specification tests based on artificial regressions abstract many specification tests can be computed with artificial linear regressions designed to be used as calculating devices to obtain test statistics and other quantities of interest this article discusses the general principles that underlie all artificial regressions and the use of such regressions to compute lagrange multiplier and other specification tests based on estimates under the null hypothesis the generality and power of artificial regressions as a means of computing test statistics is demonstrated how durbin wu hausman conditional moment and other tests that are not explicitly lagrange multiplier tests may be computed is shown and several special cases that illustrate the general results and can be useful in practice are discussed these include tests of parameter restrictions in nonlinear regression models and tests of binary choice models such as the logit and probit models

variational problems with two phases and their free boundaries the problem of minimizing vu 2 q2 x 2 v dx in an appropriate class of functions v is considered here q x 0 and a2 t x2 if v 0 any minimizer u is harmonic in u 0 and vu 2 has a jump a2 x 22 across the free boundary u 0 regularity and various properties are established for the minimizer u and for the free boundary introduction in this paper we consider the problem of minimizing j v f vv 2 q2 x x2 v dx vgk where q2 x 0 a22 ifu 0 and x2 u is lower semicontinuous at v 0 it is assumed that a2 0 and a a2 a22 0 the class k consists of all functions v in l loc s2 with vu g l2 i2 such that v u on a given open subset s of 3i2 and s is a domain in r the analogous problem for functions in k v k v 0 a e was studied in 1 in that paper it was proved that any local minimizer of j v in k is lipschitz continuous and if n 2 the free boundary 8 w 0 is analytic if q x is analytic the present variational problem is motivated by applications to the now of two liquids in models of jets and cavities these applications will be studied in other forthcoming papers 5 6 the present work is aimed at extending results of 1 in particular we shall establish nondegeneracy theorems the lipschitz continuity of the solution and some properties of the free boundary for n 2 the free boundary is proved to be continuously differentiable a new and rather powerful tool introduced in this paper is the monotonicity formula lemma 5 1 asserting that for a minimizer u if u x0 0 then

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

testing for arch in the presence of a possibly misspecified conditional mean abstract ever since the development of the autoregressive conditional heteroskedasticity arch model engle 1982 testing for the presence of arch has become a routine diagnostic one popular method of testing for arch is t times the r2 from a regression of the squared residuals on p of its lags this test has been shown to have a lagrange multiplier interpretation and is asymptotically distributed as a 2 p random variable underlying this test is the assumption of a correctly specified conditional mean in this paper we consider the properties of the arch test when there is a possibly misspecified conditional mean examples of misspecification include omitted variables structural change and parameter instability we show that in general misspecification will lead to overrejection of the null hypothesis of conditional homoskedasticity we demonstrate the use of recursive residuals to improve the fit of a first stage conditional mean regression we illustrate these results via monte carlo simulations and consider two empirical examples

random walks in modular scale free networks with multiple traps extensive empirical investigation has shown that a plethora of real networks synchronously exhibit scale free and modular structure and it is thus of great importance to uncover the effects of these two striking properties on various dynamical processes occurring on such networks in this paper we examine two cases of random walks performed on a class of modular scale free networks with multiple traps located at several given nodes we first derive a formula of the mean first passage time mfpt for a general network which is the mean of the expected time to absorption originating from a specific node averaged over all nontrap starting nodes although the computation is complex the expression of the formula is exact moreover the computational approach and procedure are independent of the number and position of the traps we then determine analytically the mfpt for the two random walks being considered the obtained analytical results are in complete agreement with the numerical ones our results show that the number and location of traps play an important role in the behavior of the mfpt since for both cases the mfpt grows as a power law function of the number of nodes but their exponents are quite different we demonstrate that the root of the difference in the behavior of mfpt is attributed to the modular and scale free topologies of the networks this work can deepen the understanding of diffusion on networks with modular and scale free architecture and motivate relevant studies for random walks running on complex random networks with multiple traps

minimizers of convex functionals arising in random surfaces we investigate c1 regularity of minimizers to f u dx in two dimensions for certain classes of non smooth convex functionals f in particular our results apply to the surface tensions that appear in recent works on random surfaces and random tilings of kenyon okounkov and others

