inverse avalanches in the abelian sandpile model we define and study the inverse of particle addition process in the abelian sandpile model we show how to obtain the unique recurrent configuration corresponding to a single particle deletion by a sequence of operations called inverse avalanches we study the probability distribution of s 1 the number of untopplings in the first inverse avalanche for a square lattice we determine prob s 1 exactly for s 1 0 1 2 and 3 for large s 1 we show that prob s 1 varies as s 1 11 8 in the direct avalanches this is related to the probability distribution of the number of sites which topple as often as the site where the particle was added these results are verified by numerical simulations

scaling properties of random walks on small world networks using both numerical simulations and scaling arguments we study the behavior of a random walker on a one dimensional small world network for the properties we study we find that the random walk obeys a characteristic scaling form these properties include the average number of distinct sites visited by the random walker the mean square displacement of the walker and the distribution of first return times the scaling form has three characteristic time regimes at short times the walker does not see the small world shortcuts and effectively probes an ordinary euclidean network in d dimensions at intermediate times the properties of the walker shows scaling behavior characteristic of an infinite small world network finally at long times the finite size of the network becomes important and many of the properties of the walker saturate we propose general analytical forms for the scaling properties in all three regimes and show that these analytical forms are consistent with our numerical simulations

noncausal gauss markov random fields parameter structure and estimation the parameter structure of noncausal homogeneous gauss markov random fields gmrf defined on finite lattices is studied for first order nearest neighbor and a special class of second order fields a complete characterization of the parameter space and a fast implementation of the maximum likelihood estimator of the field parameters are provided for general higher order fields tight bounds for the parameter space are presented and an efficient procedure for ml estimation is described experimental results illustrate the application of the approach presented and the viability of the present method in fitting noncausal models to 2 d data

investigation of the forest fire model on a small world network it is shown that the forest fire model of bak et al run on a square lattice network with additional long range interactions in the spirit of a small world network results in a scale free system reminiscent of self organized criticality without recourse to fine tuning as the number of these long range interactions is increased the cluster size distribution exponent is found to decrease in magnitude as the small world regime is entered indicating a change in its universality class it is suggested that such a model could have applicability in the study of disease spreading in human populations

clogging and self organized criticality in complex networks we propose a simple model that aims at describing in a stylized manner how local breakdowns due to imbalances or congestion propagate in real dynamical networks the model converges to a self organized critical stationary state in which the network shapes itself as a consequence of avalanches of rewiring processes depending on the model s specification we obtain either single scale or scale free networks we characterize in detail the relation between the statistical properties of the network and the nature of the critical state by computing the critical exponents the model also displays a nontrivial sudden collapse to a complete network

turbulent self organized criticality abstract in the prototype sandpile model of self organized criticality time series obtained by decomposing avalanches into waves distinguished by the number of times the avalanche seed site has toppled show intermittent fluctuations the q th moments of wave size differences possess local multiscaling and global simple scaling regimes analogous to those holding for velocity structure functions in fluid turbulence the correspondence involves identity of a basic scaling relation and of the form of relevant probability distributions the sandpile provides a qualitative analog of many features of turbulent phenomena

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

critical behavior of a small world sandpile model we have considered the abelian sandpile model on a small world network to obtain a small world network we have added a few long ranged bonds to an ordinary square lattice it is observed that the probability distribution of the avalanches has two distinct power law regimes one corresponding to the ordinary btw model and another that emerges because of the small world structure of the network

target problem on small world networks in this work we focus on reactions on small world networks swn s disordered graphs of much recent interest we study the target problem since it allows an exact solution on regular lattices on swn s we find that the decay of the targets for which we extend the formalism to disordered lattices is again related to s n the mean number of distinct sites visited in n steps although the s n vs n dependence changes here drastically in going from regular linear chains to their swn

