a polynomial time approximation algorithm for the permanent of a matrix with nonnegative entries we present a polynomial time randomized algorithm for estimating the permanent of an arbitrary n n matrix with nonnegative entries this algorithm technically a fully polynomial randomized approximation scheme computes an approximation that is with high probability within arbitrarily small specified relative error of the true value of the permanent

on the approximations of solutions to neutral sdes with markovian switching and jumps under non lipschitz conditions abstract in this paper we investigate the existence and uniqueness of solutions to neutral stochastic differential equations with markovian switching and jumps nsdewmsjs under non lipschitz conditions on the other hand we present the euler approximate solutions for nsdewmsjs and show that the convergence of the euler approximate solutions to the true solutions by applying it o formula bihari s lemma and burkholder davis gundy s lemma some examples are provided to illustrate the main results

asymptotic behavior and distributional limits of preferential attachment graphs we give an explicit construction of the weak local limit of a class of preferential attachment graphs this limit contains all local information and allows several computations that are otherwise hard for example joint degree distributions and more generally the limiting distribution of subgraphs in balls of any given radius k around a random vertex in the preferential attachment graph we also establish the finite volume corrections which give the approach to the limit ams 2000 subject classifications 60c05 60k99

combinatorial approach to the interpolation method and scaling limits in sparse random graphs we establish the existence of free energy limits for several combinatorial models on erdos renyi graph g n cn and random r regular graph g n r for a variety of models including independent sets max cut coloring and k sat we prove that the free energy both at a positive and zero temperature appropriately rescaled converges to a limit as the size of the underlying graph diverges to infinity in the zero temperature case this is interpreted as the existence of the scaling limit for the corresponding combinatorial optimization problem for example as a special case we prove that the size of a largest independent set in these graphs normalized by the number of nodes converges to a limit w h p this resolves an open problem which was proposed by aldous some open problems as one of his six favorite open problems it was also mentioned as an open problem in several other places conjecture 2 20 in wormald in surveys in combinatorics 1999 canterbury 1999 239 298 cambridge univ press bollobas and riordan random structures algorithms 39 2011 1 38 janson and thomason combin probab comput 17 2008 259 264 and aldous and steele in probability on discrete structures 2004 1 72 springer our approach is based on extending and simplifying the interpolation method of guerra and toninelli comm math phys 230 2002 71 79 and franz and leone j stat phys 111 2003 535 564 among other applications this method was used to prove the existence of free energy limits for viana bray and k sat models on erdos renyi graphs the case of zero temperature was treated by taking limits of positive temperature models we provide instead a simpler combinatorial approach and work with the zero temperature case optimization directly both in the case of erdos renyi graph g n cn and random regular graph g n r in addition we establish the large deviations principle for the satisfiability property of the constraint satisfaction problems coloring k sat and nae k sat for the g n cn random graph model

approximating the minimum spanning tree weight in sublinear time we present a probabilistic algorithm that given a connected graph g represented by adjacency lists of average degree d with edge weights in the set 1 w and given a parameter 0 eps 1 2 estimates in time o dw varepsilon 2 log frac dw varepsilon the weight of the minimum spanning tree mst of g with a relative error of at most eps note that the running time does not depend on the number of vertices in g we also prove a nearly matching lower bound of omega dw varepsilon 2 on the probe and time complexity of any approximation algorithm for mst weight the essential component of our algorithm is a procedure for estimating in time o d eps 2 log frac d varepsilon the number of connected components of an unweighted graph to within an additive error of varepsilon n this becomes o eps 2 log frac 1 varepsilon for d o 1 the time bound is shown to be tight up to within the log frac d varepsilon factor our connected components algorithm picks o 1 varepsilon 2 vertices in the graph and then grows local spanning trees whose sizes are specified by a stochastic process from the local information collected in this way the algorithm is able to infer with high confidence an estimate of the number of connected components we then show how estimates on the number of components in various subgraphs of g can be used to estimate the weight of its mst

spectrum of large random reversible markov chains heavy tailed weights on the complete graph we consider the random reversible markov kernel k obtained by assigning i i d non negative weights to the edges of the complete graph over n vertices and normalizing by the corresponding row sum the weights are assumed to be in the domain of attraction of an alpha stable law with alpha in 0 2 when 1 alpha 2 we show that for a suitable regularly varying sequence kappa n of index 1 1 alpha the limiting spectral distribution mu alpha of kappa n k coincides with the one of the random symmetric matrix of the un normalized weights levy matrix with i i d entries in contrast when 0 alpha 1 we show that the empirical spectral distribution of k converges without rescaling to a non trivial law wmu alpha supported on 1 1 whose moments are the return probabilities of the random walk on the poisson weighted infinite tree pwit introduced by aldous the limiting spectral distributions are given by the expected value of the random spectral measure at the root of suitable self adjoint operators defined on the pwit this characterization is used together with recursive relations on the tree to derive some properties of mu alpha and wmu alpha we also study the limiting behavior of the invariant probability measure of k

mean square stability properties of an adaptive time stepping sde solver we consider stability properties of a class of adaptive time stepping schemes based upon the milstein method for stochastic differential equations with a single scalar forcing in particular we focus upon mean square stability for a class of linear test problems with multiplicative noise we demonstrate that desirable stability properties can be induced in the numerical solution by the use of two realistic local error controls one for the drift term and one for the diffusion

group testing with random pools phase transitions and optimal strategy the problem of group testing is to identify defective items out of a set of objects by means of pool queries of the form does the pool contain at least a defective the aim is of course to perform detection with the fewest possible queries a problem which has relevant practical applications in different fields including molecular biology and computer science here we study gt in the probabilistic setting focusing on the regime of small defective probability and large number of objects p 0 and n we construct and analyze one stage algorithms for which we establish the occurrence of a non detection detection phase transition resulting in a sharp threshold overline m for the number of tests by optimizing the pool design we construct algorithms whose detection threshold follows the optimal scaling overline m propto np log p then we consider two stages algorithms and analyze their performance for different choices of the first stage pools in particular via a proper random choice of the pools we construct algorithms which attain the optimal value previously determined in mezard and toninelli arxiv 0706 3104 for the mean number of tests required for complete detection we finally discuss the optimal pool design in the case of finite p