effect of trap position on the efficiency of trapping in treelike scale free networks the conventional wisdom is that the role and impact of nodes on dynamical processes in scale free networks are not homogenous because of the presence of highly connected nodes at the tail of their power law degree distribution in this paper we explore the influence of different nodes as traps on the trapping efficiency of the trapping problem taking place on scale free networks to this end we study in detail the trapping problem in two families of deterministically growing scale free networks with treelike structure one family is non fractal the other is fractal in the first part of this work we attack a special case of random walks on the two network families with a perfect trap located at a hub i e node with the highest degree the second study addresses the case with trap distributed uniformly over all nodes in the networks for these two cases we compute analytically the mean trapping time mtt a quantitative indicator characterizing the trapping efficiency of the trapping process we show that in the non fractal scale free networks the mtt for both cases follows different scalings with the network order number of network nodes implying that trap s position has a significant effect on the trapping efficiency in contrast it is presented that for both cases in the fractal scale free networks the two leading scalings exhibit the same dependence on the network order suggesting that the location of trap has no essential impact on the trapping efficiency we also show that for both cases of the trapping problem the trapping efficiency is more efficient in the non fractal scale free networks than in their fractal counterparts

global mean first passage times of random walks on complex networks we present a general framework applicable to a broad class of random walks on complex networks which provides a rigorous lower bound for the mean first passage time of a random walker to a target site averaged over its starting position the so called global mean first passage time gmfpt this bound is simply expressed in terms of the equilibrium distribution at the target and implies a minimal scaling of the gmfpt with the network size we show that this minimal scaling which can be arbitrarily slow is realized under the simple condition that the random walk is transient at the target site and independently of the small world scale free or fractal properties of the network last we put forward that the gmfpt to a specific target is not a representative property of the network since the target averaged gmfpt satisfies much more restrictive bounds

distribution free estimation of some nonlinear panel data models this paper studies distribution free estimation of some multiplicative unobserved components panel data models one class of estimators requires only specification of the conditional mean in particular the multinomial quasi conditional maximum likelihood estimator is shown to be consistent when only the conditional mean in the unobserved effects model is correctly specified additional orthogonality conditions can be used in a method of moments framework a second class of problems specifies the conditional mean conditional variances and conditional covariances using the notion of a conditional linear predictor it is shown that specification of conditional second moments implies further orthogonality conditions in the observable data that can be exploited for efficiency gains this has applications to both count and gamma type panel data regression models

c1 regularity for infinity harmonic functions in two dimensions a continuous function open image in new window is said to be infinity harmonic if it satisfies the pde

diagnostic testing and evaluation of maximum likelihood models abstract the paper develops a unified theory of likelihood specification testing based on m estimators of auxiliary parameters the theory is sufficiently general to encompass a wide class of specification tests including moment based tests pearson type goodness of fit tests the information matrix test and the cox test the paper also presents a framework based on frechet differentiation for determining the effects of misspecification on the almost sure limits of parameter estimates and specification test statistics

scaling of average sending time on weighted koch networks random walks on weighted complex networks especially scale free networks have attracted considerable interest in the past but the efficiency of a hub sending information on scale free small world networks has been addressed less in this paper we study random walks on a class of weighted koch networks with scaling factor 0 r 1 we derive some basic properties for random walks on the weighted koch networks based on which we calculate analytically the average sending time ast defined as the average of mean first passage times mfpts from a hub node to all other nodes excluding the hub itself the obtained result displays that for 0 r 1 in large networks the ast grows as a power law function of the network order with the exponent represented by log43r 1r and for r 1 in large networks the ast grows with network order as n ln n which is larger than the linear scaling of the average receiving time defined as the average of mfpts for random walks to a given hub node averaged over all starti

testing for conditional heteroskedasticity with misspecified alternative hypotheses for the multiple linear regression model this paper examines the asymptotic behaviour of some tests for conditional heteroskedasticity when an alternative hypothesis is misspecified in the standard likelihood based framework a lagrange multiplier statistic based on a correctly specified alternative is ranked above one based on an overspecified alternative it is shown that this ranking does not hold for statistics that are robust to the form of conditional heterokurticity it is also shown that the ranking obtained in the standard likelihood based framework does not depend on a joint density being specified in its entirety the same ranking can be obtained when a type of information matrix equality holds

synchronous and asynchronous recursive random scale free nets we investigate the differences between scale free recursive nets constructed by a synchronous deterministic updating rule e g apollonian nets versus an asynchronous random sequential updating rule e g random apollonian nets we show that the dramatic discrepancies observed recently for the degree exponent in these two cases result from a biased choice of the units to be updated sequentially in the asynchronous version

