The phenomenon of residential segregation was captured by Schelling's famous segregation model where two types of agents are placed on a grid and an agent is content with her location if the fraction of her neighbors which have the same type as her is at least @math , for some @math . Discontent agents simply swap their location with a randomly chosen other discontent agent or jump to a random empty cell. We analyze a generalized game-theoretic model of Schelling segregation which allows more than two agent types and more general underlying graphs modeling the residential area. For this we show that both aspects heavily influence the dynamic properties and the tractability of finding an optimal placement. We map the boundary of when improving response dynamics (IRD), i.e., the natural approach for finding equilibrium states, are guaranteed to converge. For this we prove several sharp threshold results where guaranteed IRD convergence suddenly turns into the strongest possible non-convergence result: a violation of weak acyclicity. In particular, we show such threshold results also for Schelling's original model, which is in contrast to the standard assumption in many empirical papers. Furthermore, we show that in case of convergence, IRD find an equilibrium in @math steps, where @math is the number of edges in the underlying graph and show that this bound is met in empirical simulations starting from random initial agent placements.
Recently, a series of papers by Young @cite_24 , Zhang @cite_21 @cite_28 , @cite_13 , @cite_6 @cite_22 , @cite_18 @cite_27 and @cite_19 initiated a rigorous analysis of stochastic processes induced by Schelling's model. In these processes either two randomly chosen unhappy agents of different type swap positions @cite_24 @cite_21 @cite_28 or a randomly chosen agent changes her type with a certain probability @cite_6 @cite_18 @cite_19 @cite_27 @cite_22 . It is worth noticing that both types of processes are closely related but not identical to Schelling's original model where discontent agents move to different positions until they become content with their current location. The focus of the above mentioned works is on investigating the expected size of the obtained homogeneous regions, but it is also shown that the stochastic processes starting from a uniform random agent placement converge with high probability to a stable placement. The convergence time was considered by Mobius & Rosenblat @cite_12 who observe that the Markov chain analyzed in @cite_24 @cite_21 @cite_28 has a very high mixing time. @cite_19 show in the two-dimensional grid case a dichotomy in mixing times for high @math and very low @math values.
@cite_18 @cite_22 @cite_28 @cite_21 @cite_6 @cite_24 @cite_19 @cite_27 @cite_13 @cite_12
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Schelling's model of segregation looks to explain the way in which particles or agents of two types may come to arrange themselves spatially into configurations consisting of large homogeneous clusters, i.e. connected regions consisting of only one type. As one of the earliest agent based models studied by economists and perhaps the most famous model of self-organising behaviour, it also has direct links to areas at the interface between computer science and statistical mechanics, such as the Ising model and the study of contagion and cascading phenomena in networks. While the model has been extensively studied it has largely resisted rigorous analysis, prior results from the literature generally pertaining to variants of the model which are tweaked so as to be amenable to standard techniques from statistical mechanics or stochastic evolutionary game theory. In BK , Brandt, Immorlica, Kamath and Kleinberg provided the first rigorous analysis of the unperturbed model, for a specific set of input parameters. Here we provide a rigorous analysis of the model's behaviour much more generally and establish some surprising forms of threshold behaviour, notably the existence of situations where an level of intolerance for neighbouring agents of opposite type leads almost certainly to segregation. We introduce an extended family of continuous-domain stochastic models for sparse, piecewise-smooth signals. These are specified as solutions of stochastic differential equations, or, equivalently, in terms of a suitable innovation model; the latter is analogous conceptually to the classical interpretation of a Gaussian stationary process as filtered white noise. The two specific features of our approach are 1) signal generation is driven by a random stream of Dirac impulses (Poisson noise) instead of Gaussian white noise, and 2) the class of admissible whitening operators is considerably larger than what is allowed in the conventional theory of stationary processes. We provide a complete characterization of these finite-rate-of-innovation signals within