synchronization in scale free networks with degree correlation in this paper we study a model of synchronization process on scale free networks with degree degree correlations this model was already studied on this kind of networks without correlations by pastore y piontti et al a l pastore y piontti p a macri l a braunstein phys rev e 76 2007 046117 here we study the effects of the degree degree correlation on the behavior of the load fluctuationsw s in the steady state we found that for assortative networks there exist a specific correlation where the system is optimally synchronized in addition we found that close to this optimal value the fluctuations does not depend on the system size and therefore the system becomes fully scalable this result could be very important for some technological applications on the other hand far from the optimal correlation w s scales logarithmically with the system size

uncorrelated random networks we define a statistical ensemble of nondegenerate graphs i e graphs without multiple connections and self connections between nodes the node degree distribution is arbitrary but the nodes are assumed to be uncorrelated this completes our earlier publication phys rev 64 046118 2001 where trees and degenerate graphs were considered an efficient algorithm generating nondegenerate graphs is constructed the corresponding computer code is available on request finite size effects in scale free graphs i e those where the tail of the degree distribution falls like n beta are carefully studied we find that in the absence of dynamical internode correlations the degree distribution is cut at a degree value scaling like n gamma with gamma min 1 2 1 beta 1 where n is the total number of nodes the consequence is that independently of any specific model the internode correlations seem to be a necessary ingredient of the physics of scale free networks observed in nature

stability properties of potts neural networks with biased patterns and low loading the q state potts glass model of neural networks is extended to include biased patterns for a finite number of such patterns the existence and stability properties of the mattis states and symmetric states are discussed in detail as a function of the bias analytic results are presented for all q at zero temperature for finite temperatures numerical results are obtained for q 3 and two classes of representative bias parameters a comparison is made with the hopfield model

generalization in a multi state neural network the generalization ability of an extremely dilute feedback neural network with multi state neurons is studied by means of a deterministic noiseless parallel dynamics the overlap with any one of a macroscopic number of binary full activity concepts is determined when the network is trained with examples of variable activity according to a hebbian learning algorithm that favours stable symmetric mixture states explicit results about the phase diagram and the generalization error are obtained for a network with three state neurons which remain inactive below a threshold it is shown that the generalization ability can be considerably enhanced either by training the network with low activity examples or by means of a moderate increase in

effective edwards wilkinson equation for single file diffusion in this work we present an effective discrete edwards wilkinson equation aimed to describe the single file diffusion process the key physical properties of the system are captured defining an effective elasticity which is proportional to the single particle diffusion coefficient and to the inverse squared mean separation between particles the effective equation gives a description of single file diffusion using the global roughness of the system of particles which presents three characteristic regimes namely normal diffusion subdiffusion and saturation separated by two crossover times we show how these regimes scale with the parameters of the original system additional repulsive interaction terms are also considered and we analyze how the crossover times depend on the intensity of the additional terms finally we show that the roughness distribution can be well characterized by the edwards wilkinson universal form for the different single file diffusion processes studied here

shift invariance and surface growth the invariance of equations for self affine surface growth to reparametrization under the abelian group of shift transformations h x t to h x t l is used to bound the form of nonlinear terms and related kinetic coefficients in relaxational surface growth equations for conserved growth small but relevant diffusive terms second order in the driving can always be expected it is also shown that the asymptotic growth distributions in d 2 can be expected to be skew and are not derivable from a hamiltonian description

jamming in complex gradient networks flows of physical quantities in large complex networks natural or man made rely in general on some scalar gradients existing in the networks we investigate analytically and numerically under what conditions jamming in gradient flows can occur in random and scale free networks we find that the degree of jamming typically increases with the average connectivity k of the network a crossover phenomenon is uncovered where for kkc kc denotes a critical connectivity estimated to be about 10 scale free networks have a higher level of congestion than random networks with the same k while the opposite occurs for kkc

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

thermodynamic properties of fully connected q ising neural networks the thermodynamic and retrieval properties of fully connected q ising networks are studied in the replica symmetric mean field approximation in particular capacity gain parameter and capacity temperature phase diagrams are derived for q 3 4 and q infinity and different distributions of the stored patterns furthermore the optimal gain function is determined in order to obtain the best performance where appropriate the results are compared with the diluted and layered versions of these models

random walks in weighted networks with a perfect trap an application of laplacian spectra trapping processes constitute a primary problem of random walks which characterize various other dynamical processes taking place on networks most previous works focused on the case of binary networks while there is much less related research about weighted networks in this paper we propose a general framework for the trapping problem on a weighted network with a perfect trap fixed at an arbitrary node by utilizing the spectral graph theory we provide an exact formula for mean first passage time mfpt from one node to another based on which we deduce an explicit expression for average trapping time att in terms of the eigenvalues and eigenvectors of the laplacian matrix associated with the weighted graph where att is the average of mfpts to the trap over all source nodes we then further derive a sharp lower bound for the att in terms of only the local information of the trap node which can be obtained in some graphs moreover we deduce the att when the trap is distributed uniformly in the whole network our results show that network weights play a significant role in the trapping process to apply our framework we use the obtained formulas to study random walks on two specific networks trapping in weighted uncorrelated networks with a deep trap the weights of which are characterized by a parameter and levy random walks in a connected binary network with a trap distributed uniformly which can be looked on as random walks on a weighted network for weighted uncorrelated networks we show that the att to any target node depends on the weight parameter that is the att to any node can change drastically by modifying the parameter a phenomenon that is in contrast to that for trapping in binary networks for levy random walks in any connected network by using their equivalence to random walks on a weighted complete network we obtain the optimal exponent characterizing levy random walks which have the minimal average of atts taken over all target nodes

