost complete robustness against random node removal we also investigate in detail the robustness enhancement due to assortative degree degree correlation by introducing a joint degree degree probability matrix that interpolates between an uncorrelated network structure and the onionlike structure proposed here by tuning a single control parameter the optimal values of the control parameter that maximize the robustness against simultaneous random and targeted attacks are also determined our analytical calculations are supported by numerical simulations

the augmented multiplicative coalescent bounded size rules and critical dynamics of random graphs random graph models with limited choice have been studied extensively with the goal of understanding the mechanism of the emergence of the giant component one of the standard models are the achlioptas random graph processes on a fixed set of n vertices here at each step one chooses two edges uniformly at random and then decides which one to add to the existing configuration according to some criterion an important class of such rules are the bounded size rules where for a fixed k ge 1 all components of size greater than k are treated equally while a great deal of work has gone into analyzing the subcritical and supercritical regimes the nature of the critical scaling window the size and complexity deviation from trees of the components in the critical regime and nature of the merging dynamics has not been well understood in this work we study such questions for general bounded size rules our first main contribution is the construction of an extension of aldous s standard multiplicative coalescent process which describes the asymptotic evolution of the vector of sizes and surplus of all components we show that this process referred to as the standard augmented multiplicative coalescent amc is nearly feller with a suitable topology on the state space our second main result proves the convergence of suitably scaled component size and surplus vector for any bounded size rule to the standard amc this result is new even for the classical erd s renyi setting the key ingredients here are a precise analysis of the asymptotic behavior of various susceptibility functions near criticality and certain bounds from bhamidi et al the barely subcritical regime arxiv preprint 2012 on the size of the largest component in the barely subcritical regime

percolation on dense graph sequences in this paper we determine the percolation threshold for an arbitrary sequence of dense graphs g n let n be the largest eigenvalue of the adjacency matrix of g n and let g n p n be the random subgraph of g n obtained by keeping each edge independently with probability p n we show that the appearance of a giant component in g n p n has a sharp threshold at p n 1 n in fact we prove much more if g n converges to an irreducible limit then the density of the largest component of g n c n tends to the survival probability of a multi type branching process defined in terms of this limit here the notions of convergence and limit are those of borgs chayes lovasz s6s and vesztergombi in addition to using basic properties of convergence we make heavy use of the methods of bollobas janson and riordan who used multi type branching processes to study the emergence of a giant component in a very broad family of sparse inhomogeneous random graphs

polymer models and generalized potts kasteleyn models the well established relation between potts models withv spin values and random cluster models with intracluster bonding favored over intercluster bonding by a factorv is explored but with the random cluster model replaced by a much generalized polymer model implying a corresponding generalization of the potts model the analysis is carried out in terms a given defined functionr an entropy free energy density for the polymer model in the casev 1 expressed as a function of the density of units the aim of the analysis is to determine the analogrv ofr for general nonnegativev in terms ofr and thence to determine the critical value of density vg at which gelation occurs this critical value is independent ofv up to a valuevp the potts critical value what is principally required ofr is that it should show a certain given concave convex behavior although differentiability and another regularizing condition are required for complete conclusions under these conditions the unique evaluation ofrv in terms ofr is given in a form known to hold for integralv but not previously extended the analysis is carried out in terms of the legendre transforms of these functions in terms of which the phenomena of criticality gelation and potts criticality appear very transparently andvp is easily determined the value ofvp is 2 under mild conditions onr special interest attaches to the functionr0 which is shown to be the greatest concave minorant ofr the naturalness of the approach is demonstrated by explicit treatment of the first shell model

mean field theory for percolation models of the ising type the q 2 random cluster model is studied in the context of two mean field models the bethe lattice and the complete graph for these systems the critical exponents that are defined in terms of finite clusters have some anomalous values as the critical point is approached from the high density side which vindicates the results of earlier studies in particular the exponent which characterizes the divergence of the average size of finite clusters is 1 2 and the exponent associated with the length scale of finite clusters is 1 4 the full collection of exponents indicates an upper critical dimension of 6 the standard mean field exponents of the ising system are also present in this model 1 2 1 which implies in particular the presence of two diverging length scales furthermore the finite cluster exponents are stable to the addition of disorder which near the upper critical dimension may have interesting implications concerning the generality of the disordered system correlation length bounds