Gelfand's framework of generalized stochastic processes. We then focus on the class of scale-invariant whitening operators which correspond to unstable systems. We show that these can be solved by introducing proper boundary conditions, which leads to the specification of random, spline-type signals that are piecewise-smooth. These processes are the Poisson counterpart of fractional Brownian motion; they are nonstationary and have the same 1 ω-type spectral signature. We prove that the generalized Poisson processes have a sparse representation in a wavelet-like basis subject to some mild matching condition. We also present a limit example of sparse process that yields a MAP signal estimator that is equivalent to the popular TV-denoising algorithm. Random graph models with limited choice have been studied extensively with the goal of understanding the mechanism of the emergence of the giant component. One of the standard models are the Achlioptas random graph processes on a fixed set of (n ) vertices. Here at each step, one chooses two edges uniformly at random and then decides which one to add to the existing configuration according to some criterion. An important class of such rules are the bounded-size rules where for a fixed (K 1 ), all components of size greater than (K ) are treated equally. While a great deal of work has gone into analyzing the subcritical and supercritical regimes, the nature of the critical scaling window, the size and complexity (deviation from trees) of the components in the critical regime and nature of the merging dynamics has not been well understood. In this work we study such questions for general bounded-size rules. Our first main contribution is the construction of an extension of Aldous’s standard multiplicative coalescent process which describes the asymptotic evolution of the vector of sizes and surplus of all components. We show that this process, referred to as the standard augmented multiplicative coalescent (AMC) is ‘nearly’ Feller with a suitable topology on the state space. Our second main result proves the convergence of suitably scaled component size and surplus vector, for any bounded-size rule, to the standard AMC. This result is new even for the classical Erdős–Renyi setting. The key ingredients here are a precise analysis of the asymptotic behavior of various susceptibility functions near criticality and certain bounds from (The barely subcritical regime. Arxiv preprint, 2012) on the size of the largest component in the barely subcritical regime. Schelling’s models of segregation, first described in 1969 (Am Econ Rev 59:488–493, 1969) are among the best known models of self-organising behaviour. Their original purpose was to identify mechanisms of urban racial segregation. But his models form part of a family which arises in statistical mechanics, neural networks, social science, and beyond, where populations of agents interact on networks. Despite extensive study, unperturbed Schelling models have largely resisted rigorous analysis, prior results generally focusing on variants in which noise is introduced into the dynamics, the resulting system being amenable to standard techniques from statistical mechanics or stochastic evolutionary game theory (Young in Individual strategy and social structure: an evolutionary theory of institutions, Princeton University Press, Princeton, 1998). A series of recent papers ( in: Proceedings of the 44th annual ACM symposium on theory of computing (STOC 2012), 2012); in: 55th annual IEEE symposium on foundations of computer science, Philadelphia, 2014, J Stat Phys 158:806–852, 2015), has seen the first rigorous analyses of 1-dimensional unperturbed Schelling models, in an asymptotic framework largely unknown in statistical mechanics. Here we provide the first such analysis of 2- and 3-dimensional unperturbed models, establishing most of the phase diagram, and answering a challenge from in: Proceedings of the 44th annual ACM symposium on theory of computing (STOC 2012), 2012). In this paper, a novel algorithm to blindly separate an instantaneous linear underdetermined mixture of nonstationary sources is proposed. It means that the number of sources exceeds the number of channels of the available data. The separation is based on the working assumption that the sources are piecewise stationary with a different variance in each block. It proceeds in two steps: 1) estimating the mixing matrix, and 2) computing the optimum beamformer in each block to maximize the signal-to-interference ratio of each separated signal with respect to the remaining signals. Estimating the mixing matrix is accomplished through a specialized tensor decomposition of the set of sample covariance matrices of the received mixture in each block. It utilizes optimum weighting, which allows statistically efficient (CRB attaining) estimation provided that the data