frequency synchronization in a random oscillator network we study the frequency synchronization of a randomly coupled oscillators by analyzing the continuum limit we obtain a sufficient condition for the mean field type synchronization we especially find that the critical coupling constant k becomes 0 in the random scale free network p k ensuremath propto k ensuremath ensuremath gamma if 2l ensuremath gamma ensuremath leqslant 3 numerical simulations in finite networks are consistent with this analysis

conserved noise restricted solid on solid model on fractal substrates a conserved noise restricted solid on solid model on both a sierpinski gasket substrate and a checkerboard fractal substrate is studied the interface width w grows as t at early time t and becomes saturated at l for t l z where l is the system size we obtain 0 0788 0 377 for a sierpinski gasket and 0 100 0 516 for a checkerboard fractal the dynamic exponent z 4 79 for a sierpinski gasket and z 5 16 for a checkerboard fractal are obtained by the relation z they satisfy the scaling relations 4 2 d f z and z 2 z rw where z rw is the random walk exponent of the fractal substrate a fractional langevin equation is introduced to describe the model

the crossover phenomena in surface growth models with height dependent noise abstract in this paper we consider several known growth processes with height dependent noise this type of noise is interesting from a theoretical standpoint for example it paves the way to the derivation of the exact height distribution of the kpz equation through the hopf cole transformation in addition it may have implications for experimental growth processes using numerical methods we observe that adding such a noise to different growth processes can change their universality class or ruin the scaling laws in the case of mullins herring equation a two fold cross over is observed

dynamic scaling behaviors of linear fractal langevin type equation driven by nonconserved and conserved noise abstract in order to study the effects of the microscopic details of fractal substrates on the scaling behavior of the growth model a generalized linear fractal langevin type equation h t 1 m 1 m z r w h z r w is the dynamic exponent of random walk on substrates driven by nonconserved and conserved noise is proposed and investigated theoretically employing scaling analysis corresponding dynamic scaling exponents are obtained

static fluctuations of a thick one dimensional interface in the 1 1 directed polymer formulation numerical study we study numerically the geometrical and free energy fluctuations of a static one dimensional 1d interface with a short range elasticity submitted to a quenched random bond gaussian disorder of finite correlation length ensuremath xi g0 and at finite temperature t using the exact mapping from the static 1d interface to the 1 1 directed polymer dp growing in a continuous space we focus our analysis on the disorder free energy of the dp end point a quantity which is strictly zero in the absence of disorder and whose sample to sample fluctuations at a fixed growing time t inherit the statistical translation invariance of the microscopic disorder explored by the dp constructing a new numerical scheme for the integration of the kardar parisi zhang evolution equation obeyed by the free energy we address numerically the time and temperature dependence of the disorder free energy fluctuations at fixed finite ensuremath xi we examine on one hand the amplitude stackrel ifmmode tilde else fi d t and effective correlation length stackrel ifmmode tilde else fi ensuremath xi t of the free energy fluctuations and on the other hand the imprint of the specific microscopic disorder correlator on the large time shape of the free energy two point correlator we observe numerically the crossover to a low temperature regime below a finite characteristic temperature t c ensuremath xi as previously predicted by gaussian variational method computations and scaling arguments and extensively investigated analytically in phys rev e 87 042406 2013 finally we address numerically the time and temperature dependence of the roughness b t which quantifies the dp end point transverse fluctuations and we show how the amplitude stackrel ifmmode tilde else fi d ensuremath infty t ensuremath xi controls the different regimes experienced by b t in agreement with the analytical predictions of a dp toy model approach

learning and forgetting on asymmetric diluted neural networks it is possible to construct diluted asymmetric models of neural networks for which the dynamics can be calculated exactly we test several learning schemes in particular models for which the values of the synapses remain bounded and depend on the history our analytical results on the relative efficiencies of the various learning schemes are qualitatively similar to the corresponding ones obtained numerically on fully connected symmetric networks

interface depinning in a disordered medium numerical results abstract we propose a lattice model to study the dynamics of a driven interface in a medium with random pinning forces the critical exponents characterizing the depinning transition are determined numerically in 1 1 and 2 1 dimensions our findings are compared with recent numerical and analytical results for a langevin equation with quenched noise which is expected to be in the same universality class

synchronization in weighted uncorrelated complex networks in a noisy environment optimization and connections with transport efficiency motivated by synchronization problems in noisy environments we study the edwards wilkinson process on weighted uncorrelated scale free networks we consider a specific form of the weights where the strength and the associated cost of a link is proportional to kikj beta with ki and kj being the degrees of the nodes connected by the link subject to the constraint that the total edge cost is fixed we find that in the mean field approximation on uncorrelated scale free graphs synchronization is optimal at beta 1 numerical results based on exact numerical diagonalization of the corresponding network laplacian confirm the mean field results with small corrections to the optimal value of beta employing our recent connections between the edwards wilkinson process and resistor networks and some well known connections between random walks and resistor networks we also pursue a naturally related problem of optimizing performance in queue limited communication networks utilizing local weighted routing schemes

random walk access times on partially disordered complex networks an effective medium theory an analytic effective medium theory is constructed to study the mean access times for random walks on hybrid disordered structures formed by embedding complex networks into regular lattices considering transition rates f that are different for steps across lattice bonds from the rates f across network shortcuts the theory is developed for structures with arbitrary shortcut distributions and applied to a class of partially disordered traversal enhanced networks in which shortcuts of fixed length are distributed randomly with finite probability numerical simulations are found to be in excellent agreement with predictions of the effective medium theory on all aspects addressed by the latter access times for random walks on these partially disordered structures are compared to those on small world networks which on average appear to provide the most effective means of decreasing access times uniformly across the network