replica bounds for diluted non poissonian spin systems in this paper we extend replica bounds and free energy subadditivity arguments to diluted spin glass models on graphs with arbitrary non poissonian degree distribution the new difficulties specific of this case are overcome introducing an interpolation procedure that stresses the relation between interpolation methods and the cavity method as a byproduct we obtain self averaging identities that generalize the ghirlanda guerra ones to the multi overlap case

sharp metastability threshold for two dimensional bootstrap percolation in the bootstrap percolation model sites in an l by l square are initially independently declared active with probability p at each time step an inactive site becomes active if at least two of its four neighbours are active we study the behaviour as p 0 and l simultaneously of the probability i l p that the entire square is eventually active we prove that i l p 1 if and i l p 0 if where 2 18 we prove the same behaviour with the same threshold for the probability j l p that a site is active by time l in the process on the infinite lattice the same results hold for the so called modified bootstrap percolation model but with threshold 2 6 the existence of the thresholds settles a conjecture of aizenman and lebowitz 3 while the determination of their values corrects numerical predictions of adler stauffer and aharony 2

pairs of sat assignments in random boolean formulae we investigate geometrical properties of the random k satisfiability problem using the notion of x satisfiability a formula is x satisfiable is there exist two sat assignments differing in nx variables we show the existence of a sharp threshold for this property as a function of the clause density for large enough k we prove that there exists a region of clause density below the satisfiability threshold where the landscape of hamming distances between sat assignments experiences a gap pairs of sat assignments exist at small x and around x 12 but they do not exist at intermediate values of x this result is consistent with the clustering scenario which is at the heart of the recent heuristic analysis of satisfiability using statistical physics analysis the cavity method and its algorithmic counterpart the survey propagation algorithm our method uses elementary probabilistic arguments first and second moment methods and might be useful in other problems of computational and physical interest where similar phenomena appear

on pseudo homogeneous triangular norms triangular conorms and proper uninorms in this paper quasi homogeneous t norms are generalized and then pseudo homogeneous t norms are introduced and characterized it is shown that only some particular strict t norms and the minimum t norm t m are pseudo homogeneous the concepts of pseudo homogeneous t conorms and proper uninorms are given as well different from t norms only the maximum t conorm s m is pseudo homogeneous and any proper uninorm is not pseudo homogeneous

shift in the velocity of a front due to a cutoff we consider the effect of a small cutoff ensuremath varepsilon on the velocity of a traveling wave in one dimension simulations done over more than ten orders of magnitude as well as a simple theoretical argument indicate that the effect of the cutoff ensuremath varepsilon is to select a single velocity that converges when ensuremath varepsilon ensuremath rightarrow 0 to the one predicted by the marginal stability argument for small ensuremath varepsilon the shift in velocity has the form k mathrm ln ensuremath varepsilon ensuremath 2 and our prediction for the constant k agrees very well with the results of our simulations a very similar logarithmic shift appears in more complicated situations in particular in finite size effects of some microscopic stochastic systems our theoretical approach can also be extended to give a simple way of deriving the shift in position due to initial conditions in the fisher kolmogorov or similar equations

controllability matching ratio and graph convergence there is an important parameter in control theory which is closely related to the directed matching ratio of the network as shown in the paper of liu et al nature 473 167 173 2011 we give proofs of two main statements of liu et al 2011 on the directed matching ratio which were based on numerical results and heuristics from statistical physics first we show that the directed matching ratio of directed random networks given by a fix sequence of degrees is concentrated around its mean we also examine the convergence of the directed matching ratio of a random directed graph sequence that converges in the local weak sense and generalize the result of elek and lippner proc am math soc 138 8 2939 2947 2010 we prove that the mean of the directed matching ratio converges to the properly defined matching ratio parameter of the limiting graph we further show the almost sure convergence of the matching ratios for the most widely used families of scale free networks which was the main motivation of liu et al 2011

the αth moment stability for the stochastic pantograph equation in this paper we investigate the ath moment asymptotical stability of the analytic solution and the numerical methods for the stochastic pantograph equation by using the razumikhin technique especially the linear stochastic pantograph equations and the semi implicit euler method applying them are considered the convergence result of the semi implicit euler method is obtained the stability conditions of the analytic solution of those equations and the numerical method are given finally some experiments are given

disentangling giant component and finite cluster contributions in sparse random matrix spectra we describe a method for disentangling giant component and finite cluster contributions to sparse random matrix spectra using sparse symmetric random matrices defined on erd ifmmode mbox h o else h o fi s r enyi graphs as an example and test bed our methods apply to sparse matrices defined in terms of arbitrary graphs in the configuration model class as long as they have finite mean degree

recovery thresholds in the sparse planted matching problem we consider the statistical inference problem of recovering an unknown perfect matching hidden in a weighted random graph by exploiting the information arising from the use of two different distributions for the weights on the edges inside and outside the planted matching a recent work has demonstrated the existence of a phase transition in the large size limit between a full and a partial recovery phase for a specific form of the weights distribution on fully connected graphs we generalize and extend this result in two directions we obtain a criterion for the location of the phase transition for generic weights distributions and possibly sparse graphs exploiting a technical connection with branching random walk processes as well as a quantitatively more precise description of the critical regime around the phase transition