sensor networks with random links topology design for distributed consensus in a sensor network in practice the communication among sensors is subject to 1 errors that can cause failures of links among sensors at random times 2 costs and 3 constraints such as power data rate or communication since sensors and networks operate under scarce resources the paper studies the problem of designing the topology i e assigning the probabilities of reliable communication among sensors or of link failures to maximize the rate of convergence of average consensus when the link communication costs are taken into account and there is an overall communication budget constraint we model the network as a bernoulli random topology and establish necessary and sufficient conditions for mean square sense mss and almost sure a s convergence of average consensus when network links fail in particular a necessary and sufficient condition is for the algebraic connectivity of the mean graph topology to be strictly positive with these results we show that the topology design with random link failures link communication costs and a communication cost constraint is a constrained convex optimization problem that can be efficiently solved for large networks by semidefinite programming techniques simulations demonstrate that the optimal design improves significantly the convergence speed of the consensus algorithm and can achieve the performance of a non random network at a fraction of the communication cost

sandpiles on watts strogatz type small worlds abstract we study a one dimensional sandpile model in small world networks with long range links either by introducing them randomly to fixed connection topology quenched randomness or to temporary connection topology annealed randomness between cells to allow a grain to topple from a cell to a neighbouring or distant cell these models are investigated both analytically and by computer simulations and they show self organized criticality unlike the original one dimensional sandpile model the simulations also show that the distribution of avalanche size undergoes a transition from a non critical to a critical regime in addition we have found that for annealed and quenched randomness there is a scaling for the size distribution of avalanches with a single power law exponent which is the same as that found for the standard sandpile model in higher dimensions we also show that the average number of grains in the system follows power law behaviour as a function of the probability of long range links with different exponents for the annealed and quenched systems

moment analysis of the probability distribution of different sandpile models we reconsider the moment analysis of the bak tang wiesenfeld and the stochastic sandpile model introduced by manna j phys a 24 l363 1991 in two and three dimensions in contrast to recently performed investigations our analysis reveals that the models are characterized by different scaling behavior i e they belong to different universality classes

dissipative sandpile models with universal exponents we consider a dissipative variant of the stochastic abelian sandpile model on a two dimensional lattice the boundaries are closed and the dissipation is due to the fact that each toppled grain is removed from the lattice with probability epsilon it is shown that the scaling properties of this model are in the universality class of the stochastic abelian models with conservative dynamics and open boundaries in particular the dissipation rate epsilon can be adjusted according to a suitable function epsilon f l such that the avalanche size distribution will coincide with that of the conservative model on a finite lattice of size l

mechanisms of avalanche dynamics and forms of scaling in sandpiles abstract the probability distributions of several avalanche quantities of the bak tang wiesenfeld sandpile in two dimensions display multifractal scaling this explains why this prototype model remained extremely problematic and controversial for such a long time after its introduction other models like the manna model obey instead standard finite size scaling and fall in different universality classes an analysis of time series for the waves into which avalanches can be decomposed shows that the different forms of scaling can be ascribed to the respective autocorrelation functions the intermittency of the bak tang wiesenfeld sandpile is due to this autocorrelation being long range in time unlike in the manna case a coarse graining of the wave time series elucidates these differences and the mechanism leading to multifractality

a generalized levinson algorithm for covariance extension with application to multiscale autoregressive modeling efficient computation of extensions of banded partially known covariance matrices is provided by the classical levinson algorithm one contribution of this paper is the introduction of a generalization of this algorithm that is applicable to a substantially broader class of extension problems this generalized algorithm can compute unknown covariance elements in any order that satisfies certain graph theoretic properties which we describe this flexibility which is not provided by the classical levinson algorithm is then harnessed in a second contribution of this paper the identification of a multiscale autoregressive mar model for the maximum entropy me extension of a banded partially known covariance matrix the computational complexity of mar model identification is an order of magnitude below that of explicitly computing a full covariance extension and is comparable to that required to build a standard autoregressive ar model using the classical levinson algorithm

combining and updating of local estimates and regional maps along sets of one dimensional tracks in this paper we consider the problem of combining and updating estimates that may have been generated in a distributed fashion or may represent estimates generated at different times of the same process sample path the first of these cases has applications in decentralized estimation while the second has applications in updating maps of spatially distributed random quantities given measurements along several tracks the method of solution for the second problem uses the result of the first and the similarity in the formulation and solution of these problems emphasizes the conceptual similarity between many problems in decentralized control and in the analysis of random fields