determining global mean first passage time of random walks on vicsek fractals using eigenvalues of laplacian matrices the family of vicsek fractals is one of the most important and frequently studied regular fractal classes and it is of considerable interest to understand the dynamical processes on this treelike fractal family in this paper we investigate discrete random walks on the vicsek fractals with the aim to obtain the exact solutions to the global mean first passage time gmfpt defined as the average of first passage time fpt between two nodes over the whole family of fractals based on the known connections between fpts effective resistance and the eigenvalues of graph laplacian we determine implicitly the gmfpt of the vicsek fractals which is corroborated by numerical results the obtained closed form solution shows that the gmfpt approximately grows as a power law function with system size number of all nodes with the exponent lies between 1 and 2 we then provide both the upper bound and lower bound for gmfpt of general trees and show that the leading behavior of the upper bound is the square of system size and the dominating scaling of the lower bound varies linearly with system size we also show that the upper bound can be achieved in linear chains and the lower bound can be reached in star graphs this study provides a comprehensive understanding of random walks on the vicsek fractals and general treelike networks

trapping in scale free networks with hierarchical organization of modularity a wide variety of real life networks share two remarkable generic topological properties scale free behavior and modular organization and it is natural and important to study how these two features affect the dynamical processes taking place on such networks in this paper we investigate a simple stochastic process trapping problem a random walk with a perfect trap fixed at a given location performed on a family of hierarchical networks that exhibit simultaneously striking scale free and modular structure we focus on a particular case with the immobile trap positioned at the hub node having the largest degree using a method based on generating functions we determine explicitly the mean first passage time mfpt for the trapping problem which is the mean of the node to trap first passage time over the entire network the exact expression for the mfpt is calculated through the recurrence relations derived from the special construction of the hierarchical networks the obtained rigorous formula corroborated by extensive direct numerical calculations exhibits that the mfpt grows algebraically with the network order concretely the mfpt increases as a power law function of the number of nodes with the exponent much less than 1 we demonstrate that the hierarchical networks under consideration have more efficient structure for transport by diffusion in contrast with other analytically soluble media including some previously studied scale free networks we argue that the scale free and modular topologies are responsible for the high efficiency of the trapping process on the hierarchical networks

specification testing and quasi maximum likelihood estimation abstract this paper develops robust regression based conditional moment tests for models estimated by quasi maximum likelihood using a density in the linear exponential family the tests are relatively simple to compute while being robust to distributional assumptions other than those being explicitly tested the conditional mean tests assume only that the conditional mean is correctly specified under the null hypothesis but are asymptotically equivalent to standard forms of the tests if the conditional variance is correctly specified tests for conditional variance misspecification assume only that the first two moments are correctly specified under the null hypothesis an empirical application to individual arrest data is included

econometric models for count data with an application to the patents r d relationship this paper focuses on developing and adapting statistical models of counts nonnegative integers in the context of panel data and using them to analyze the relationship between patents and r d expenditures since a variety of other economic data come in the form of repeated counts of some individual actions or events the methodology should have wide applications the statistical models we develop are applications and generalizations of the poisson distribution two important issues are i given the panel nature of our data how can we allow for separate persistent individual fixed or random effects ii how does one introduce the equivalent of disturbances in the equation into the analysis of poisson and other discrete probability functions the first problem is solved by conditioning on the total sum of outcomes over the observed years while the second problem is solved by introducing an additional source of randomness allowing the poisson parameter to be itself randomly distributed and compounding the two distributions lastly we develop a test statistic for the presence of serial correlation when fixed effects estimators are used in nonlinear conditional models

a heteroscedasticity consistent covariance matrix estimator for time series regressions abstract this paper provides a covariance matrix estimator for the ordinary least squares coefficients of a linear time series model which is consistent even when the disturbances are heteroscedastic this estimator does not require a formal model of the heteroscedasticity one can also obtain a direct test of heteroscedasticity although monte carlo experiments show that it may have low power