spin glass model for the c dismantling problem c dismantling cd problem aims at finding the minimum vertex set d of a graph g v e after removing which the remaining graph will break into connected components with the size not larger than c in this paper we introduce a spin glass model with c 1 integer value states into the cd problem and then study the properties of this spin glass model by the belief propagation bp equations under the replica symmetry ansatz we give the lower bound rho c of the relative size of d with finite c on regular random graphs and erdos renyi random graphs we find rho c will decrease gradually with growing c and it converges to rho infty as c to infty the cd problem is called dismantling problem when c is a small finite fraction of v therefore rho infty is also the lower bound of the dismantling problem when v to infty to reduce the computation complexity of the bp equations taking the knowledge of the probability of a random selected vertex belonging to a remaining connected component with the size a the original bp equations can be simplified to one with only three states when c to infty the simplified bp equations are very similar to the bp equations of the feedback vertex set spin glass model h j zhou eur phys j b 86 455 2013 at last we develop two practical belief propagation guide decimation algorithms based on the original bp equations cd bpd and the simplified bp equations scd bpd to solve the cd problem on a certain graph our bpd algorithms and two other state of art heuristic algorithms are applied on various random graphs and some real world networks computation results show that the cd bpd is the best in all tested algorithms in the case of small c but considering the performance and computation consumption we recommend using scd bpd for the network with small clustering coefficient when c is large

stability analysis of numerical schemes for stochastic differential equations stochastic differential equations sdes represent physical phenomena dominated by stochastic processes as for deterministic ordinary differential equations odes various numerical schemes are proposed for sdes in this paper we study the stability of numerical schemes for scalar sdes with respect to the mean square norm which we call ms stability we will show some figures of the ms stability domain or regions for some numerical schemes and present numerical results which confirm it this notion is an extension of absolute stability in numerical methods for odes

preserving exponential mean square stability in the simulation of hybrid stochastic differential equations positive results are derived concerning the long time dynamics of fixed step size numerical simulations of stochastic differential equation systems with markovian switching euler maruyama and implicit theta method discretisations are shown to capture exponential mean square stability for all sufficiently small time steps under appropriate conditions moreover the decay rate as measured by the second moment lyapunov exponent can be reproduced arbitrarily accurately new finite time convergence results are derived as an intermediate step in this analysis we also show however that the mean square a stability of the theta method does not carry through to this switching scenario the proof techniques are quite general and hence have the potential to be applied to other numerical methods

the mathematics of infectious diseases many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases threshold theorems involving the basic reproduction number r 0 the contact number sigma and the replacement number r are reviewed for the classic sir epidemic and endemic models similar results with new expressions for r 0 are obtained for mseir and seir endemic models with either continuous age or age groups values of r 0 and sigma are estimated for various diseases including measles in niger and pertussis in the united states previous models with age structure heterogeneity and spatial structure are surveyed

identifying optimal targets of network attack by belief propagation for a network formed by nodes and undirected links between pairs of nodes the network optimal attack problem aims at deleting a minimum number of target nodes to break the network down into many small components this problem is intrinsically related to the feedback vertex set problem that was successfully tackled by spin glass theory and an associated belief propagation guided decimation bpd algorithm zhou eur phys j b 86 455 2013 in the present work we apply the bpd algorithm which has approximately linear time complexity to the network optimal attack problem and demonstrate that it has much better performance than a recently proposed collective information algorithm morone and makse nature 524 65 2015 for different types of random networks and real world network instances the bpd guided attack scheme often induces an abrupt collapse of the whole network which may make it very difficult to defend

metastable behavior for bootstrap percolation on regular trees we examine bootstrap percolation on a regular b 1 ary tree with initial law given by bernoulli p the sites are updated according to the usual rule a vacant site becomes occupied if it has at least occupied neighbors occupied sites remain occupied forever it is known that when b 2 the limiting density q q p of occupied sites exhibits a jump at some p t p t b 0 1 from q t q p t p t we investigate the metastable behavior associated with this transition explicitly we pick p p t h with h 0 and show that as h 0 the system lingers around the critical state for time order h 1 2 and then passes to fully occupied state in time o 1 the law of the entire configuration observed when the occupation density is q q t 1 converges as h 0 to a well defined measure

t stability of the semi implicit euler method for delay differential equations with multiplicative noise the paper deals with the t stability of the semi implicit euler method for delay differential equations with multiplicative noise a difference equation is obtained by applying the numerical method to a linear test equation in which the wiener increment is approximated by a discrete random variable with two point distribution the conditions under which the method is t stable are considered and the numerical experiments are given

network dismantling abstract we study the network dismantling problem which consists of determining a minimal set of vertices in which removal leaves the network broken into connected components of subextensive size for a large class of random graphs this problem is tightly connected to the decycling problem the removal of vertices leaving the graph acyclic exploiting this connection and recent works on epidemic spreading we present precise predictions for the minimal size of a dismantling set in a large random graph with a prescribed light tailed degree distribution building on the statistical mechanics perspective we propose a three stage min sum algorithm for efficiently dismantling networks including heavy tailed ones for which the dismantling and decycling problems are not equivalent we also provide additional insights into the dismantling problem concluding that it is an intrinsically collective problem and that optimal dismantling sets cannot be viewed as a collection of individually well performing nodes

numerical integration of multiplicative noise stochastic differential equations a two stage runge kutta algorithm for vector ito and by transform also stratonovich stochastic differential equations with multiplicative noise has been developed the method is second order accurate but for vanishing drift the algorithm yields a martingale independent of step size several examples are included to illustrate our method a discussion of errors shows that sample size can be as important as truncation

the complexity of counting in sparse regular and planar graphs we show that a number of graph theoretic counting problems remain cal np hard indeed cal p complete in very restricted classes of graphs in particular we prove that the problems of counting matchings vertex covers independent sets and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree we obtain corollaries about counting cliques in restricted classes of graphs and counting satisfying assignments to restricted classes of monotone 2 cnf formulae to achieve these results a new interpolation based reduction technique which preserves properties such as constant degree is introduced