inhomogeneous continuum random trees and the entrance boundary of the additive coalescent regard an element of the set of ranked discrete distributions x1 x2 x1 x2 0 i x i 1 as a fragmentation of unit mass into clusters of masses x i the additive coalescent is the valued markov process in which pairs of clusters of masses x i x j merge into a cluster of mass x i x j at rate x i x j aldous and pitman 1998 showed that a version of this process starting from time with infinitesimally small clusters can be constructed from the brownian continuum random tree of aldous 1991 1993 by poisson splitting along the skeleton of the tree in this paper it is shown that the general such process may be constructed analogously from a new family of inhomogeneous continuum random trees

least squares solutions of bilinear equations abstract the problem of finding a least squares solution for a system of bilinear equations is investigated sufficient conditions to have a unique minimum are given in the case of random inputs three methods the normalized iterative method the over parameterization method and the numerical method are presented for solving the problem along with their convergence properties simulation examples are provided

mixing properties of the swendsen wang process on the complete graph and narrow grids we consider the mixing properties of the swendsen wang process for the two state potts model or ising model on the complete n vertex graph kn and for the q state model on an a n grid where a is bounded as n

mean field theory of the potts gas we consider a gas of classical particles in r d having q distinct colours interacting via a mean field potts potential and subject to an external field a colour independent molecular interaction of mean field type is also admitted in contrast to the usual lattice potts model the potts gas exhibits the specific volume v as a parameter in addition to colour ordering the v dependence of the potts gas is studied in detail and the complete phase diagram is derived it turns out that for q 3 the transition to colour ordering implies a jump of density for q 2 this transition is continuous but may become discontinuous under the influence of a suitable molecular interaction

brownian excursions critical random graphs and the multiplicative coalescent let b t s 0 s be reflecting inhomogeneous brownian motion with drift t s at time s started with b t 0 0 consider the random graph script g sign n n 1 tn 4 3 whose largest components have size of order n 2 3 normalizing by n 2 3 the asymptotic joint distribution of component sizes is the same as the joint distribution of excursion lengths of b t corollary 2 the dynamics of merging of components as t increases are abstracted to define the multiplicative coalescent process the states of this process are vectors x of nonnegative real cluster sizes x i and clusters with sizes x i and x j merge at rate x i x j the multiplicative coalescent is shown to be a feller process on l 2 the random graph limit specifies the standard multiplicative coalescent which starts from infinitesimally small clusters at time the existence of such a process is not obvious

spectral statistics of erdős renyi graphs ii eigenvalue spacing and the extreme eigenvalues we consider the ensemble of adjacency matrices of erd s renyi random graphs i e graphs on n vertices where every edge is chosen independently and with probability p p n we rescale the matrix so that its bulk eigenvalues are of order one under the assumption p n gg n 2 3 we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum more precisely we prove 1 that the eigenvalue spacing of the erd s renyi graph in the bulk of the spectrum has the same distribution as that of the gaussian orthogonal ensemble and 2 that the second largest eigenvalue of the erd s renyi graph has the same distribution as the largest eigenvalue of the gaussian orthogonal ensemble as an application of our method we prove the bulk universality of generalized wigner matrices under the assumption that the matrix entries have at least 4 e moments

a bias corrected estimator for nonlinear systems with output error type model structures parametric identification of linear time invariant lti systems with output error oe type of noise model structures has a well established theoretical framework different algorithms like instrumental variables based approaches or prediction error methods pems have been proposed in the literature to compute a consistent parameter estimate for linear oe systems although the prediction error method provides a consistent parameter estimate also for nonlinear output error noe systems it requires to compute the solution of a nonconvex optimization problem therefore an accurate initialization of the numerical optimization algorithms is required otherwise they may get stuck in a local minimum and as a consequence the computed estimate of the system might not be accurate in this paper we propose an approach to obtain in a computationally efficient fashion a consistent parameter estimate for output error systems with polynomial nonlinearities the performance of the method is demonstrated through a simulation example