pseudo maximum likelihood methods applications to poisson models pseudo maximum likelihood techniques are applied to basic poisson models and to poisson models with specification errors in the latter case it is shown that consistent and asymptotically normal estimators can be obtained without specifying the p d f of the disturbances these estimators are compared both from the finite sample and the asymptotic point of view quasi generalized pml estimators which asymptotically dominate all pml estimators are also proposed finally bivariate and panel data poisson models are discussed the analysis of economic behavior often leads to the study of characteristics taking a small number of positive values the classical linear model is not adapted to explain how such discrete variables depend on other quantitative or qualitative variables the reasons are similar to those usually given in the case of an endogenous qualitative variable the shape of the observation set does not correspond to a linear model the assumption of normality of the disturbances cannot be made since the endogenous variables take a small number of values with strictly positive probabilities and the prediction formulae which are deduced from a linear model give impossible values in the models considered in the literature to describe discrete variables cox and lewis 2 el sayyad 3 frome kutner and beauchamp 4 gilbert 5 hausman hall and griliches 8 see also lancaster 12 the endogenous variable is assumed to have a poisson distribution conditional upon the exogenous variables the parameter of this distribution is a function of the values of the exogenous variables the choice of such a model is justified if the dependent variable counts the occurrence of a given event during a fixed period and if the usual assumptions of the poisson process are satisfied for instance the model is adapted to describe daily numbers of oil tankers arrivals in a port the number of accidents at work by factory or the number of patents applied for and received by firms hausman hall and griliches 8

matrix exponential garch i propose a new multivariate garch specification that maintains positive definiteness of the conditional covariance matrix the idea is to specify the dynamics in the matrix logarithm of the conditional covariance because the matrix exponential transformation ensures positive definiteness the dynamics can be specified without the positive definiteness constraint this affords a variety of specifications and in particular we can specify element by element the dynamics of the matrix logarithm i discuss specifications with leverage effects estimation with multivariate gaussian and t distributions and diagnostics that evaluate the appropriateness of the matrix exponential specification

likelihood based estimation in a panel setting robustness redundancy and validity of copulas abstract this paper considers the estimation of likelihood based models in a panel setting that is we have panel data and for each time period separately we have a correctly specified model that could be estimated by mle we want to allow non independence over time this paper shows how to improve on the qmle it then considers mle based on joint distributions constructed using copulas it discusses the efficiency gain from using the true copula and shows that knowledge of the true copula is redundant only if the variance matrix of the relevant set of moment conditions is singular it also discusses the question of robustness against misspecification of the copula and proposes a test of the validity of the copula gmm methods are argued to be useful analytically and also for reasons of efficiency if the copula is robust but not correct

criterion for the emergence of explosive synchronization transitions in networks of phase oscillators the emergence of explosive synchronization transitions in networks of phase oscillators recently has become one of the most interesting topics it is widely believed that the large frequency mismatch of a pair of oscillators also known as disassortativity in frequency is a direct cause of an explosive synchronization it is found that besides the disassortativity in frequency the disassortativity in node degree also shows up in connection with the first order synchronization transition in this paper we simulate the kuramoto model on top of a family of networks with different degree degree and frequency frequency correlation patterns results show that only when the degrees and natural frequencies of the network s nodes are both disassortative can an explosive synchronization occur

almost monotonicity formulas for elliptic and parabolic operators with variable coefficients in this paper we extend the results of caffarelli jerison and kenig ann of math 2 155 2002 and caffarelli and kenig amer j math 120 1998 by establishing an almost monotonicity estimate for pairs of continuous functions satisfying u 0 lu 1 u u 0 in an infinite strip global version or a finite parabolic cylinder localized version where l is a uniformly parabolic operator lu l a b c u div a x s u b x s u c x s u s u with double dini continuous a and uniformly bounded b and c we also prove the elliptic counterpart of this estimate this closes the gap between the known conditions in the literature both in the elliptic and parabolic case imposed on u in order to obtain an almost monotonicity estimate at the end of the paper we demonstrate how to use this new almost monotonicity formula to prove the optimal c 1 1 regularity in a fairly general class of quasi linear obstacle type free boundary problems