approximating the minimum vertex cover in sublinear time and a connection to distributed algorithms for a given graph g over n vertices let opt g denote the size of an optimal solution in g of a particular minimization problem e g the size of a minimum vertex cover a randomized algorithm will be called an a approximation algorithm with an additive error for this minimization problem if for any given additive error parameter e 0 it computes a value opt such that with probability at least 2 3 it holds that opt g opt a opt g en assume that the maximum degree or average degree of g is bounded in this case we show a reduction from local distributed approximation algorithms for the vertex cover problem to sublinear approximation algorithms for this problem this reduction can be modified easily and applied to other optimization problems that have local distributed approximation algorithms such as the dominating set problem we also show that for the minimum vertex cover problem the query complexity of such approximation algorithms must grow at least linearly with the average degree d of the graph this lower bound holds for every multiplicative factor a and small constant e as long as d o n a in particular this means that for dense graphs it is not possible to design an algorithm whose complexity is o n

random generation of combinatorial structures from a uniform distribution abstract the class of problems involving the random generation of combinatorial structures from a uniform distribution is considered uniform generation problems are in computational difficulty intermediate between classical existence and counting problems it is shown that exactly uniform generation of efficiently verifiable combinatorial structures is reducible to approximate counting and hence is within the third level of the polynomial hierarchy natural combinatorial problems are presented which exhibit complexity gaps between their existence and generation and between their generation and counting versions it is further shown that for self reducible problems almost uniform generation and randomized approximate counting are inter reducible and hence of similar complexity

numerical treatment of stochastic differential equations we define general runge kutta approximations for the solution of stochastic differential equations sde these approximations are proved to converge in quadratic mean to the solution of an sde with a corrected drift the explicit form of the correction term is given concerning the order of convergence we show that in general it is impossible for the quadratic mean of the one step error to be of an order greater than o h 3 this order is attained e g by the stochastic analogue of heun s method in the n dimensional case the highest order of convergence is in general only o h 2 attained by euler s method the order o h 3 can only be reached if nabla x sigma r sigma s nabla x sigma s sigma r for the diffusion matrix

geometric classifications of homogeneous production functions in this paper we completely classify homogeneous production functions with an arbitrary number of inputs whose production hypersurfaces are flat as an immediate consequence we obtain a complete classification of homogeneous production functions with two inputs whose production surfaces are developable

random k satisfiability problem from an analytic solution to an efficient algorithm we study the problem of satisfiability of randomly chosen clauses each with k boolean variables using the cavity method at zero temperature we find the phase diagram for the k 3 case we show the existence of an intermediate phase in the satisfiable region where the proliferation of metastable states is at the origin of the slowdown of search algorithms the fundamental order parameter introduced in the cavity method which consists of surveys of local magnetic fields in the various possible states of the system can be computed for one given sample these surveys can be used to invent new types of algorithms for solving hard combinatorial optimizations problems one such algorithm is shown here for the k 3 satisfiability problem with very good performances

exponential stability in p th mean of solutions and of convergent euler type solutions of stochastic delay differential equations one concept of the stability of a solution of an evolutionary equation relates to the sensitivity of the solution to perturbations in the initial data there are other stability concepts notably those concerned with persistent perturbations results are presented on the stability in p th mean of solutions of stochastic delay differential equations with multiplicative noise and of stochastic delay difference equations the difference equations are of a type found in numerical analysis and we employ our results to obtain mean square stability criteria for the solution of the euler maruyama discretization of stochastic delay differential equations the analysis proceeds as follows we show that an inequality of halanay type derivable via comparison theory can be employed to derive conditions for p th mean stability of a solution we then produce a discrete analogue of the halanay type theory that permits us to develop a p th mean stability analysis of analogous stochastic difference equations the application of the theoretical results is illustrated by deriving mean square stability conditions for solutions and numerical solutions of a constant coefficient linear test equation

on some geometric properties of quasi product production models abstract a production function f is called quasi sum if there are continuous strict monotone functions f h 1 h n with f 0 such that f x f h 1 x 1 h n x n cf aczel and maksa 1996 1 a quasi sum production function is called quasi linear if at most one of f h 1 h n is a nonlinear function for a production function f the graph of f is called the production hypersurface of f in this paper we obtain a very simple necessary and sufficient condition for a quasi sum production function f to be quasi linear in terms of graph of f moreover we completely classify quasi sum production functions whose production hypersurfaces have vanishing gauss kronecker curvature

rumors in a network who s the culprit we provide a systematic study of the problem of finding the source of a rumor in a network we model rumor spreading in a network with the popular susceptible infected si model and then construct an estimator for the rumor source this estimator is based upon a novel topological quantity which we term rumor centrality we establish that this is a maximum likelihood ml estimator for a class of graphs we find the following surprising threshold phenomenon on trees which grow faster than a line the estimator always has nontrivial detection probability whereas on trees that grow like a line the detection probability will go to 0 as the network grows simulations performed on synthetic networks such as the popular small world and scale free networks and on real networks such as an internet as network and the u s electric power grid network show that the estimator either finds the source exactly or within a few hops of the true source across different network topologies we compare rumor centrality to another common network centrality notion known as distance centrality we prove that on trees the rumor center and distance center are equivalent but on general networks they may differ indeed simulations show that rumor centrality outperforms distance centrality in finding rumor sources in networks which are not tree like

on the phase transition in random simplicial complexes it is well known that the g n p model of random graphs undergoes a dramatic change around p 1 it is here that the random graph almost surely contains cycles and here it first acquires a giant i e order n connected component several years ago linial and meshulam have introduced theyd n p model a probability space of n vertex d dimensional simplicial complexes where y1 n p coincides with g n p within this model we prove a naturald dimensional analog of these graph theoretic phenomena specifically we determine the exact threshold for the nonvanishing of the real d th homology of complexes from yd n p we also compute the real betti numbers ofyd n p forp c n finally we establish the emergence of giant shadow at this threshold for d 1 a giant shadow and a giant component are equivalent unlike the case for graphs for d 2 the emergence of the giant shadow is a first order phase transition