continuity of the temperature in the two phase stefan problem the classical two phase stefan problem a model for ice water melting gives rise to a singular nonlinear partial differential equation which admits a unique weak solution here we prove that this solution and therefore the temperature in the stefan problem are continuous

symmetry based determination of space time functions in nonequilibrium growth processes we study the space time correlation and response functions in nonequilibrium growth processes described by linear stochastic langevin equations exploiting exclusively the existence of space and time dependent symmetries of the noiseless part of these equations we derive expressions for the universal scaling functions of two time quantities which are found to agree with the exact expressions obtained from the stochastic equations of motion the usefulness of the space time functions is illustrated through the investigation of two atomistic growth models the family model and the restricted family model which are shown to belong to a unique universality class in 1 1 and 2 1 space dimensions this corrects earlier studies which claimed that in 2 1 dimensions the two models belong to different universality classes

two state representations of three state neural networks the authors investigate to what extent the dynamical behaviour of nets consisting of three state neurons can be realised in nets using two state neurons they find that the generalisation from a two state neuron to a three state neuron cannot introduce any new behaviour in the deterministic case but can do so for noisy nets even in the zero noise limit for those noisy three state nets which have a hamiltonian description and a two state representation they investigate the relationship between the three state hamiltonian and the two state hamiltonian and find an interesting level splitting phenomenon

retrieval and chaos in layered q ising neural networks using a probabilistic approach we study the parallel dynamics of theq ising layered networks for arbitraryq by introducing auxiliary thermal fields we express the stochastic dynamics within the gain function formulation of the deterministic dynamics evolution equations are derived for arbitraryq at both zero and finite temperatures an explicit analysis of the fixed point equations is carried out for bothq 3 andq the retrieval properties are discussed in terms of the gain parameter the storage capacity and the temperature using the time evolution of the distance between two network configurations we investigate the possibility of microscopic chaos chaotic behavior is always present for arbitrary finiteq however in the limitq the existence of chaos depends on the parameters of the system

kinetic roughening global quantities and fluctuation dissipation relations abstract growth processes and interface fluctuations can be studied through the properties of global quantities we here discuss a global quantity that not only captures better the roughness of an interface than the widely studied surface width but that is also directly conjugate to an experimentally accessible parameter thereby allowing us to study in a consistent way the global response of the system to a global change of external conditions exploiting the full analyticity of the linear edwards wilkinson and mullins herring equations we study in detail various two time functions related to that quantity this quantity fulfills the fluctuation dissipation theorem when considering steady state equilibrium fluctuations

persistence in nonequilibrium surface growth persistence probabilities of the interface height in 1 1 and 2 1 dimensional atomistic solid on solid stochastic models of surface growth are studied using kinetic monte carlo simulations with emphasis on models that belong to the molecular beam epitaxy mbe universality class both the initial transient and the long time steady state regimes are investigated we show that for growth models in the mbe universality class the nonlinearity of the underlying dynamical equation is clearly reflected in the difference between the measured values of the positive and negative persistence exponents in both transient and steady state regimes for the mbe universality class the positive and negative persistence exponents in the steady state are found to be 0 66 0 02 and 0 78 0 02 respectively in 1 1 dimensions and 0 76 0 02 and 0 85 0 02 respectively in 2 1 dimensions the noise reduction technique is applied on some of the 1 1 dimensional models in order to obtain accurate values of the persistence exponents we show analytically that a relation between the steady state persistence exponent and the dynamic growth exponent found earlier to be valid for linear models should be satisfied by the smaller of the two steady state persistence exponents in the nonlinear models our numerical results for the persistence exponents are consistent with this prediction we also find that the steady state persistence exponents can be obtained from simulations over times that are much shorter than that required for the interface to reach the steady state the dependence of the persistence probability on the system size and the sampling time is shown to be described by a simple scaling form

the roles of stability and symmetry in the dynamics of neural networks the authors study the retrieval phase of spin glass like neural networks considering that the dynamics should depend only on gauge invariant quantities they propose that two such parameters characterising the symmetry of the neural net s connections and the stabilities of the patterns are responsible for most of the dynamical effects this is supported by a numerical study of the shape of the basins of attraction for a one pattern neural network model the effects of stability and symmetry on the short time dynamics of this model are studied analytically and the full dynamics for vanishing symmetry is shown to be exactly solvable

on the overlap dynamics of multi state neural networks with a finite number of patterns neural networks with multi state neurons are studied in the case of low loading for symmetric couplings satisfying a certain positivity condition a lyapunov function is shown to exist in the space of overlaps between the instantaneous microscopic state of the system and the learned patterns furthermore an algorithm is derived for zero temperature to determine all the fixed points as an illustration the three state model is worked out explicitly for hebbian couplings for finite temperature the time evolution of the overlap is studied for couplings which need not be symmetric the stability properties are discussed in detail for the three state model for asymmetric couplings limit cycle behaviour is shown to be possible

learning grey toned patterns in neural networks the problem of learning multi state patterns in neural networks is investigated an analysis of the space of couplings gardner approach yields the distribution of local fields the critical storage capacity alpha c and the minimum number of errors for an overloaded network for noisy local fields the classification error is minimized if the local fields of the patterns are allowed to lie in intervals of finite width a fast converging adaptive learning algorithm is presented which finds the coupling matrix of optimal stability