self organized criticality on small world networks abstract we study the btw height model of self organized criticality on a square lattice with some long range connections giving to the lattice the character of small world network we find that as function of the fraction p of long ranged bonds the power law of the avalanche size and lifetime distribution changes following a crossover scaling law with crossover exponents 2 3 and 1 for size and lifetime respectively

characteristics of deterministic and stochastic sandpile models in a rotational sandpile model rotational constraint representing a local external bias generally has a nontrivial effect on the critical behavior of lattice statistical models in equilibrium critical phenomena in order to study the effect of rotational bias in an out of equilibrium situation like self organized criticality a two state quasideterministic rotational sandpile model is developed here imposing rotational constraint on the flow of sand grains an extended set of critical exponents are estimated to characterize the avalanche properties at the nonequilibrium steady state of the model the probability distribution functions are found to obey usual finite size scaling supported by negative time autocorrelation between the toppling waves the model exhibits characteristics of both deterministic and stochastic sandpile models

exact results for the one dimensional self organized critical forest fire model we present the analytic solution of the self organized critical soc forest fire model in one dimension proving soc in systems without conservation laws by analytic means under the condition that the system is in the steady state and very close to the critical point we calculate the probability that a string of ital n neighboring sites is occupied by a given configuration of trees the critical exponent describing the size distribution of forest clusters is exactly tau 2 and does not change under certain changes of the model rules computer simulations confirm the analytic results

local variational principle concerning entropy of a sofic group action abstract recently lewis bowen introduced a notion of entropy for measure preserving actions of countable sofic groups admitting a generating measurable partition with finite entropy and then david kerr and hanfeng li developed an operator algebraic approach to actions of countable sofic groups not only on a standard probability space but also on a compact metric space and established the global variational principle concerning measure theoretic and topological entropy in this sofic context by localizing these two kinds of entropy in this paper we prove a local version of the global variational principle for any finite open cover of the space and show that these local measure theoretic and topological entropies coincide with their classical counterparts when the acting group is an infinite amenable group

robust distributed estimation using the embedded subgraphs algorithm we propose a new iterative distributed approach for linear minimum mean square error lmmse estimation in graphical models with cycles the embedded subgraphs algorithm esa decomposes a loopy graphical model into a number of linked embedded subgraphs and applies the classical parallel block jacobi iteration comprising local lmmse estimation in each subgraph involving inversion of a small matrix followed by an information exchange between neighboring nodes and subgraphs our primary application is sensor networks where the model encodes the correlation structure of the sensor measurements which are assumed to be gaussian the resulting lmmse estimation problem involves a large matrix inverse which must be solved in network with distributed computation and minimal intersensor communication by invoking the theory of asynchronous iterations we prove that esa is robust to temporary communication faults such as failing links and sleeping nodes and enjoys guaranteed convergence under relatively mild conditions simulation studies demonstrate that esa compares favorably with other recently proposed algorithms for distributed estimation simulations also indicate that energy consumption for iterative estimation increases substantially as more links fail or nodes sleep thus somewhat surprisingly sensor network energy conservation strategies such as low powered transmission and aggressive sleep schedules could actually prove counterproductive our results can be replicated using matlab code from www dsp rice edu software

dynamical and spatial aspects of sandpile cellular automata the bak tang and wiesenfeld cellular automaton is simulated in 1 2 3 4 and 5 dimensions we define a new set of scaling exponents by introducing the concept of conditional expectation values scaling relations are derived and checked numerically and the critical dimension is discussed we address the problem of the mass dimension of the avalanches and find that the avalanches are noncompact for dimensions larger than 2 the scaling of the power spectrum derives from the assumption that the instantaneous dissipation rate of the individual avalanches obeys a simple scaling relation primarily the results of our work show that the flow of sand down the slope does not have a 1 f power spectrum in any dimension although the model does show clear critical behavior with scaling exponents depending on the dimension the power spectrum behaves as 1 f 2 in all the dimensions considered