higher order implicit strong numerical schemes for stochastic differential equations higher order implicit numerical methods which are suitable for stiff stochastic differential equations are proposed these are based on a stochastic taylor expansion and converge strongly to the corresponding solution of the stochastic differential equation as the time step size converges to zero the regions of absolute stability of these implicit and related explicit methods are also examined

the composite euler method for stiff stochastic differential equations in this paper we present the composite euler method for the strong solution of stochastic differential equations driven by d dimensional wiener processes this method is a combination of the semi implicit euler method and the implicit euler method at each step either the semi implicit euler method or the implicit euler method is used in order to obtain better stability properties we give criteria for selecting the semi implicit euler method or the implicit euler method for the linear test equation the convergence properties of the composite euler method depend on the criteria for selecting the methods numerical results suggest that the convergence properties of the composite euler method applied to nonlinear sdes is the same as those applied to linear equations the stability properties of the composite euler method are shown to be far superior to those of the euler methods and numerical results show that the composite euler method is a very promising method

on some geometric properties of the generalized ces production functions abstract in this paper we prove that the generalized ces production function has constant return to scale if and only if the corresponding hypersurface is developable moreover we establish that this production function has decreasing increasing return to scale if and only if the corresponding hypersurface has positive negative gaussian curvature these results are a generalization of some recent results concerning the generalized cobb douglas production functions g e vilcu a geometric perspective on the generalized cobb douglas production functions appl math lett 24 2011 777 783

matchings on infinite graphs elek and lippner proc am math soc 138 8 2939 2947 2010 showed that the convergence of a sequence of bounded degree graphs implies the existence of a limit for the proportion of vertices covered by a maximum matching we provide a characterization of the limiting parameter via a local recursion defined directly on the limit of the graph sequence interestingly the recursion may admit multiple solutions implying non trivial long range dependencies between the covered vertices we overcome this lack of correlation decay by introducing a perturbative parameter temperature which we let progressively go to zero this allows us to uniquely identify the correct solution in the important case where the graph limit is a unimodular galton watson tree the recursion simplifies into a distributional equation that can be solved explicitly leading to a new asymptotic formula that considerably extends the well known one by karp and sipser for erd s renyi random graphs

convergence and stability of the split step θ method for stochastic differential equations in this paper we construct a new split step method for solving stochastic differential equations namely the split step q method under lipschitz and linear growth conditions we establish a mean square convergence theory of split step q approximate solutions moreover the mean square stability of the method for a linear test equation with real parameters is considered and the real mean square stability region is plotted finally numerical results are presented to demonstrate the efficiency of the split step q method

the parisi formula in this chapter we prove the parisi formula which gives the limiting value of the free energy per site for the sherrington kirkpatrick model at each temperature starting with the famous result of guerra that the parisi formula provided a bound for this free energy we make full use of poisson dirichlet cascades we then use the same techniques to prove in a strong sense that in presence of an external field the overlaps are positive

metastability effects in bootstrap percolation bootstrap percolation models or equivalently certain types of cellular automata exhibit interesting finite volume effects these are studied at a rigorous level the authors find that for an initial configuration obtained by placing particles independently with probability p or 2 the density of the bootstrapped final configurations in the sequence of cubes l 2 l 2 d typically undergoes an abrupt transition as l is increased from being close to 0 to the value 1 with l fixed at a large value the mean final density as a function of p changes from 0 to 1 around a value which varies only slowly with l the pertinent parameter being lambda p1 d 1 ln l the driving mechanism is the capture of a critical droplet this behaviour is analogous to the decay of a metastable state near a first order phase transition for which the analysis offers some suggestive ideas

bounds for diluted mean fields spin glass models in an important recent paper 2 s franz and m leone prove rigorous lower bounds for the free energy of the diluted p spin model and the k sat model at any temperature we show that the results for these two models are consequences of a single general principle our calculations are significantly simpler than those of 2 even in the replica symmetric case

extremal paths on a random cayley tree we investigate the statistics of extremal path s both the shortest and the longest from the root to the bottom of a cayley tree the lengths of the edges are assumed to be independent identically distributed random variables drawn from a distribution rho l besides the number of branches from any node is also random exact results are derived for arbitrary distribution rho l in particular for the binary 0 1 distribution rho l pdelta l 1 1 p delta l 0 we show that as p increases the minimal length undergoes an unbinding transition from a localized phase to a moving phase at the critical value p p c 1 b 1 where b is the average branch number of the tree as the height n of the tree increases the minimal length saturates to a finite constant in the localized phase p p c where the velocity v min p is determined via a front selection mechanism at p p c the minimal length grows with n in an extremely slow double logarithmic fashion the length of the maximal path on the other hand increases linearly as v max p n for all p the maximal and minimal velocities satisfy a general duality relation v min p v max 1 p 1 which is also valid for directed paths on finite dimensional lattices

mean square stability and dissipativity of two classes of theta methods for systems of stochastic delay differential equations in this paper we first study the mean square stability of numerical methods for stochastic delay differential equations under a coupled condition on the drift and diffusion coefficients this condition admits that the diffusion coefficient can be highly nonlinear i e it does not necessarily satisfy a linear growth or global lipschitz condition it is proved that for all positive stepsizes the classical stochastic theta method with q 0 5 is asymptotically mean square stable and the split step theta method with q 0 5 is exponentially mean square stable conditional stability results for the methods with q 0 5 and prove that the method possesses a bounded absorbing set in mean square independent of initial data

an analysis of stability of milstein method for stochastic differential equations with delay this paper deals with the adapted milstein method for solving linear stochastic delay differential equations it is proved that the numerical method is mean square ms stable under suitable conditions the obtained result shows that the method preserves the stability property of a class of linear constant coefficient problems this is also verified by several numerical examples