generation of uncorrelated random scale free networks uncorrelated random scale free networks are useful null models to check the accuracy and the analytical solutions of dynamical processes defined on complex networks we propose and analyze a model capable of generating random uncorrelated scale free networks with no multiple and self connections the model is based on the classical configuration model with an additional restriction on the maximum possible degree of the vertices we check numerically that the proposed model indeed generates scale free networks with no two and three vertex correlations as measured by the average degree of the nearest neighbors and the clustering coefficient of the vertices of degree k respectively complex networks constitute a general framework for the topological characterization of many natural and technological systems whose complexity prevents a more detailed microscopic description f1 3g within this framework these systems are represented in terms of networks or graphs f4g in which vertices stand for the units composing the system while edges among vertices represent the interactions or relations between pairs of units the focus is thus shift to the topological characterization of the representative network a task which is largely more feasible and yields nevertheless a noticeable amount of information on the structure and properties of the original system the empirical analysis of many real complex networks has unveiled the presence of several typical properties widely found in systems belonging to a large variety of realms one of the most relevant is given by the scale free nature of the degree distribution pskd f1 3 5g defined as the probability that a randomly chosen vertex has degree k si e it is connected to other k verticesd in mathematical terms the scale free property translates into a power law function of the form psk d k g

dynamical surface structures in multiparticle correlated surface growths we investigate the scaling properties of the interface fluctuation width for the q mer and q particle correlated deposition evaporation models these models are constrained with a global conservation law that the particle number at each height is conserved modulo q in equilibrium the stationary roughness is anomalous but universal with the roughness exponent alpha 1 3 while the early time evolution shows nonuniversal behavior with the growth exponent beta varying with models and q nonequilibrium surfaces display diverse growing and stationary behaviors the q mer model shows a faceted structure while the q particle correlated model shows a macroscopically grooved structure

a 2 1 dimensional growth process with explicit stationary measures we introduce a class of 2 1 2 1 dimensional stochastic growth processes that can be seen as irreversible random dynamics of discrete interfaces irreversible means that the interface has an average nonzero drift interface configurations correspond to height functions of dimer coverings of the infinite hexagonal or square lattice the model can also be viewed as an interacting driven particle system and in the totally asymmetric case the dynamics corresponds to an infinite collection of mutually interacting hammersley processes when the dynamical asymmetry parameter p q p q equals zero the infinite volume gibbs measures with given slope are stationary and reversible when p qp q are not reversible any more but remarkably they are still stationary in such stationary states we find that the average height function at any given point xx grows linearly with time tt with a nonzero speed eqx t e hx t hx 0 v teqx t e hx t hx 0 v t while the typical fluctuations of qx t qx t are smaller than any power of tt as t t in the totally asymmetric case of p 0 q 1p 0 q 1 and on the hexagonal lattice the dynamics coincides with the anisotropic kpz growth model introduced by a borodin and p l ferrari in j stat mech theory exp 2009 2009 p02009 comm math phys 325 603 684 for a suitably chosen integrable initial condition that is very far from the stationary state they were able to determine the hydrodynamic limit and a clt for interface fluctuations on scale logt log t exploiting the fact that in that case certain space time height correlations can be computed exactly in the same setting they proved that asymptotically for t t the local statistics of height fluctuations tends to that of a gibbs state which led to the prediction that gibbs states should be stationary

on the distribution of surface extrema in several one and two dimensional random landscapes we study here a standard next nearest neighbor nnn model of ballistic growth on one and two dimensional substrates focusing our analysis on the probability distribution function p m l of the number m of maximal points i e local peaks of growing surfaces our analysis is based on two central results i the proof presented here of the fact that uniform one dimensional ballistic growth process in the steady state can be mapped onto rise and descent sequences in the ensemble of random permutation matrices and ii the fact established in ref g oshanin and r voituriez j phys a math gen 37 6221 2004 that different characteristics of rise and descent patterns in random permutations can be interpreted in terms of a certain continuous space hammersley type process for one dimensional system we compute p m l exactly and also present explicit results for the correlation function characterizing the enveloping surface for surfaces grown on 2d substrates we pursue similar approach considering the ensemble of permutation matrices with long ranged correlations determining exactly the first three cumulants of the corresponding distribution function we define it in the scaling limit using an expansion in the edgeworth series and show that it converges to a gaussian function as l

network synchronization diffusion and the paradox of heterogeneity many complex networks display strong heterogeneity in the degree connectivity distribution heterogeneity in the degree distribution often reduces the average distance between nodes but paradoxically may suppress synchronization in networks of oscillators coupled symmetrically with uniform coupling strength here we offer a solution to this apparent paradox our analysis is partially based on the identification of a diffusive process underlying the communication between oscillators and reveals a striking relation between this process and the condition for the linear stability of the synchronized states we show that for a given degree distribution the maximum synchronizability is achieved when the network of couplings is weighted and directed and the overall cost involved in the couplings is minimum this enhanced synchronizability is solely determined by the mean degree and does not depend on the degree distribution and system size numerical verification of the main results is provided for representative classes of small world and scale free networks

storing an extensive number of grey toned patterns in a neural network using multistate neurons grey toned patterns are pictures composed of pixels of several shades of grey the ability of neural networks using multistate neurons to store such patterns is systematically investigated if conventional generalizations of hopfield networks using analogue or soft neurons are considered it is impossible to stabilize these grey tones nevertheless it is shown that it can be done with networks that use neurons which have only a discrete set of possible activities this is demonstrated for the pseudo inverse rule for the synaptic couplings where only the stability of the patterns shrinks with increasing number q of grey tones one wants to store if the patterns are uncorrelated one can use the hebb rule and in this case the mean field theory is presented applying this rule the storage capacity decreases as q 2 with the number of grey tones

on the parallel dynamics of the q state potts and q ising neural networks using a probabilistic approach the parallel dynamics of theq state potts andq ising neural networks are studied at zero and at nonzero temperatures evolution equations are derived for the first time step and arbitraryq these formulas constitute recursion relations for the exact parallel dynamics of the extremely diluted asymmetric versions of these networks an explicit analysis including dynamical capacity temperature diagrams and the temperature dependence of the overlap is carried out forq 3 both types of models are compared