kalman filtering with intermittent observations motivated by navigation and tracking applications within sensor networks we consider the problem of performing kalman filtering with intermittent observations when data travel along unreliable communication channels in a large wireless multihop sensor network the effect of communication delays and loss of information in the control loop cannot be neglected we address this problem starting from the discrete kalman filtering formulation and modeling the arrival of the observation as a random process we study the statistical convergence properties of the estimation error covariance showing the existence of a critical value for the arrival rate of the observations beyond which a transition to an unbounded state error covariance occurs we also give upper and lower bounds on this expected state error covariance

scaling of waves in the bak tang wiesenfeld sandpile model n with aen ae0 n a similar ansatzes should according to this view hold for the number of topplings s which difiers from a because sites can topple more than once in an avalanche and for the radius and duration of avalanches but recent inves tigations 10 12 show that the two dimensional btw model may be characterized by a multifractal behavior where difierent moments of a are governed by exponents aen which are not linear in n and are indeed not related to each other for difierent n difierent reasons for this have been proposed in 10 11 and in 12 notice that multi fractality of avalanches can be proven for certain variants of the 1 d sandpile model 14 deviations from pure power laws had been seen already in early simulations but were usually interpreted as fl nite size efiects due to avalanches which touch the bound ary of the lattice to illustrate that this is most likely not true and that there is a real problem with simple scaling we show in fig 1 the ratio ps x l pa x l of the integrated distributions ps x l r 1 x dx 0 ps x 0 l for d 2 and for difierent values of l in these simula tions cylindrical boundary conditions were used open at y l 2 and periodic at x 0 l and data were col lected only for avalanches starting at y 0 in this way we hope to have minimized boundary efiects also since we do not make separate flts to ps x l and pa x l we have none of the uncertainties inherent in such flts due

sandpile on directed small world networks we numerically investigate the avalanche dynamics of the bak tang wiesenfeld sandpile model on directed small world networks we find that the avalanche size and duration distribution follow a power law for all rewiring probabilities p specially we find that approaching the thermodynamic limit l the values of critical exponents do not depend on p and are consistent with the mean field solution in euclidean space for any p 0 in addition we measure the dynamic exponent in the relation between avalanche size and avalanche duration and find that the values of the dynamic exponents are also consistent with the mean field values for any p 0

rare events and breakdown of simple scaling in the abelian sandpile model due to intermittency and conservation the abelian sandpile in two dimensions obeys multifractal rather than finite size scaling in the thermodynamic limit a vanishingly small fraction of large avalanches dominates the statistics and a constant gap scaling is recovered in higher moments of the toppling distribution thus rare events shape most of the scaling pattern and preserve a meaning for effective exponents which can be determined on the basis of numerical and exact results

critical properties of a dissipative sandpile model on small world networks a dissipative sandpile model is constructed and studied on small world networks swns swns are generated by adding extra links between two arbitrary sites of a two dimensional square lattice with different shortcut densities ensuremath phi three regimes are identified regular lattice rl for ensuremath phi ensuremath lesssim 2 ensuremath 12 swn for 2 ensuremath 12 l ensuremath phi l0 1 and random network rn for ensuremath phi ensuremath ge 0 1 in the rl regime the sandpile dynamics is characterized by the usual bak tang and weisenfeld btw type correlated scaling whereas in the rn regime it is characterized by mean field scaling on swns both scaling behaviors are found to coexist small compact avalanches below a certain characteristic size s c are found to belong to the btw universality class whereas large sparse avalanches above s c are found to belong to the mean field universality class a scaling theory for the coexistence of two scaling forms on a swn is developed and numerically verified though finite size scaling is not valid for the dissipative sandpile model on rls or on swns it is found to be valid on rns for the same model finite size scaling on rns appears to be an outcome of super diffusive sand transport and uncorrelated toppling waves