typology of phase transitions in bayesian inference problems many inference problems undergo phase transitions as a function of the signal to noise ratio a prominent example of this phenomenon being found in the stochastic block model sbm that generates a random graph with a hidden community structure some of these phase transitions affect the information theoretic optimal but possibly computationally expensive estimation procedure others concern the behavior of efficient iterative algorithms a computational gap opens when the phase transitions for these two aspects do not coincide leading to a hard phase in which optimal inference is computationally challenging in this paper we refine this description in two ways from a qualitative perspective we emphasize the existence of more generic phase diagrams with a hybrid hard phase in which it is computationally easy to reach a nontrivial inference accuracy but computationally hard to match the information theoretic optimal one we support this discussion by quantitative expansions of the functional cavity equations that describe inference problems on sparse graphs around their trivial solution these expansions shed light on the existence of hybrid hard phases for a large class of planted constraint satisfaction problems and on the question of the tightness of the kesten stigum ks bound for the associated tree reconstruction problem our results show that the instability of the trivial fixed point is not generic evidence for the bayes optimality of the message passing algorithms we clarify in particular the status of the symmetric sbm with four communities and of the tree reconstruction of the associated potts model in the assortative ferromagnetic case the ks bound is always tight whereas in the disassortative antiferromagnetic case we exhibit an explicit criterion involving the degree distribution that separates a large degree regime where the ks bound is tight and a low degree regime where it is not we also investigate the sbm with two communities of different sizes also known as the asymmetric ising model and describe quantitatively its computational gap as a function of its asymmetry as well as a version of the sbm with two groups of q 1 and q 2 communities we complement this study with numerical simulations of the belief propagation iterative algorithm confirming that its behavior on large samples is well described by the cavity method

asymptotically efficient runge kutta methods for a class of itoˆ and stratonovich equations in certain applications of stochastic differential equations approximate solutions must be found that depend only on samples of the driving process it is known that the order of convergence of such approximations is limited and that some are asymptotically efficient in the sense that they minimize the leading coefficient in the expansion of mean square errors as power series in the sample step size this article develops asymptotically efficient runge kt tta methods that involve evaluations either of the coefficients of an ito equation or of the coefficients of the corresponding stratonovich equation simpler approximations which converge with the maximum possible order but which are not asymptotically efficient are also defined the runge kutta methods for ito equations differ from those designed for ordinary differential equations in that they involve terms in the square root of the sample step size the approximations are tested along with the classical euler method on five examples the simulatio

numerical approximations to the stationary solutions of stochastic differential equations in this paper we investigate the possibility of approximating the stationary solution of a stochastic differential equation sde we start with the random dynamical system generated by the sde with the multiplicative noise we prove that the pullback flow has a stationary point however the stationary point is not constructible explicitly therefore we look at the numerical approximation we prove that the discrete time random dynamical system also has a stationary point finally we prove mean square convergence of the approximate stationary solution to the exact stationary solution as the time step diminishes as well as almost surely convergence when the time step is rational

combinatorial model and bounds for target set selection the adoption of everyday decisions in public affairs fashion movie going and consumer behavior is now thoroughly believed to migrate in a population through an influential network the same diffusion process when being imitated by intention is called viral marketing this process can be modeled by a directed graph g v e with a threshold t v for every vertex v v where v becomes active once at least t v of its in neighbors are already active a perfect target set is a set of vertices whose activation will eventually activate the entire graph and the perfect target set selection problem ptss asks for the minimum such initial set it is known chen 2008 6 that ptss is hard to approximate even for some special cases such as bounded degree graphs or majority thresholds we propose a combinatorial model for this dynamic activation process and use it to represent ptss and its variants by linear integer programs this allows one to use standard integer programming solvers for solving small size ptss instances we also show combinatorial lower and upper bounds on the size of the minimum perfect target set our upper bound implies that there are always perfect target sets of size at most v 2 and 2 v 3 under majority and strict majority thresholds respectively both in directed and undirected graphs this improves the bounds of 0 727 v and 0 7732 v found recently by chang and lyuu 2010 5 for majority and strict majority thresholds in directed graphs and matches their bound under majority thresholds in undirected graphs furthermore our proof is much simpler and we observe that some of these bounds are tight one interesting and perhaps surprising implication of our lower bound for undirected graphs is that it is easy to get a constant factor approximation for ptss for relatively balanced graphs e g bounded degree graphs nearly regular graphs with a more than majority threshold that is t v deg v for every v v and some constant 1 2 whereas no polylogarithmic approximation exists for more than majority graphs

spin glass phase transitions and minimum energy of the random feedback vertex set problem a feedback vertex set fvs of an undirected graph contains vertices from every cycle of this graph constructing a fvs of sufficiently small cardinality is very difficult in the worst cases but for random graphs this problem can be efficiently solved by converting it into an appropriate spin glass model h j zhou eur phys j b 86 455 2013 in the present work we study the spin glass phase transitions and the minimum energy density of the random fvs problem by the first step replica symmetry breaking 1rsb mean field theory for both regular random graphs and erdos renyi graphs we determine the inverse temperature beta 1 at which the replica symmetric mean field theory loses its local stability the inverse temperature beta d of the dynamical clustering phase transition and the inverse temperature beta s of the static condensation phase transition these critical inverse temperatures all change with the mean vertex degree in a nonmonotonic way and beta d is distinct from beta s for regular random graphs of vertex degrees k 60 while beta d are identical to beta s for erdos renyi graphs at least up to mean vertex degree c 512 we then derive the zero temperature limit of the 1rsb theory and use it to compute the minimum fvs cardinality