surface growth on percolation networks by a conserved noise restricted solid on solid growth model surface growth by the conserved noise restricted solid on solid model is investigated on diluted lattices i e on percolation networks that are embedded in two spatial dimensions the growth exponent and the roughness exponent are defined respectively by the mean square surface width via w 2 t t 2 and the mean square saturated width via w sat 2 l l 2 where l is the system size these are measured on both an infinite network and a backbone network and the results are compared with power counting predictions obtained using the fractional langevin equation while the monte carlo results on deterministic fractal substrates show excellent agreement with the predictions d h kim and j m kim phys rev e 84 011105 2011 the results on critical percolation networks deviate by 8 12 from these predictions

on the parallel dynamics of the diluted clock neural network abstract we present explicit formulae for one step evolution of the overlaps in a clock neural network for the parallel dynamics these formulae constitute recursion relations for the exact dynamics in the case of extreme dilution the fixed point equation for this exactly soluble model is investigated for varying temperature t capacity parameter and the number q of the clock states

interface growth with competing surface currents the authors examine the dynamics of interface growth under the combined influence of a stabilizing and a destabilizing surface current in 1 1 dimensions it has been shown that the surface is weakly stabilized if a newly deposited particle can move locally to maximize the total number of nearest neighbour contacts with the surface curvature driven current the introduction of a weak destabilizing surface tension term results in an almost periodic grooved surface the width of the surface grows linearly in time and does not reach a steady state it appears that such a formation cannot be described by a continuous langevin equation

numerical study of a non equilibrium interface model we have carried out extensive computer simulations of one dimensional models related to the low noise solid on solid non equilibrium interface of a two dimensional anchored toom model with unbiased and biased noise for the unbiased case the computed fluctuations of the interface in this limit provide new numerical evidence for the logarithmic correction to the subnormal l frac 1 2 variance which was predicted by the dynamic renormalization group calculations on the modified edwards wilkinson equation in the biased case the simulations are in close quantitative agreement with the predictions of the collective variable approximation cva which gives the same l frac 2 3 behavior of the variance as the kpz equation

traversal times for random walks on small world networks we study the mean traversal time for a class of random walks on newman watts small world networks in which steps around the edge of the network occur with a transition rate that is different from the rate for steps across small world connections when f f the mean time to traverse the network exhibits a transition associated with percolation of the random graph i e small world part of the network and a collapse of the data onto a universal curve this transition was not observed in earlier studies in which equal transition rates were assumed for all allowed steps we develop a simple self consistent effective medium theory and show that it gives a quantitatively correct description of the traversal time in all parameter regimes except the immediate neighborhood of the transition as is characteristic of most effective medium theories

renormalization of stochastic lattice models basic formulation we describe a general method for the multiscale analysis of stochastic lattice models beginning with a lattice langevin formulation of site fluctuations we derive stochastic partial differential equations by regularizing the transition rules of the model subsequent coarse graining is accomplished by calculating renormalization group rg trajectories from initial conditions determined by the regularized atomistic models the rg trajectories correspond to hierarchies of continuum equations describing lattice models over expanding length and time scales these continuum equations retain a quantitative connection over different scales as well as to the underlying atomistic dynamics this provides a systematic method for the derivation of continuum equations from the transition rules of lattice models for any length and time scales as an illustration we consider the one dimensional 1d wolf villain wv model europhys lett 13 389 1990 the rg analysis of this model which we develop in detail is generic and can be applied to a wide range of conservative lattice models the rg trajectory of the 1d wv model shows a complex crossover sequence of linear and nonlinear stochastic differential equations which is in excellent agreement with kinetic monte carlo simulations of this model we conclude by discussing possible applications of the multiscale method described here to other nonequilibrium systems

theory of resistor networks the two point resistance the resistance between two arbitrary nodes in a resistor network is obtained in terms of the eigenvalues and eigenfunctions of the laplacian matrix associated with the network explicit formulae for two point resistances are deduced for regular lattices in one two and three dimensions under various boundary conditions including that of a mobius strip and a klein bottle the emphasis is on lattices of finite sizes we also deduce summation and product identities which can be used to analyse large size expansions in two and higher dimensions

a mechanism to synchronize fluctuations in scale free networks using growth models abstract in this paper we study the steady state of the fluctuations of the surface for a model of surface growth with relaxation to any of its lower nearest neighbors sram f family j phys a 19 1986 l441 in scale free networks it is known that for euclidean lattices this model belongs to the same universality class as the model of surface relaxation to the minimum srm for the srm model it was found that for scale free networks with broadness the steady state of the fluctuations scales with the system size n as a constant for 3 and has a logarithmic divergence for 3 a l pastore y piontti p a macri l a braunstein phys rev e 76 2007 046117 it was also shown c e la rocca l a braunstein p a macri phys rev e 77 2008 046120 that this logarithmic divergence is due to non linear terms that arises from the topology of the network in this paper we show that the fluctuations for the sram model scale as in the srm model we also derive analytically the evolution equation for this model for any kind of complex graphs and find that as in the srm model non linear terms appear due to the heterogeneity and the lack of symmetry of the network in spite of that the two models have the same scaling but the srm model is more efficient to synchronize systems

transport in weighted networks partition into superhighways and roads transport in weighted networks is dominated by the minimum spanning tree mst the tree connecting all nodes with the minimum total weight we find that the mst can be partitioned into two distinct components having significantly different transport properties characterized by centrality the number of times a node or link is used by transport paths one component superhighways is the infinite incipient percolation cluster for which we find that nodes or links with high centrality dominate for the other component roads which includes the remaining nodes low centrality nodes dominate we find also that the distribution of the centrality for the infinite incipient percolation cluster satisfies a power law with an exponent smaller than that for the entire mst the significance of this finding is that one can improve significantly the global transport by improving a tiny fraction of the network the superhighways