sandpile models and random walkers on finite lattices abelian sandpile models both deterministic such as the bak tang wiesenfeld btw model p bak c tang and k wiesenfeld phys rev lett 59 381 1987 and stochastic such as the manna model s s manna j phys a 24 l363 1991 are studied on finite square lattices with open boundaries the avalanche size distribution p l n is calculated for a range of system sizes l the first few moments of this distribution are evaluated numerically and their dependence on the system size is examined the sandpile models are conservative in the sense that grains are conserved in the bulk and can leave the system only through the boundaries it is shown that the conservation law provides an interesting connection between the sandpile models and random walk models using this connection it is shown that the average avalanche sizes l for the btw and manna models are equal to each other and both are equal to the average path length of a random walker starting from a random initial site on the same lattice of size l this is in spite of the fact that the sandpile models with deterministic btw and stochastic manna toppling rules exhibit different critical exponents indicating that they belong to different universality classes

distributed sensor localization in random environments using minimal number of anchor nodes the paper introduces diloc a distributed iterative algorithm to locate m sensors with unknown locations in rm m ges 1 with respect to a minimal number of m 1 anchors with known locations the sensors and anchors nodes in the network exchange data with their neighbors only no centralized data processing or communication occurs nor is there a centralized fusion center to compute the sensors locations diloc uses the barycentric coordinates of a node with respect to its neighbors these coordinates are computed using the cayley menger determinants i e the determinants of matrices of internode distances we show convergence of diloc by associating with it an absorbing markov chain whose absorbing states are the states of the anchors we introduce a stochastic approximation version extending diloc to random environments i e when the communications among nodes is noisy the communication links among neighbors may fail at random times and the internodes distances are subject to errors we show a s convergence of the modified diloc and characterize the error between the true values of the sensors locations and their final estimates given by diloc numerical studies illustrate diloc under a variety of deterministic and random operating conditions

simultaneous recursive displacement estimation and restoration of noisy blurred image sequences we develop a recursive model based maximum a posteriori map estimator that simultaneously estimates the displacement vector field dvf and the intensity field from a noisy blurred image sequence current motion compensated spatio temporal noise filters treat the estimation of the dvf as a preprocessing step generally no attempt is made to verify the accuracy of these estimates prior to their use in the filter by simultaneously estimating these two fields we establish a link between the two estimators it is through this link that the dvf estimate and its corresponding accuracy information are shared with the other intensity estimator and vice versa to model the dvf and the intensity field we use coupled gauss markov cgm models a cgm model consists of two levels an upper level which is made up of several submodels with various characteristics and a lower level or line field which governs the transitions between the submodels the cgm models are well suited for estimating the displacement and intensity fields since the resulting estimates preserve the boundaries between the stationary areas present in both fields detailed line fields are proposed for the modeling of these boundaries which also take into account the correlations that exist between these two fields a kalman type estimator results followed by a decision criterion for choosing the appropriate set of line fields several experiments using noisy and noisy blurred image sequences demonstrate the superior performance of the proposed algorithm with respect to prediction error and mean square error

sandpile models with and without an underlying spatial structure we present a simple mean field model for the sandpile model introduced by bak tang and wiesenfeld btw phys rev lett 59 381 1987 in the mean field model we are able to pinpoint the process of self organization as well as the emerging scale invariance displayed as a power law distribution of avalanche sizes we discuss the btw sandpile model on a lattice and show that the dynamical behavior can be expressed as a transport problem this implies that the average avalanche size scales with the system size and additional heuristic arguments related to the transport properties more than indicate the origin of the power law behavior we review recent work in which scaling relations and additional constraints between the various critical exponents are addressed we demonstrate that some of the proposed relations are inconsistent we present a coherent theory in which the scaling relations along with additional constraints leave only one exponent unknown

distributing the kalman filter for large scale systems this paper presents a distributed kalman filter to estimate the state of a sparsely connected large scale n dimensional dynamical system monitored by a network of n sensors local kalman filters are implemented on nl dimensional subsystems nl lt n obtained by spatially decomposing the large scale system the distributed kalman filter is optimal under an lth order gauss markov approximation to the centralized filter we quantify the information loss due to this lth order approximation by the divergence which decreases as l increases the order of the approximation l leads to a bound on the dimension of the subsystems hence providing a criterion for subsystem selection the approximated centralized riccati and lyapunov equations are computed iteratively with only local communication and low order computation by a distributed iterate collapse inversion dici algorithm we fuse the observations that are common among the local kalman filters using bipartite fusion graphs and consensus averaging algorithms the proposed algorithm achieves full distribution of the kalman filter nowhere in the network storage communication or computation of n dimensional vectors and matrices is required only nl lt n dimensional vectors and matrices are communicated or used in the local computations at the sensors in other words knowledge of the state is itself distributed