obey the assumed Gaussian piecewise stationary model. In simulations, performance of the algorithm is successfully tested on blind separation of 16 speech signals from nine linear instantaneous mixtures of these signals. This paper considers the effect of stochasticity on the quality of convergence of genetic algorithms (GAs). In many problems, the variance of building-block fitness or so-called collateral noise is the major source of variance, and a population-sizing equation is derived to ensure that average signal-to-collateral-noise ratios are favorable to the discrimination of the best building blocks required to solve a problem of bounded deception. The sizing relation is modified to permit the inclusion of other sources of stochasticity, such as the noise of selection, the noise of genetic operators, and the explicit noise or nondeterminism of the objective function. In a test suite of five functions, the sizing relation proves to be a conservative predictor of average correct convergence, as long as all major sources of noise are considered in the sizing calculation. These results suggest how the sizing equation may be viewed as a coarse delineation of a boundary between what a physicist might call two distinct phases of GA behavior. At low population sizes the GA makes many errors of decision, and the quality of convergence is largely left to the vagaries of chance or the serial fixup of flawed results through mutation or other serial injection of diversity. At large population sizes, GAs can reliably discriminate between good and bad building blocks, and parallel processing and recombination of building blocks lead to quick solution of even difficult deceptive problems. Additionally, the paper outlines a number of extensions to this work, including the development of more refined models of the relation between generational average error and ultimate convergence quality, the development of online methods for sizing populations via the estimation of populations via the estimation of population-sizing parameters, and the investigation of populationsizing in the context of niching and other schemes designed for use in problems with high cardinality solution sets. The paper also discusses how these results may one day lead to rigorous proofs of convergence for recombinative G As operating on problems of bounded description. We study the statistical properties of the endpoint of stochastic gradient descent (SGD). We approximate SGD as a stochastic differential equation (SDE) and consider its Boltzmann Gibbs equilibrium distribution under the assumption of isotropic variance in loss gradients.. Through this analysis, we find that three factors – learning rate, batch size and the variance of the loss gradients – control the trade-off between the depth and width of the minima found by SGD, with wider minima favoured by a higher ratio of learning rate to batch size. In the equilibrium distribution only the ratio of learning rate to batch size appears, implying that it’s invariant under a simultaneous rescaling of each by the same amount. We experimentally show how learning rate and batch size affect SGD from two perspectives: the endpoint of SGD and the dynamics that lead up to it. For the endpoint, the experiments suggest the endpoint of SGD is similar under simultaneous rescaling of batch size and learning rate, and also that a higher ratio leads to flatter minima, both findings are consistent with our theoretical analysis. We note experimentally that the dynamics also seem to be similar under the same rescaling of learning rate and batch size, which we explore showing that one can exchange batch size and learning rate in a cyclical learning rate schedule. Next, we illustrate how noise affects memorization, showing that high noise levels lead to better generalization. Finally, we find experimentally that the similarity under simultaneous rescaling of learning rate and batch size breaks down if the learning rate gets too large or the batch size gets too small. Abstract This paper considers monochromatic radiative transfer in a diffusive three dimensional random binary mixture. The absorption coefficient, along any line-of-sight is a homogeneous Markov process, which is described by a three-dimensional Kubo-Anderson process. The transfer equation is solved numerically by Monte-Carlo simulations on a massively parallel computer (a Connection Machine) by attaching one or several photons to each processor. The implementation of the simulations on the machine is discussed in detail, in particular the association between photons and processors and the storage of the data concerning the photons and the realizations of the statistics. With a CM-2 having 8000 processors, it is possible, with an adequate strategy, to follow simultaneously millions of photons in hundreds of realizations and to reach optical thicknesses up to 