relaxation function and dynamic exponent for discrete growth models abstract we develop a new method of measuring the dynamic exponent z for discrete growth models starting from a sinusoidal initial surface of a selected wavelength l we consider a relaxation function r t l which is a quantity similar to the autocorrelation function of the surface height typically the relaxation function decays exponentially following e g t l where l is the relaxation time and it depends on the wavelength l the dynamic exponent z is measured by the relation l l z we find that g x scales as x 1 0 for the family model and as x 1 5 for the restricted solid on solid model the advantages of the method are also discussed

neural networks that use three state neurons the dynamics of a neural network which uses three state neurons 1 0 and 1 is solved exactly in the limit of non symmetric and highly dilute synapses recursion relations for the activity the fraction of non zero neurons and overlap of the network with a given pattern are derived which have three generic kinds of fixed points a retrieval fixed point a chaotic fixed point which corresponds to non zero activity but no overlap and a zero fixed point where all the neurons go to the 0 state as the non retrieval fixed points both have activities different from the retrieval fixed point one can easily tell whether a pattern has been recovered an analysis of which fixed points occur as a function of the thresholds and the storage ratio of the system yields remarkably rich phase diagrams optimising the threshold level can be very important especially when low activity patterns are stored a similar analysis can be applied to biased networks using two state 1 0 neurons finally one finds that mixture states which have an overlap with two patterns can be stabilised by a threshold in the networks using three state neurons this property allows larger higher activity memories to be naturally constructed out of smaller ones

synchronous versus sequential updating in the three state ising neural network with variable dilution abstract the q 3 state ising neural network with synchronous updating and variable dilution is discussed starting from the appropriate hamiltonians the thermodynamic and retrieval properties are examined using replica mean field theory the appearance and properties of two cycles are discussed capacity temperature phase diagrams are derived for several values of the pattern activity and different gradations of dilution and the information content is calculated it is found that the asymptotic behaviour is rather similar to that for sequential updating the retrieval region is enhanced marginally but the spin glass region is visibly enlarged only the presence of self coupling can enlarge the retrieval region substantially the dynamics of the network is studied for general q and both synchronous and sequential updating using an extension of the generating function technique the differences with the signal to noise approach are outlined typical flow diagrams for the q 3 overlap order parameter are presented

a finite size scaling investigation for q state hopfield models storage capacity and basins of attraction the storage capacity of a q state hopfield network is determined via finite size scaling for parallel dynamics and q or 8 the results are in good agreement with theoretical predictions by rieger the basins of attraction and other associative memory properties are discussed for q 4 6 a self controlling q state model with improved basins of attraction is proposed

load distribution in weighted complex networks we study the load distribution in weighted networks by measuring the effective number of optimal paths passing through a given vertex the optimal path along which the total cost is minimum crucially depends on the cost distribution function p c c in the strong disorder limit where p c c ensuremath sim c ensuremath 1 the load distribution follows a power law both in the erd ifmmode mbox h o else h o fi s r enyi er random graphs and in the scale free sf networks and its characteristics are determined by the structure of the minimum spanning tree the distribution of loads at vertices with a given vertex degree also follows the sf nature similar to the whole load distribution implying that the global transport property is not correlated to the local structural information finally we measure the effect of disorder by the correlation coefficient between vertex degree and load finding that it is larger for er networks than for sf networks

generalized survival in equilibrium step fluctuations we investigate the dynamics of a generalized survival probability s t r defined with respect to an arbitrary reference level r rather than the average in equilibrium step fluctuations the exponential decay at large time scales of the generalized survival probability is numerically analyzed s t r is shown to exhibit simple scaling behavior as a function of system size l sampling time ensuremath delta t and the reference level r the generalized survival time scale ensuremath tau s r associated with s t r is shown to decay exponentially as a function of r

extreme fluctuations in stochastic network coordination with time delays we study the effects of uniform time delays on the extreme fluctuations in stochastic synchronization and coordination problems with linear couplings in complex networks we obtain the average size of the fluctuations at the nodes from the behavior of the underlying modes of the network we then obtain the scaling behavior of the extreme fluctuations with system size as well as the distribution of the extremes on complex networks and compare them to those on regular one dimensional lattices for large complex networks when the delay is not too close to the critical one fluctuations at the nodes effectively decouple and the limit distributions converge to the fisher tippett gumbel density in contrast fluctuations in low dimensional spatial graphs are strongly correlated and the limit distribution of the extremes is the airy density finally we also explore the effects of nonlinear couplings on the stability and on the extremes of the synchronization landscapes

a central limit theorem for the kpz equation we consider the kpz equation in one space dimension driven by a stationary centred space time random field which is sufficiently integrable and mixing but not necessarily gaussian we show that in the weakly asymmetric regime the solution to this equation considered at a suitable large scale and in a suitable reference frame converges to the hopf cole solution to the kpz equation driven by space time gaussian white noise while the limiting process depends only on the integrated variance of the driving field the diverging constants appearing in the definition of the reference frame also depend on higher order moments

superroughening by linear growth equations with spatiotemporally correlated noise we give an extensive study on a class of interfacial superroughening processes with finite lateral system size in 1 1 dimensions described by linear growth equations with spatiotemporally power law decaying correlated noise since some of these processes have an extremely long relaxation time we first develop a very efficient method capable of simulating the interface morphology of these growth processes even in very late time we numerically observe that this class of superrough growth processes indeed gradually develops macroscopic structures with the lateral size comparable to the lateral system size through the rigorous analytical study of the equal time height difference correlation function the different time height difference correlation function and the local width we explicitly evaluate not only the leading anomalous dynamic scaling term but also all the subleading anomalous dynamic scaling terms which dominate over the ordinary dynamic scaling term moreover the relation between the macroscopic structure formation and anomalous interfacial roughening of the superrough growth processes is analytically investigated in detail