minimal contagious sets in random regular graphs the bootstrap percolation or threshold model is a dynamic process modelling the propagation of an epidemic on a graph where inactive vertices become active if their number of active neighbours reach some threshold we study an optimization problem related to it namely the determination of the minimal number of active sites in an initial configuration that leads to the activation of the whole graph under this dynamics with and without a constraint on the time needed for the complete activation this problem encompasses in special cases many extremal characteristics of graphs like their independence decycling or domination number and can also be seen as a packing problem of repulsive particles we use the cavity method including the effects of replica symmetry breaking an heuristic technique of statistical mechanics many predictions of which have been confirmed rigorously in the recent years we have obtained in this way several quantitative conjectures on the size of minimal contagious sets in large random regular graphs the most striking being that 5 regular random graph with a threshold of activation of 3 resp 6 regular with threshold 4 have contagious sets containing a fraction 1 6 resp 1 4 of the total number of vertices equivalently these numbers are the minimal fraction of vertices that have to be removed from a 5 regular resp 6 regular random graph to destroy its 3 core we also investigated survey propagation like algorithmic procedures for solving this optimization problem on single instances of random regular graphs

spectrum of non hermitian heavy tailed random matrices let x jk j k 1 be i i d complex random variables such that x jk is in the domain of attraction of an alpha stable law with 0 alpha 2 our main result is a heavy tailed counterpart of girko s circular law namely under some additional smoothness assumptions on the law of x jk we prove that there exists a deterministic sequence a n n 1 alpha and a probability measure mu alpha on c depending only on alpha such that with probability one the empirical distribution of the eigenvalues of the rescaled matrix a n 1 x jk 1 j k n converges weakly to mu alpha as n tends to infinity our approach combines aldous steele s objective method with girko s hermitization using logarithmic potentials the underlying limiting object is defined on a bipartized version of aldous poisson weighted infinite tree recursive relations on the tree provide some properties of mu alpha in contrast with the hermitian case we find that mu alpha is not heavy tailed

max product for maximum weight matching convergence correctness and lp duality max product belief propagation bp is an iterative message passing algorithm for finding the maximum a posteriori map assignment of a discrete probability distribution specified by a graphical model despite the spectacular success of the algorithm in many application areas such as iterative decoding and combinatorial optimization which involve graphs with many cycles theoretical results about both the correctness and convergence of the algorithm are known in only a few cases see section i for references in this paper we prove the correctness and convergence of max product for finding the maximum weight matching mwm in bipartite graphs even though the underlying graph of the mwm problem has many cycles somewhat surprisingly we show that the max product algorithm converges to the correct mwm as long as the mwm is unique we provide a bound on the number of iterations required and show that for a graph of size n the computational cost of the algorithm scales as o n3 which is the same as the computational cost of the best known algorithms for finding the mwm we also provide an interesting relation between the dynamics of the max product algorithm and the auction algorithm which is a well known distributed algorithm for solving the mwm problem

the rank of diluted random graphs we investigate the rank of the adjacency matrix of large diluted random graphs for a sequence of graphs gn n 0 converging locally to a galton watson tree t gwt we provide an explicit formula for the asymptotic multiplicity of the eigenvalue 0 in terms of the degree generating function of t in the first part we show that the adjacency operator associated with t is always self adjoint we analyze the associated spectral measure at the root and characterize the distribution of its atomic mass at 0 in the second part we establish a sufficient condition on for the expectation of this atomic mass to be precisely the normalized limit of the dimension of the kernel of the adjacency matrices of gn n 0 our proofs borrow ideas from analysis of algorithms functional analysis random matrix theory and statistical physics

almost sure and moment exponential stability of euler maruyama discretizations for hybrid stochastic differential equations positive results are derived concerning the long time dynamics of numerical simulations of stochastic differential equation systems with markovian switching euler maruyama discretizations are shown to capture almost sure and moment exponential stability for all sufficiently small timesteps under appropriate conditions

option pricing with markov modulated dynamics markov modulated models for equity prices have recently been extensively studied in the literature in this paper we apply some old results on the wiener hopf factorization of markov processes to a range of option pricing problems for such models the first example is the perpetual american put where the exact numerical solution is obtained without discretizing any pde we then show how the methodology of rogers and stapleton finance stoch 2 1997 pp 3 17 can be used to tackle finite horizon problems and illustrate the methodology by pricing european american single barrier and double barrier options under markov modulated dynamics

replica bounds for optimization problems and diluted spin systems in this paper we generalize to the case of diluted spin models and random combinatorial optimization problems a technique recently introduced by guerra cond mat 0205123 to prove that the replica method generates variational bounds for disordered systems we analyze a family of models that includes the viana bray model the diluted p spin model or random xor sat problem and the random k sat problem showing that the replica cavity method at the various levels of approximation provides systematic schemes to obtain lower bounds of the free energy at all temperatures and of the ground state energy in the case of k sat and xor sat it thus gives upper bounds of the satisfiability threshold our analysis underlines deep connections with the cavity method which are not evident in the long range case

approximation of upper lyapunov exponents of bilinear stochastic differential systems a bilinear stochastic differential system is considered whose solution starting from x is denoted by x t x it is supposed that its upper lyapunov exponent lambda exists the purpose of the paper is to propose an efficient algorithm to approximate lambda it is shown that for markov chains bar x p h x p in mathbb n defined by several approximation schemes the lyapunov exponents bar lambda h are well defined and can be computed estimates are given in terms of the discretization step h of the theoretical error lambda bar lambda h then these results are applied to an engineering problem the stability of the motion of helicopter rotor blades in a turbulent wind