100 with dispersions of order 10 -2 for the reflection and transmission coeficients. The simultations are validated, in the case of the one-dimensional rod geometry, by comparison with the exact analytical solution, constructed by averaging the solution of the non-stochastic problem (diffusion in a rod of given optical thickness) over the probability density of the optical thickness. The latter obeys a stochastic Liouville equation which is solved by a Green's function method. The influence of the dimension of the Kubo-Anderson process is studied for the case of a slab and it is shown that a slab consisting in a pile of layers (1D process) is more transparent than one which consists in a stack of lumps (3D process). A strategy for improving the efficiency of Monte-Carlo simulations, based on the distribution of the lengths of the individuals steps of the photons, is presented and discussed. We consider the Glauber dynamics for the 2D Ising model in a box of side L, at inverse temperature β and random boundary conditions τ whose distribution P either stochastically dominates the extremal plus phase (hence the quotation marks in the title) or is stochastically dominated by the extremal minus phase. A particular case is when P is concentrated on the homogeneous configuration identically equal to + (equal to −). For β large enough we show that for any ( >0 ) there exists ( c=c( , ) ) such that the corresponding mixing time T mix satisfies ( lim _ L , P (T_ mix exp ( cL^ ) ) =0 ). In the non-random case τ ≡ + (or τ ≡ −), this implies that ( T_ mix exp ( cL^ ) ). The same bound holds when the boundary conditions are all + on three sides and all − on the remaining one. The result, although still very far from the expected Lifshitz behavior T mix = O(L 2), considerably improves upon the previous known estimates of the form ( T_ mix exp ( c L^ 12 + ) ). The techniques are based on induction over length scales, combined with a judicious use of the so-called “censoring inequality” of Y. Peres and P. Winkler, which in a sense allows us to guide the dynamics to its equilibrium measure. We generalize the standard site percolation model on the @math -dimensional lattice to a model on random tessellations of @math . We prove the uniqueness of the infinite cluster by adapting the Burton-Keane argument burton1989density , develop two frameworks that imply the non-triviality of the phase transition and show that large classes of random tessellations fit into one of these frameworks. Our focus is on a very general approach that goes well beyond the typical Poisson driven models. The most interesting examples might be Voronoi tessellations induced by determinantal processes or certain classes of Gibbs processes introduced in schreiber2013 . In a second paper we will investigate first passage percolation on random tessellations. In this paper, we present a method for recognising an agent's behaviour in dynamic, noisy, uncertain domains, and across multiple levels of abstraction. We term this problem on-line plan recognition under uncertainty and view it generally as probabilistic inference on the stochastic process representing the execution of the agent's plan. Our contributions in this paper are twofold. In terms of probabilistic inference, we introduce the Abstract Hidden Markov Model (AHMM), a novel type of stochastic processes, provide its dynamic Bayesian network (DBN) structure and analyse the properties of this network. We then describe an application of the Rao-Blackwellised Particle Filter to the AHMM which allows us to construct an efficient, hybrid inference method for this model. In terms of plan recognition, we propose a novel plan recognition framework based on the AHMM as the plan execution model. The Rao-Blackwellised hybrid inference for AHMM can take advantage of the independence properties inherent in a model of plan execution, leading to an algorithm for online probabilistic plan recognition that scales well with the number of levels in the plan hierarchy. This illustrates that while stochastic models for plan execution can be complex, they exhibit special structures which, if exploited, can lead to efficient plan recognition algorithms. We demonstrate the usefulness of the AHMM framework via a behaviour recognition system in a complex spatial environment using distributed video surveillance data. We provide evidence that for some values of the parameters a simple agent-based model, describing herding behavior, yields signals with 1 f power spectral density. We derive a non-linear stochastic differential equation for the ratio of number of agents and show, that it has the form proposed earlier for modeling of 1 fnoise with different exponents �. The non- linear terms in the transition probabilities, quantifying the herding behavior, are crucial to the appearance of 1 f noise. Thus, the herding dynamics can be seen as