joint distributions of partial and global maxima of a brownian bridge we analyze the joint distributions and temporal correlations between the partial maximum m and the global maximum m achieved by a brownian bridge on the subinterval and on the entire interval respectively we determine three probability distribution functions the joint distribution of both maxima the distribution p m of the partial maximum and the distribution of the gap between the maxima we present exact results for the moments of these distributions and quantify the temporal correlations between m and m by calculating the pearson correlation coefficient

synchronization transition in scale free networks clusters of synchrony we study the synchronization transition in scale free networks that display power law asymptotic behaviors in their degree distributions the critical coupling strength and the order parameter critical exponent derived by the mean field approach depend on the degree exponent ensuremath lambda which implies a close connection between structural organization and the emergence of dynamical order in complex systems we also derive the finite size scaling behavior of the order parameter finding that the giant cluster of synchronized nodes is formed in different ways between scale free networks with 2l ensuremath lambda l3 and those with ensuremath lambda g3

pattern selectivity in neural networks as a means of understanding basin structures the authors study the problem of learning and retrieving for a pair of correlated patterns within an extensive number of uncorrelated patterns for networks where learning may be treated as an optimization process with respect to an arbitrary cost function this formulation is then applied to several specific examples where they study the pattern selectivity of these systems i e their ability to differentiate correlated patterns and investigate the process of basin shrinking with increasing loading levels they define and discuss several different retrieval phases whose existence depends on the competitive interplay of the loading level and the pattern correlation discussion of asymptotic retrieval is restricted to dilute asymmetric networks

the architecture of complex weighted networks networked structures arise in a wide array of different contexts such as technological and transportation infrastructures social phenomena and biological systems these highly interconnected systems have recently been the focus of a great deal of attention that has uncovered and characterized their topological complexity along with a complex topological structure real networks display a large heterogeneity in the capacity and intensity of the connections these features however have mainly not been considered in past studies where links are usually represented as binary states i e either present or absent here we study the scientific collaboration network and the world wide air transportation network which are representative examples of social and large infrastructure systems respectively in both cases it is possible to assign to each edge of the graph a weight proportional to the intensity or capacity of the connections among the various elements of the network we define appropriate metrics combining weighted and topological observables that enable us to characterize the complex statistical properties and heterogeneity of the actual strength of edges and vertices this information allows us to investigate the correlations among weighted quantities and the underlying topological structure of the network these results provide a better description of the hierarchies and organizational principles at the basis of the architecture of weighted networks

mathematics of random growing interfaces we establish a thermodynamic limit and gaussian fluctuations for the height and surface width of the random interface formed by the deposition of particles on surfaces the results hold for the standard ballistic deposition model as well as the surface relaxation model in the off lattice setting the results are proved with the aid of general limit theorems for stabilizing functionals of marked poisson point processes

roughening of the interfaces in 1 1 dimensional two component surface growth with an admixture of random deposition we simulate competitive two component growth on a one dimensional substrate of l sites one component is a poisson type deposition that generates kardar parisi zhang kpz correlations the other is random deposition rd we derive the universal scaling function of the interface width for this model and show that the rd admixture acts as a dilatation mechanism to the fundamental time and height scales but leaves the kpz correlations intact this observation is generalized to other growth models it is shown that the flat substrate initial condition is responsible for the existence of an early nonscaling phase in the interface evolution the length of this initial phase is a nonuniversal parameter but its presence is universal we introduce a method to measure the length of this initial nonscaling phase in application to parallel and distributed computations the important consequence of the derived scaling is the existence of the upper bound for the desynchronization in a conservative update algorithm for parallel discrete event simulations it is shown that such algorithms are generally scalable in a ring communication topology

correlations in a generalized elastic model fractional langevin equation approach the generalized elastic model gem provides the evolution equation which governs the stochastic motion of several many body systems in nature such as polymers membranes and growing interfaces on the other hand a probe tracer particle in these systems performs a fractional brownian motion due to the spatial interactions with the other system s components the tracer s anomalous dynamics can be described by a fractional langevin equation fle with a space time correlated noise we demonstrate that the description given in terms of gem coincides with that furnished by the relative fle by showing that the correlation functions of the stochastic field obtained within the fle framework agree with the corresponding quantities calculated from the gem furthermore we show that the fox h function formalism appears to be very convenient to describe the correlation properties within the fle approach

conservative model for synchronization problems in complex networks in this paper we study the scaling behavior of the interface fluctuations roughness for a discrete model with conservative noise on complex networks conservative noise is a noise which has no external flux of deposition on the surface and the whole process is due to the diffusion it was found that in euclidean lattices the roughness of the steady state w s does not depend on the system size here we find that for scale free networks of n nodes characterized by a degree distribution p k approximately k lambda w s is independent of n for any lambda this behavior is very different than the one found by pastore y piontti phys rev e 76 046117 2007 for a discrete model with nonconservative noise which implies an external flux where w s approximately ln n for lambda 3 and was explained by nonlinear terms in the analytical evolution equation for the interface la rocca phys rev e 77 046120 2008 in this work we show that in these processes with conservative noise the nonlinear terms are not relevant to describe the scaling behavior of w s

limits on relief through constrained exchange on random graphs abstract agents are represented by nodes on a random graph e g small world each agent is endowed with a zero mean random value that may be either positive or negative all agents attempt to find relief i e to reduce the magnitude of that initial value to zero if possible through exchanges the exchange occurs only between the agents that are linked a constraint that turns out to dominate the results the exchange process continues until pareto equilibrium is achieved only 40 90 of the agents achieved relief on small world graphs with mean degree between 2 and 40 even fewer agents achieved relief on scale free like graphs with a truncated power law degree distribution the rate at which relief grew with increasing degree was slow only at most logarithmic for all of the graphs considered viewed in reverse the fraction of nodes that achieve relief is resilient to the removal of links