mixing time of near critical random graphs let mathcal c 1 be the largest component of the erd s renyi random graph mathcal g n p the mixing time of random walk on mathcal c 1 in the strictly supercritical regime p c n with fixed c 1 was shown to have order log2n by fountoulakis and reed and independently by benjamini kozma and wormald in the critical window p 1 e n where e3n is bounded nachmias and peres proved that the mixing time on mathcal c 1 is of order n however it was unclear how to interpolate between these results and estimate the mixing time as the giant component emerges from the critical window indeed even the asymptotics of the diameter of mathcal c 1 in this regime were only recently obtained by riordan and wormald as well as the present authors and kim in this paper we show that for p 1 e n with e3n and o n the mixing time on mathcal c 1 is with high probability of order n log2 in addition we show that this is the order of the largest mixing time over all components both in the slightly supercritical and in the slightly subcritical regime i e p 1 e n with as above

universality for distances in power law random graphs we survey the recent work on phase transition and distances in various random graph models with general degree sequences we focus on inhomogeneous random graphs the configuration model and affine preferential attachment models and pay special attention to the setting where these random graphs have a power law degree sequence this means that the proportion of vertices with degree k in large graphs is approximately proportional to k for some 1 since many real networks have been empirically shown to have power law degree sequences these random graphs can be seen as more realistic models for real complex networks than classical random graphs such as the erd s renyi random graph it is often suggested that the behavior of random graphs should have a large amount of universality meaning in this case that random graphs with similar degree sequences share similar behavior we survey the available results on graph distances in power law random graphs that are consistent with this prediction

the swendsen wang process does not always mix rapidly the swendsen wang process provides one possible dynamics for the q state potts model computer simulations of this process are widely used to estimate the expectations of various observables random variables of a potts system in the equilibrium or gibbs distribution the legitimacy of such simulations depends on the rate of convergence of the process to equilibrium as measured by the mixing time empirical observations suggest that the mixing time of the swendsen wang process is short in many instances of practical interest although proofs of this desirable behavior are known only for some very special cases nevertheless we show that there are occasions when the mixing time of the swendsen wang process is exponential in the size of the system this undesirable behavior is related to the phenomenon of first order phase transitions in potts systems with q 2 states

graphical representations and cluster algorithms ii we continue the study initiated in part i of graphical representations and cluster algorithms for various models in or related to statistical mechanics for certain models e g the blume emery griffths model and various generalizations we develop fortuin kasteleyn type representations which lead immediately to swendsen wang type algorithms for other models e g the random cluster model that are defined by a graphical representation we develop cluster algorithms without reference to an underlying spin system in all cases phase transitions are related to percolation or incipient percolation in the graphical representation which via the ic algorithm allows for the rapid simulation of these systems at the transition point pertinent examples include the continuum widom rowlinson model the restricted 1 step solid on solid model and the xy model

brief paper revisiting hammerstein system identification through the two stage algorithm for bilinear parameter estimation the two stage algorithm tsa has been extensively used and adapted for the identification of hammerstein systems it is essentially based on a particular formulation of hammerstein systems in the form of bilinearly parameterized linear regressions this paper has been motivated by a somewhat contradictory fact though the optimality of the tsa has been established by bai in 1998 only in the case of some special weighting matrices the unweighted tsa is usually used in practice it is shown in this paper that the unweighted tsa indeed gives the optimal solution of the weighted nonlinear least squares problem formulated with a particular weighting matrix this provides a theoretical justification of the unweighted tsa and also leads to a generalization of the obtained result to the case of colored noise with noise whitening numerical examples of identification of hammerstein systems are presented to validate the theoretical analysis

identification of hammerstein systems without explicit parameterisation of non linearity this article proposes a new approach to identification of hammerstein systems where a non linearity precedes a linear dynamic system driven by piece wise constant inputs the proposed approach does not require an explicit parameterisation of the non linearity moreover the non linearity does not have to be static but could be the one with finite memories like backlash by exploiting input s piecewise constant property the denominator of the linear system described by an arx model is consistently identified from the information of the output only next a subspace direct equalisation method estimates the unmeasurable inner signal based on the resulted denominator estimate and output measurements contrary to the existing blind approaches the numerator of the linear system is not required which leads to a significant improvement of removing an error propagation on the basis of the estimated inner signal the measured input and output the non linearity and linear system are obtained separately the

identification scheme for hammerstein output error models with bounded noise this technical note presents a method for the identification of hammerstein output error hoe models corrupted by bounded noise the method is based on an iterative scheme using two optimal bounding ellipsoid obe type algorithms the proposed approach allows the joint estimation of the linear part and of the nonlinear part with a low computational burden a numerical example is given so as to show the efficiency of the method