mean field solution of the small world network model the small world network model is a simple model of the structure of social networks which possesses characteristics of both regular lattices and random graphs the model consists of a one dimensional lattice with a low density of shortcuts added between randomly selected pairs of points these shortcuts greatly reduce the typical path length between any two points on the lattice we present a mean field solution for the average path length and for the distribution of path lengths in the model this solution is exact in the limit of large system size and either a large or small number of shortcuts

matrices with banded inverses inversion algorithms and factorization of gauss markov processes the paper considers the inversion of full matrices whose inverses are l banded we derive a nested inversion algorithm for such matrices applied to a tridiagonal matrix the algorithm provides its explicit inverse as an element wise product hadamard product of three matrices when related to gauss markov random processes gmrp this result provides a closed form factored expression for the covariance matrix of a first order gmrp this factored form leads to the interpretation of a first order gmrp as the product of three independent processes a forward independent increments process a backward independent increments process and a variance stationary process we explore the nonuniqueness of the factorization and design it so that the forward and backward factor processes have minimum energy we then consider the issue of approximating general nonstationary gaussian processes by gauss markov processes under two optimality criteria the kullback leibler distance and maximum entropy the problem reduces to approximating general covariances by covariance matrices whose inverses are banded our inversion result is an efficient algorithmic solution to this problem we evaluate the information loss between the original process and its gauss markov approximation

optimal control of spatially distributed systems in this paper we study the structural properties of optimal control of spatially distributed systems such systems consist of an infinite collection of possibly heterogeneous linear control systems that are spatially interconnected via certain distant dependent coupling functions over arbitrary graphs we study the structural properties of optimal control problems with infinite horizon linear quadratic criteria by analyzing the spatial structure of the solution to the corresponding operator lyapunov and riccati equations the key idea of the paper is the introduction of a special class of operators called spatially decaying sd these operators are a generalization of translation invariant operators used in the study of spatially invariant systems we prove that given a control system with a state space representation consisting of sd operators the solution of operator lyapunov and riccati equations are sd furthermore we show that the kernel of the optimal state feedback for each subsystem decays in the spatial domain with the type of decay e g exponential polynomial or logarithmic depending on the type of coupling between subsystems

the abelian sandpile and related models abstract the abelian sandpile model is the simplest analytically tractable model of self organized criticality this paper presents a brief review of known results about the model the abelian group structure of the algebra of operators allows an exact calculation of many of its properties in particular when there is a preferred direction one can calculate all the critical exponents characterizing the distribution of avalanche sizes in all dimensions for the undirected case the model is related to q 0 state potts model this enables exact calculation of some exponents in two dimensions and there are some conjectures about others we also discuss a generalization of the model to a network of communicating reactive processors this includes sandpile models with stochastic toppling rules as a special case we also consider a nondashabelian stochastic variant which lies in a different universality class related to directed percolation

some results and a conjecture for manna s stochastic sandpile model we present some analytical results for the stochastic sandpile model studied earlier by manna in this model the operators corresponding to particle addition at different sites commute the eigenvalues of operators satisfy a system of coupled polynomial equations for an l l square we construct a nontrivial toppling invariant and hence a ladder operator which acting on eigenvectors of the evolution operator gives new eigenvectors with different eigenvalues for periodic boundary conditions in one direction one more toppling invariant can be constructed we show that there are many forbidden subconfigurations and only an exponentially small fraction of all stable configurations are recurrent we obtain rigorous lower and upper bounds for the minimum number of particles in a recurrent configuration and conjecture a formula for its exact value for finite size rectangles