a microscopic explanation of the proposed non-linear stochastic differential equations generatingsignals with 1 fspectrum. Wealsoconsiderthepossiblefeedbackofmacroscopicstateonmicroscopictransitionprobabilities strengthening the non-linearity of equations and providing more opportunities in themodeling of processes exhibiting power-law statistics. Copyright c �EPLA, 2011 A stochastic Forward-Backward algorithm with a constant step is studied. At each time step, this algorithm involves an independent copy of a couple of random maximal monotone operators. Defining a mean operator as a selection integral, the differential inclusion built from the sum of the two mean operators is considered. As a first result, it is shown that the interpolated process obtained from the iterates converges narrowly in the small step regime to the solution of this differential inclusion. In order to control the long term behavior of the iterates, a stability result is needed in addition. To this end, the sequence of the iterates is seen as a homogeneous Feller Markov chain whose transition kernel is parameterized by the algorithm step size. The cluster points of the Markov chains invariant measures in the small step regime are invariant for the semiflow induced by the differential inclusion. Conclusions regarding the long run behavior of the iterates for small steps are drawn. It is shown that when the sum of the mean operators is demipositive, the probabilities that the iterates are away from the set of zeros of this sum are small in Ces aro mean. The ergodic behavior of these iterates is studied as well. Applications of the proposed algorithm are considered. In particular, a detailed analysis of the random proximal gradient algorithm with constant step is performed. Preface. Part I: Extensions of Basic Models. 1. The Solidarity of Markov Renewal Processes R. Pyke. 2. A Generalization of Semi-Markov Processes M. Iosifescu. 3. Quasi-stationary Phenomena for Semi-Markov Processes M. Gyllenberg, D.S. Silvestrov. 4. Semi-Markov Random Walks V.S. Korolyuk. 5. Diffusion Approximation for Processes with Semi-Markov Switches V.V. Anisimov. 6. Approximations for Semi-Markov Single Ion Channel Models S.M. Pitts. Part II: Statistical Estimation. 7. Log-likelihood in Stochastic Processes G.G. Rousas, D. Bhattacharya. 8. Some Asymptotic Results and Exponential Approximation in Semi-Markov Models G.G. Roussas, D. Bhattacharya. 9. Markov Renewal Processes and Exponential Families V.T. Stefanov. 10. On Homogeneity of Two Semi-Markov Samples L. Afanasyeva, P. Radchenko. 11. Product-Type Estimator of Convolutions I. Gertsbakh, I. Spungin. 12. Failure Rate Estimation of Semi-Markov Systems B. Ouhbi, N. Limnios. 13. Estimation for Semi-Markov Manpower Models in a Stochastic Environment S. McClean, E. Montgomery. 14. Semi-Markov Models for Lifetime Data Analysis R. Perez-Ocon, et al Part III: Non-Homogeneous Models. 15. Continuous Time Non Homogeneous Semi-Markov Systems A.A. Papadopoulou, P.C.G. Vassiliou. 16. The Perturbed Non-Homogeneous Semi-Markov System P.C.G. Vassiliou, H. Tsakiridou. Part IV: Queueing Systems Theory. 17. Semi-Markov Queues with Heavy Tails S. Asmussen.18. MR Modelling of Poisson Traffic at Intersections Having Separate Turn Lanes R. Gideon, R. Pyke. Part V: Financial Models. 19. Stochastic Stability and Optimal Control in Insurance Mathematics A. Swishchuk. 20. Option Pricing with Semi-Markov Volatility J. Janssen, et al Part VI: Controlled Processes & Maintenance. 21. Applications of Semi-Markov Processes in Reliability and Maintenance M. Abdel-Hameed. 22. Controlled Queueing Systems with Recovery Functions T. Dohi, et al Part VII: Chromatography & Fluid Mechanics. 23. Continuous Semi-Markov Models for Chromatography B.P. Harlamov. 24. The Stress Tensor of the Closed Semi-Markov System. Energy and Entropy G.M. Tsaklidis. Index. Volatility clustering, evaluated through slowly decaying auto-correlations, Hurst effect or 1 f noise for absolute returns, is a characteristic property ofmost financial assets return time series (1999). Statistical analysis alone is not able to provide a definite answer for the presence or absence of long-range dependence phenomenon in stock returns or volatility, unless economic mechanisms are proposed to understand the origin of such phenomenon Cont (2005); (1999). Whether results of statistical analysis correspond to longrange dependence is a difficult question and subject to an ongoing statistical debate Cont (2005). Extensive empirical analysis of the financial market data, supporting the idea that the longrange volatility correlations arise from trading activity, provides valuable background for further development of the long-ranged memory stochastic models (2003); (2001). The power-law behavior of the auto-regressive conditional duration process Sato (2004) based on the random multiplicative process