generalization and chaos in a layered neural network the generalization performance of a multi state and a graded response layered attractor neural network trained with examples of low activity is established exactly for monotonic and non monotonic input output functions complex behaviour is found which goes from fixed point attractors to chaos through a cascade of bifurcations depending on an appropriate threshold or cut off parameter the effect of the irregular behaviour on the generalization curves is explicitly demonstrated and phase diagrams for the recognition ratio of concepts in terms of the threshold cut off exhibit ordered generalization disordered paramagnetic or self sustained activity and chaotic phases

evolution equation for a model of surface relaxation in complex networks during the last few years the study of complex networks has moved its focus from the study of their topology to the dynamic processes occurring on the underlying network this is because many physical and dynamic processes use complex networks as substrates recently many studies of dynamic processes on networks such as epidemic spreading 2 traffic flow 3 4 cascading failure 5 and synchronization 6 7 have demonstrated the importance of the topology of the substrate network in the dynamic process there exists much evidence that many real networks possess a scale free sf degree distribution characterized by a power law tail

crossover and finite size effects in the 1 1 dimensional kardar parisi zhang equation crossover scaling of the surface width in the kardar parisi zhang equation for surface growth is studied numerically by means of a perturbative solution of the discretized equation and by comparison with the exact solution of the corresponding linear equation the finite size effects due to the spatial discretization are carefully analyzed the dependence on the nonlinearity of both the finite size and asymptotic scaling forms is then investigated

schramm loewner evolution theory of the asymptotic behaviors of 2 1 dimensional wolf villain model abstract although extensive analytical and numerical work has focus on investigating the 2 1 dimensional wolf villain wv model some problems concerning its asymptotical behaviors such as the universality class to which it belongs remain controversial the schramm loewner evolution sle theory is an attractive approach to describe the fluctuation phenomena and random processes and has also been applied to the analysis of the stochastic growth of surfaces in this work we applied sle theory to the analysis of the saturated surface to conduct in depth research with a new perspective on the asymptotical behaviors of the 2 1 dimensional wv model on the basis of the analysis of the saturated surface contour lines we determine that the diffusion coefficient calculated for 2 1 dimensional wv model is 2 91 0 01 and that for family model is 2 88 0 01 accordingly we can conclude from the view of sle theory that the 2 1 dimensional wv model similar to the family model also belongs to the edwards wilkinson universality class

the surface statistics of a granular aggregate the problem of the surface fluctuations in a settled granular material is posed a simple model is given which describes the process by which a particle settles and comes to rest on the existing surface of the packing and from this a set of langevin equations for the fourier modes of the surface are derived these equations imply that the fourier amplitudes behave like the velocities of a set of independent brownian particles we show that this results in logarithmically divergent surface fluctuations if the flux of particles onto the surface is random the divergence being removed by a more accurate description of the settling material for example by having the granules fall through a sieve

discrete surface growth process as a synchronization mechanism for scale free complex networks we consider the discrete surface growth process with relaxation to the minimum f family j phys a 19 l441 1986 as a possible synchronization mechanism on scale free networks characterized by a degree distribution p k ensuremath sim k ensuremath ensuremath lambda where k is the degree of a node and ensuremath lambda its broadness and compare it with the usually applied edward wilkinson process ew s f edwards and d r wilkinson proc r soc london ser a 381 17 1982 in spite of both processes belonging to the same universality class for euclidean lattices in this work we demonstrate that for scale free networks with exponents ensuremath lambda l3 the scaling behavior of the roughness in the saturation cannot be explained by the ew process moreover we show that for these ubiquitous cases the edward wilkinson process enhances spontaneously the synchronization when the system size is increased this nonphysical result is mainly due to finite size effects due to the underlying network contrarily the discrete surface growth process does not present this flaw and is applicable for every ensuremath lambda

mapping spatial persistent large deviations of nonequilibrium surface growth processes onto the temporal persistent large deviations of stochastic random walk processes spatial persistent large deviations probability of surface growth processes governed by the edwards wilkinson dynamics px x s with 1 or s or 1 is mapped isomorphically onto the temporal persistent large deviations probability pt t s associated with the stochastic markovian random walk problem we show using numerical simulations that the infinite family of spatial persistent large deviations exponents thetax s characterizing the power law decay of px x s agrees as predicted on theoretical grounds by phys rev lett 86 3700 2001 with the numerical measurements of thetat s the continuous family of exponents characterizing the long time power law behavior of pt t s we also discuss the simulations of the spatial persistence probability corresponding to a discrete model in the mullins herring universality class where our discrete simulations do not agree well with the theoretical predictions perhaps because of the severe finite size corrections which are known to strongly inhibit the manifestation of the asymptotic continuum behavior in discrete models involving large values of the dynamical exponent and the associated extremely slow convergence to the asymptotic regime

numerical solution of the kardar parisi zhang equation in one two and three dimensions abstract on a course grained level a family of microscopic growth processes may be described by a stochastic differential equation which is solved numerically for surface dimensions d 1 2 and 3 dimensional analysis shows that the spatial discretization parameter has the meaning of an effective coupling constant the numerical stability of the euler integration scheme is discussed for the strong coupling exponents defined by surface width time the following effective values were obtained d 1 0 330 0 004 and d 2 0 24 0 005 considering the width and its ensemble fluctuations at constant dimensionless time the transition between strong and weak coupling phases is located in d 3 for the largest coupling for which reliable data are available we obtain an effective exponent close to the best estimates on discrete models d 3 0 17