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

scaling of random spreading in small world networks in this study we have carried out computer simulations of random walks on watts strogatz type small world networks and measured the mean number of visited sites and the return probabilities these quantities were found to obey scaling behavior with intuitively reasoned exponents as long as the probability p of having a long range bond was sufficiently low

numerical determination of the avalanche exponents of the bak tang wiesenfeld model we consider the bak tang wiesenfeld sandpile model on a two dimensional square lattice of lattice sizes up to l 4096 a detailed analysis of the probability distribution of the size area duration and radius of the avalanches will be given to increase the accuracy of the determination of the avalanche exponents we introduce a new method for analyzing the data which reduces the finite size effects of the measurements the exponents of the avalanche distributions differ slightly from previous measurements and estimates obtained from a renormalization group approach

self organized random walks and stochastic sandpile from linear to branched avalanches abstract in a model of self organized criticality unstable sites discharge to just one of their neighbours for constant discharge ratio and for a certain range of values of the input energy avalanches are simple branchless polya random walks and their scaling properties can be derived exactly if fluctuates widely enough avalanches become branched due to multiple discharges and behave like those of the stochastic sandpile at the threshold for branched behaviour peculiar scaling and anomalous diffusive transport are observed

from waves to avalanches two different mechanisms of sandpile dynamics time series resulting from wave decomposition show the existence of different correlation patterns for avalanche dynamics for the d 2 bak tang wisenfeld model long range correlations determine a modification of the wave size distribution under coarse graining in time and multifractal scaling for avalanches in the manna model the distribution of avalanche coincides with that of waves which are uncorrelated and obey finite size scaling a result expected also for the d 3 bak tang wiesenfeld sandpile

theoretical studies of self organized criticality these notes are intended to provide a pedagogical introduction to the abelian sandpile model of self organized criticality and its related models the abelian group the algebra of particle addition operators the burning test for recurrent states equivalence to the spanning trees problem are described the exact solution of the directed version of the model in any dimension is explained the model s equivalence to scheidegger s model of river basins takayasu s aggregation model and the voter model is discussed for the undirected case the solution for one dimensional lattices and the bethe lattice is briefly described known results about the two dimensional case are summarized generalization to the abelian distributed processors model is discussed time dependent properties and the universality of critical behavior in sandpiles are briefly discussed i conclude by listing some still unsolved problems

erratum small world networks evidence for a crossover picture phys rev lett 82 3180 1999 we have performed new calculations using the breadth first search algorithm 1 2 we are now able to study systems with sizes up to n 5500 as shown in fig 1 we now find t o 1 in agreement with the simple argument given in our letter but different from the originally reported numerical result st 0 67 6 0 10d the reason for the incorrect numerical result reported initially is the small system sizes we studied which did not allow us to reach the asymptotic regime we thank a barrat 3 and m e j newman and d j watts 4 for alerting us to the possibility of an error on our estimate of t we also thank m argollo de menezes for directing us to the breadth first search algorithm

universality classes in isotropic abelian and non abelian sandpile models universality in isotropic abelian and non abelian sandpile models is examined using extensive numerical simulations to characterize the critical behavior we employ an extended set of critical exponents geometric features of the avalanches as well as scaling functions describing the time evolution of average quantities such as the area and size during the avalanche comparing between the abelian bak tang wiesenfeld model p bak c tang and k wiensenfeld phys rev lett 59 381 1987 and the non abelian models introduced by manna s s manna j phys a 24 l363 1991 and zhang y c zhang phys rev lett 63 470 1989 we find strong indications that each one of these models belongs to a distinct universality class

sandpile model on an optimized scale free network on euclidean space deterministic sandpile models are studied on a cost optimized barabasi albert ba scale free network whose nodes are the sites of a square lattice for the optimized ba network the sandpile model has the same critical behaviour as the btw sandpile whereas for the un optimized ba network the critical behaviour is mean field like

block matrices with l block banded inverse inversion algorithms block banded matrices generalize banded matrices we study the properties of positive definite full matrices p whose inverses a are l block banded we show that for such matrices the blocks in the l block band of p completely determine p namely all blocks of p outside its l block band are computed from the blocks in the l block band of p we derive fast inversion algorithms for p and its inverse a that when compared to direct inversion are faster by two orders of magnitude of the linear dimension of the constituent blocks we apply these inversion algorithms to successfully develop fast approximations to kalman bucy filters in applications with high dimensional states where the direct inversion of the covariance matrix is computationally unfeasible