and it’s special case the self-modulation process Takayasu (2003), exhibiting 1 f fluctuations, supported the idea of stochastic modeling with a power-law probability density function (PDF) and long-range memory. Thus the agent based economic models Kirman & Teyssiere (2002); Lux & Marchesi (2000) as well as the stochastic models Borland (2004); (2008; 2010); Queiros (2007) exhibiting long-range dependence phenomenon in volatility or trading volume are of great interest and remain an active topic of research. Properties of stochastic multiplicative point processes have been investigated analytically and numerically and the formula for the power spectrum has been derived Gontis & Kaulakys (2004). In themore recent papers (2006); Kaulakys A Ruseckas & Kaulakys (2010) the general form of the multiplicative stochastic differential equation (SDE) was derived in agreement with the model earlier proposed in Gontis & Kaulakys (2004). Since Gontis & Kaulakys (2004) a model of trading activity, based on a SDE driven Poisson-like process, was presented (2008) and in the most recent paper (2010) we proposed a double stochastic model, whose return time series yield two power-law statistics, i.e., the PDF and the power spectral density (PSD) of absolute return, mimicking the empirical data for the one-minute trading return in the NYSE. In this chapter we present theoretical arguments and empirical evidence for the non-linear double stochastic model of return in financial markets. With empirical data from NYSE and Vilnius Stock Exchange (VSE) demonstrating universal scaling of return statistical properties, 27 We study the problem of computing maximin share allocations, a recently introduced fairness notion. Given a set of n agents and a set of goods, the maximin share of an agent is the best she can guarantee to herself, if she is allowed to partition the goods in any way she prefers, into n bundles, and then receive her least desirable bundle. The objective then is to find a partition, where each agent is guaranteed her maximin share. Such allocations do not always exist, hence we resort to approximation algorithms. Our main result is a 2 3-approximation that runs in polynomial time for any number of agents and goods. This improves upon the algorithm of Procaccia and Wang (2014), which is also a 2 3-approximation but runs in polynomial time only for a constant number of agents. To achieve this, we redesign certain parts of the algorithm in Procaccia and Wang (2014), exploiting the construction of carefully selected matchings in a bipartite graph representation of the problem. Furthermore, motivated by the apparent difficulty in establishing lower bounds, we undertake a probabilistic analysis. We prove that in randomly generated instances, maximin share allocations exist with high probability. This can be seen as a justification of previously reported experimental evidence. Finally, we provide further positive results for two special cases arising from previous works. The first is the intriguing case of three agents, where we provide an improved 7 8-approximation. The second case is when all item values belong to 0, 1, 2 , where we obtain an exact algorithm. These lecture notes highlight the mathematical and computational structure relating to the formulation of, and development of algorithms for, the Bayesian approach to inverse problems in differential equations. This approach is fundamental in the quantification of uncertainty within applications in volving the blending of mathematical models with data. The finite dimensional situation is described first, along with some motivational examples. Then the development of probability measures on separable Banach space is undertaken, using a random series over an infinite set of functions to construct draws; these probability measures are used as priors in the Bayesian approach to inverse problems. Regularity of draws from the priors is studied in the natural Sobolev or Besov spaces implied by the choice of functions in the random series construction, and the Kolmogorov continuity theorem is used to extend regularity considerations to the space of Holder continuous functions. Bayes’ theorem is de rived in this prior setting, and here interpreted as finding conditions under which the posterior is absolutely continuous with respect to the prior, and determining a formula for the Radon-Nikodym derivative in terms of the likelihood of the data. Having established the form of the posterior, we then describe various properties common to it in the infinite dimensional setting. These properties include well-posedness, approximation theory, and the existence of maximum a posteriori estimators. We then describe measure-preserving dynamics, again on the infinite dimensional space, including Markov chain-Monte C arlo and sequential Monte Carlo methods, and measure-preserving reversible stochastic differential equations. By formulating the theory and algorithms on the underlying infinite dimensional space, we obtain a framework suitable for rigorous analysis of the accuracy of reconstructions, of computational complexity, as well as naturally constructing algorithms which perform well under mesh refinement, since they are inherently well-defined in infinite dimensions. By symmetry, P has eigenvalues 1 = I03 > I381 > ?