explicit determination of mean first passage time for random walks on deterministic uniform recursive trees the determination of mean first passage time mfpt for random walks in networks is a theoretical challenge and is a topic of considerable recent interest within the physics community in this paper according to the known connections between mfpt effective resistance and the eigenvalues of graph laplacian we first study analytically the mfpt between all node pairs of a class of growing treelike networks which we term deterministic uniform recursive trees durts since one of its particular cases is a deterministic version of the famous uniform recursive tree the interesting quantity is determined exactly through the recursive relation of the laplacian spectra obtained from the special construction of durts the analytical result shows that the mfpt between all couples of nodes in durts varies as n text text ln text n for large networks with node number n second we study trapping on a particular network of durts focusing on a special case with the immobile trap positioned at a node having largest degree we determine exactly the average trapping time att that is defined as the average of fpt from all nodes to the trap in contrast to the scaling of the mfpt the leading behavior of att is a linear function of n interestingly we show that the behavior for att of the trapping problem is related to the trapping location which is in comparison with the phenomenon of trapping on fractal t graph although both networks exhibit tree structure finally we believe that the methods could open the way to exactly calculate the mfpt and att in a wide range of deterministic media

inverse probability weighted estimation for general missing data problems abstract i study inverse probability weighted m estimation under a general missing data scheme examples include m estimation with missing data due to a censored survival time propensity score estimation of the average treatment effect in the linear exponential family and variable probability sampling with observed retention frequencies i extend an important result known to hold in special cases estimating the selection probabilities is generally more efficient than if the known selection probabilities could be used in estimation for the treatment effect case the setup allows a general characterization of a double robustness result due to scharfstein et al 1999 rejoinder journal of the american statistical association 94 1135 1146

scaling theory of transport in complex biological networks transport is an important function in many network systems and understanding its behavior on biological social and technological networks is crucial for a wide range of applications however it is a property that is not well understood in these systems probably because of the lack of a general theoretical framework here based on the finding that renormalization can be applied to bionetworks we develop a scaling theory of transport in self similar networks we demonstrate the networks invariance under length scale renormalization and we show that the problem of transport can be characterized in terms of a set of critical exponents the scaling theory allows us to determine the influence of the modular structure on transport in metabolic and protein interaction networks we also generalize our theory by presenting and verifying scaling arguments for the dependence of transport on microscopic features such as the degree of the nodes and the distance between them using transport concepts such as diffusion and resistance we exploit this invariance and we are able to explain based on the topology of the network recent experimental results on the broad flow distribution in metabolic networks

generalized method of moments specification testing abstract this paper analyzes the asymptotic power properties of specification tests which are based on a finite set of moment conditions it shows that any such test may fail against general misspecification that causes estimator inconsistency the mutual asymptotic equivalence of maximal degree of freedom tests is shown and the form of optimal tests against specific forms of misspecification is derived applications to testing for exogeneity of a set of instrumental variables are presented

geometric fractal growth model for scale free networks we introduce a deterministic model for scale free networks whose degree distribution follows a power law with the exponent ensuremath gamma at each time step each vertex generates its offspring whose number is proportional to the degree of that vertex with proportionality constant m ensuremath 1 mg1 we consider the two cases first each offspring is connected to its parent vertex only forming a tree structure second it is connected to both its parent and grandparent vertices forming a loop structure we find that both models exhibit power law behaviors in their degree distributions with the exponent ensuremath gamma 1 mathrm ln 2m ensuremath 1 mathrm ln m thus by tuning m the degree exponent can be adjusted in the range 2l ensuremath gamma l3 we also solve analytically a mean shortest path distance d between two vertices for the tree structure showing the small world behavior that is d ensuremath sim mathrm ln n mathrm ln overline k where n is system size and overline k is the mean degree finally we consider the case that the number of offspring is the same for all vertices and find that the degree distribution exhibits an exponential decay behavior

exact encounter times for many random walkers on regular and complex networks the exact mean time between encounters of a given particle in a system consisting of many particles undergoing random walks in discrete time is calculated on both regular and complex networks analytical results are obtained both for independent walkers where any number of walkers can occupy the same site and for walkers with an exclusion interaction when no site can contain more than one walker these analytical results are then compared with numerical simulations showing very good agreement

ising model on networks with an arbitrary distribution of connections we find the exact critical temperature t c of the nearest neighbor ferromagnetic ising model on an equilibrium random graph with an arbitrary degree distribution p k we observe an anomalous behavior of the magnetization magnetic susceptibility and specific heat when p k is fat tailed or loosely speaking when the fourth moment of the distribution diverges in infinite networks when the second moment becomes divergent t c approaches infinity the phase transition is of infinite order and size effect is anomalously strong

generalized adjacency and laplacian spectra of the weighted corona graphs abstract recently corona product graphs have attracted a great deal of attention from scientific communities such as coding theory frequency assignment mathematical chemistry meanwhile weight factor gradually became a research hotpoint because of closing to reality hence we introduce the weighted corona graphs in this paper the weighted corona product graphs are the in depth and promotion of the corona product graphs linking the spectra between the initial graph and the product graph then we deduce the complete information of generalized adjacency and laplacian eigenvalues of the weighted product graph with different structures and we generalize the results obtained previously into product corona graph with m 1 iterations

influences of degree inhomogeneity on average path length and random walks in disassortative scale free networks various real life networks exhibit degree correlations and heterogeneous structure with the latter being characterized by power law degree distribution p k k where the degree exponent describes the extent of heterogeneity in this paper we study analytically the average path length apl of and random walks rws on a family of deterministic networks recursive scale free trees rsfts with negative degree correlations and various 2 1 ln 3 ln 2 with an aim to explore the impacts of structure heterogeneity on the apl and rws we show that the degree exponent has no effect on the apl d of rsfts in the full range of d behaves as a logarithmic scaling with the number of network nodes n i e d ln n which is in sharp contrast to the well known double logarithmic scaling d ln ln n previously obtained for uncorrelated scale free networks with 2 3 in addition we present that some scaling efficiency exponents of random walks are reliant on the degree exponent

two phase anisotropic free boundary problems and applications to the bellman equation in 2d we prove lipschitz continuity of solutions to a class of rather general two phase anisotropic free boundary problems in 2d and we classify global solutions as a consequence we obtain c 2 1 regularity of solutions to the bellman equation in 2d

degree dependent intervertex separation in complex networks we study the mean length l k of the shortest paths between a vertex of degree k and other vertices in growing networks where correlations are essential in a number of deterministic scale free networks we observe a power law correction to a logarithmic dependence l k a ln n k gamma 1 2 ck gamma 1 n in a wide range of network sizes here n is the number of vertices in the network gamma is the degree distribution exponent and the coefficients a and c depend on a network we compare this law with a corresponding l k dependence obtained for random scale free networks growing through the preferential attachment mechanism in stochastic and deterministic growing trees with an exponential degree distribution we observe a linear dependence on degree l k approximately a ln n ck we compare our findings for growing networks with those for uncorrelated graphs

random walks on complex trees we study the properties of random walks on complex trees we observe that the absence of loops is reflected in physical observables showing large differences with respect to their looped counterparts first both the vertex discovery rate and the mean topological displacement from the origin present a considerable slowing down in the tree case second the mean first passage time mfpt displays a logarithmic degree dependence in contrast to the inverse degree shape exhibited in looped networks this deviation can be ascribed to the dominance of source target topological distance in trees to show this we study the distance dependence of a symmetrized mfpt and derive its logarithmic profile obtaining good agreement with simulation results these unique properties shed light on the recently reported anomalies observed in diffusive dynamical systems on trees

degree mixing and the enhancement of synchronization in complex weighted networks real networks often consist of local units interacting with each other by means of heterogeneous connections in many cases furthermore such networks feature degree mixing properties i e the tendency of nodes with high degree with low degree to connect with connectivity peers with highly connected nodes such degree degree correlations may have an important influence in the spreading of information or infectious agents on a network we explore the role played by these correlations for the synchronization of networks of coupled dynamical systems using a stochastic optimization technique we find that the value of degree mixing providing optimal conditions for synchronization depends on the weighted coupling scheme we also show that a minimization of the assortative coefficient may induce a strong destabilization of the synchronous state we illustrate our findings for weighted networks with scale free and random topologies

pseudofractal scale free web we find that scale free random networks are excellently modeled by simple deterministic graphs our graph has a discrete degree distribution degree is the number of connections of a vertex which is characterized by a power law with exponent gamma 1 ln 3 ln 2 properties of this compact structure are surprisingly close to those of growing random scale free networks with gamma in the most interesting region between 2 and 3 we succeed to find exactly and numerically with high precision all main characteristics of the graph in particular we obtain the exact shortest path length distribution for a large network ln n 1 the distribution tends to a gaussian of width approximately sqrt ln n centered at l approximately ln n we show that the eigenvalue spectrum of the adjacency matrix of the graph has a power law tail with exponent 2 gamma