> I 31xI- 1 2 -1. This paper develops methods for getting upper and lower bounds on 8i3 by comparison with a second reversible chain on the same state space. This extends the ideas introduced in Diaconis and Saloff-Coste (1993), where random walks on finite groups were considered. The bounds involve geometric properties such as the diameter and covering number of an associated graph along the lines of Diaconis and Stroock (1991). The main application gives a sharp upper bound on the second eigenvalue of the symmetric exclusion process. Thus, let S0 be a connected undirected graph with n vertices. For simplicity, we assume in this introduction that SW is regular. To start, r unlabelled particles are placed in an initial configuration, 1 < r < n. At each step, a particle is chosen at random; then one of the neighboring sites of this particle is chosen at random. If the neighboring site is unoccupied, the chosen particle is moved there; if the neighboring site is occupied, the system stays as it was. This is a reversible Markov chain on the r-sets of 1, 2, . . ., n with uniform stationary distribution. Liggett (1985) gives background and motivation (he focuses on infinite systems). Fill (1991) gives bounds on the second eigenvalue of the labeled exclusion process on the finite circle ZZn 1 We study this chain by comparison with a second Markov chain on r-sets that proceeds by picking a particle at random, picking an unoccupied site at random (not necessarily a neighboring site) and moving the particle to the unoccupied site. This is a well studied chain (the Bernoulli-Laplace model for diffusion). Its eigenvalues are known. We show that the comparison techniques apply to give upper bounds on the eigenvalues of the exclusion .We consider a class of stochastic processing networks. Assume that the networks satisfy a complete resource pooling condition. We prove that each maximum pressure policy asymptotically minimizes the workload process in a stochastic processing network in heavy traffic. We also show that, under each quadratic holding cost structure, there is a maximum pressure policy that asymptotically minimizes the holding cost. A key to the optimality proofs is to prove a state space collapse result and a heavy traffic limit theorem for the network processes under a maximum pressure policy. We extend a framework of Bramson [Queueing Systems Theory Appl. 30 (1998) 89–148] and Williams [Queueing Systems Theory Appl. 30 (1998b) 5–25] from the multiclass queueing network setting to the stochastic processing network setting to prove the state space collapse result and the heavy traffic limit theorem. The extension can be adapted to other studies of stochastic processing networks. 1. Introduction. This paper is a continuation of Dai and Lin (2005), in which maximum pressure policies are shown to be throughput optimal for a class of stochastic processing networks. Throughput optimality is an important, first-order objective for many networks, but it ignores some key secondary performance measures like queueing delays experienced by jobs in these networks. In this paper we show that maximum pressure policies enjoy additional optimality properties; they are asymptotically optimal in minimizing a certain workload or holding cost of a stochastic processing network. Stochastic processing networks have been introduced in a series of three papers by Harrison (2000, 2002, 2003). In Dai and Lin (2005) and this paper we consider a special class of Harrison’s model. This class of stochastic processing networks is much more general than multiclass queueing networks that have been a subject of intensive study in the last 20 years; see, for example, Harrison (1988), Williams Agent-based modeling of zapping behavior of viewers, television commercial allocation, and advertisement markets by Hiroyuki Kyan and Jun-ichi Inoue.- Agent based modeling of Housing asset bubble: A simple utility function based investigation by Kausik Gangopadhyay and Kousik Guhathakurta.- Urn model-based adaptive Multi-arm clinical trials: A stochastic approximation approach by Sophie Laruelle and Gilles Pages.- Logistic modeling of a Religious Sect features by Marcel Ausloos.- Characterizing financial crisis by means of the three states random field Ising model by Mitsuaki Murota and Jun-ichi Inoue.- Themes and applications of kinetic exchange models: Redux by Asim Ghosh, Anindya S. 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