statistically consistent coarse grained simulations for critical phenomena in complex networks we propose a degree based coarse graining approach that not just accelerates the evaluation of dynamics on complex networks but also satisfies the consistency conditions for both equilibrium statistical distributions and nonequilibrium dynamical flows for the ising model and susceptible infected susceptible epidemic model we introduce these required conditions explicitly and further prove that they are satisfied by our coarse grained network construction within the annealed network approximation finally we numerically show that the phase transitions and fluctuations on the coarse grained network are all in good agreements with those on the original one

random walks and diameter of finite scale free networks abstract dynamical scalings for the end to end distance r e e and the number of distinct visited nodes n v of random walks rws on finite scale free networks sfns are studied numerically r e e shows the dynamical scaling behavior r e e l t l n g t l z where l is the average minimum distance between all possible pairs of nodes in the network n is the number of nodes is the degree exponent of the sfn and t is the step number of rws especially r e e l t in the limit t satisfies the relation r e e l d where d is the diameter of network with d l ln n for 3 or d l ln ln n for 3 based on the scaling relation r e e we also find that the scaling behavior of the diameter of networks can be measured very efficiently by using rws

first passage properties of the erdos renyi random graph we study the mean time for a random walk to traverse between two arbitrary sites of the erd s renyi random graph we develop an effective medium approximation that predicts that the mean first passage time between pairs of nodes as well as all moments of this first passage time are insensitive to the fraction p of occupied links this prediction qualitatively agrees with numerical simulations away from the percolation threshold near the percolation threshold the statistically meaningful quantity is the mean transit rate namely the inverse of the first passage time this rate varies non monotonically with p near the percolation transition much of this behaviour can be understood by simple heuristic arguments

stratification of free boundary points for a two phase variational problem abstract in this paper we prove the local lipschitz regularity of the minimizers of the two phase bernoulli type free boundary problem arising from the minimization of the functional j u u p p u 0 p u 0 1 p here r n is a bounded smooth domain and are positive constants such that p p 0 furthermore we show that for p 1 the free boundary has locally finite perimeter and the set of non smooth points of the free boundary is of zero n 1 dimensional hausdorff measure for this our approach is new even for the classical case p 2

some new monotonicity theorems with applications to free boundary problems the monotonicity theorem of alt caffarelli and friedman acf lemma 5 1 plays a central role in the existence and regularity theory of two phase free boundary problems it has been extended for example to variable coefficient operators c3 lemma 1 and to eigenvalue problems fl in this paper we reformulate the theorem so that it applies to inhomogeneous equations in which the right hand side of the equation need not vanish at the free boundary of particular interest is the case in which au takes two different constant values on the set where u 0 and on the set where u 0 the prandtl batchelor

inverted berezinskii kosterlitz thouless singularity and high temperature algebraic order in an ising model on a scale free hierarchical lattice small world network we have obtained exact results for the ising model on a hierarchical lattice incorporating three key features characterizing many real world networks a scale free degree distribution a high clustering coefficient and the small world effect by varying the probability p of long range bonds the entire spectrum from an unclustered non small world network to a highly clustered small world system is studied using the self similar structure of the network we obtain analytic expressions for the degree distribution p k and clustering coefficient c for all p as well as the average path length ensuremath ell for p 0 and 1 the ferromagnetic ising model on this network is studied through an exact renormalization group transformation of the quenched bond probability distribution using up to 562 phantom rule 0 2em 0ex 500 renormalized probability bins to represent the distribution for pl0 494 we find power law critical behavior of the magnetization and susceptibility with critical exponents continuously varying with p and exponential decay of correlations away from t c for p ensuremath ge 0 494 in fact where the network exhibits small world character the critical behavior radically changes we find a highly unusual phase transition namely an inverted berezinskii kosterlitz thouless singularity between a low temperature phase with nonzero magnetization and finite correlation length and a high temperature phase with zero magnetization and infinite correlation length with power law decay of correlations throughout the phase approaching t c from below the magnetization and the susceptibility respectively exhibit the singularities of mathrm exp ensuremath c sqrt t c ensuremath t and mathrm exp d sqrt t c ensuremath t with c and d positive constants with long range bond strengths decaying with distance we see a phase transition with power law critical singularities for all p and evaluate an unusually narrow critical region and important corrections to power law behavior that depend on the exponent characterizing the decay of long range interactions