front propagation into unstable states universal algebraic convergence towards uniformly translating pulled fronts fronts that start from a local perturbation and propagate into a linearly unstable state come in two classes pulled fronts and pushed fronts the term pulled front expresses that these fronts are pulled along by the spreading of linear perturbations about the unstable state accordingly their asymptotic speedv equals the spreading speed of perturbations whose dynamics is governed by the equations linearized about the unstable state the central result of this paper is the analysis of the convergence of asymptotically uniformly traveling pulled fronts towardsv we show that when such fronts evolve from sufficiently steep initial conditions which initially decay faster than e x forx 1 t hey have auniversal relaxation behavioras timet 1 the velocity of a pulled front always relaxes algebraically like v t d v 3 2t c 3 p d d 2 t 3 2 c o 1 t 2 the parametersv and d are determined through a saddle point analysis from the equation of motion linearized about the unstable invaded state this front velocity is independent of the precise value of the front amplitude which one tracks to measure the front position the interior of the front is essentially slaved to the leading edge and develops universally as x t d8v t x rt dt 0 v t 0 c o 1 t 2 where8v x vt is a uniformly translating front solution with velocity v v our result which can be viewed as a general center manifold result for pulled front propagation is derived in detail for the well known nonlinear diffusion equation of type t d 2 x c 3 where the invaded unstable state is d 0 even for this simple case the subdominant t 3 2 term extends an earlier result of bramson our analysis is then generalized to more general sets of partial differential equations with higher spatial or temporal derivatives to pdes with memory kernels and also to difference equations such as those that occur in numerical finite difference codes our universal result for pulled fronts thus implies independence i of the level curve which is used to track the front position ii of the precise nonlinearities iii of the precise form of the linear operators in the dynamical equation and iv of the precise initial conditions as long as they are sufficiently steep the only remainders of the explicit form of the dynamical equation are the nonlinear solutions 8v and the three saddle point parameters v and d as our simulations confirm all our analytical predictions in every detail it can be concluded that we have a complete analytical understanding of the propagation mechanism and relaxation behavior of pulled fronts if they are uniformly translating for t 1 an immediate consequence of the slow algebraic relaxation is that the standard moving boundary approximation breaks down for weakly curved pulled fronts in two or three dimensions in addition to our main result for pulled fronts we also discuss the propagation and convergence of fronts emerging from

asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications in this paper we extend our previous work on the stochastic block model a commonly used generative model for social and biological networks and the problem of inferring functional groups or communities from the topology of the network we use the cavity method of statistical physics to obtain an asymptotically exact analysis of the phase diagram we describe in detail properties of the detectability undetectability phase transition and the easy hard phase transition for the community detection problem our analysis translates naturally into a belief propagation algorithm for inferring the group memberships of the nodes in an optimal way i e that maximizes the overlap with the underlying group memberships and learning the underlying parameters of the block model finally we apply the algorithm to two examples of real world networks and discuss its performance

whom to befriend to influence people alice wants to join a new social network and influence its members to adopt a new product or idea each person v in the network has a certain threshold t v for activation i e adoption of the product or idea if v has at least t v activated neighbors then v will also become activated if alice wants to activate the entire social network whom should she befriend we study the problem of finding the minimum number of links that alice should form to people in the network in order to activate the entire social network this minimum links problem has applications in viral marketing and the study of epidemics we show that the solution can be quite different from the related and widely studied target set selection problem we prove that the minimum links problem is np complete in fact it is hard to approximate to within an epsilon ln n factor for some constant epsilon even for graphs with degree 3 and with threshold at most 2 in contrast we give linear time algorithms to solve the problem for trees cycles and cliques and give precise bounds on the number of links needed

mean square and asymptotic stability of the stochastic theta method stability analysis of numerical methods for ordinary differential equations odes is motivated by the question for what choices of stepsize does the numerical method reproduce the characteristics of the test equation we study a linear test equation with a multiplicative noise term and consider mean square and asymptotic stability of a stochastic version of the theta method we extend some mean square stability results in saito and mitsui siam j numer anal 33 1996 pp 2254 2267 in particular we show that an extension of the deterministic a stability property holds we also plot mean square stability regions for the case where the test equation has real parameters for asymptotic stability we show that the issue reduces to finding the expected value of a parametrized random variable we combine analytical and numerical techniques to get insights into the stability properties for a variant of the method that has been proposed in the literature we obtain precise analytic expressions for the asymptotic stability region this allows us to prove a number of results the technique introduced is widely applicable and we use it to show that a fully implicit method suggested by kloeden and platen numerical solution of stochastic differential equations springer verlag 1992 has an asymptotic stability extension of the deterministic a stability property we also use the approach to explain some numerical results reported in milstein platen and schurz siam j numer anal 35 1998 pp 1010 1019

broken replica symmetry bounds in the mean field spin glass model by using a simple interpolation argument in previous work we have proven the existence of the thermodynamic limit for mean field disordered models including the sherrington kirkpatrick model and the derrida p spin model here we extend this argument in order to compare the limiting free energy with the expression given by the parisi ansatz and including full spontaneous replica symmetry breaking our main result is that the quenched average of the free energy is bounded from below by the value given in the parisi ansatz uniformly in the size of the system moreover the difference between the two expressions is given in the form of a sum rule extending our previous work on the comparison between the true free energy and its replica symmetric sherrington kirkpatrick approximation we give also a variational bound for the infinite volume limit of the ground state energy per site

explicit approximations for nonlinear switching diffusion systems in finite and infinite horizons abstract focusing on hybrid diffusion dynamics involving continuous dynamics as well as discrete events this article investigates the explicit approximations for nonlinear switching diffusion systems modulated by a markov chain different kinds of easily implementable explicit schemes have been proposed to approximate the dynamical behaviors of switching diffusion systems with locally lipschitz continuous drift and diffusion coefficients in both finite and infinite intervals without additional restriction conditions except those which guarantee the exact solutions possess their dynamical properties the numerical solutions converge strongly to the exact solutions in finite horizon moreover realize the approximation of long time dynamical properties including the moment boundedness stability and ergodicity some simulations and examples are provided to support the theoretical results and demonstrate the validity of the approach

ms stability of the euler maruyama method for stochastic differential delay equations the aim of this paper is to investigate the stability of the euler maruyama method for the stochastic differential equations with time delay the definition of ms stability of numerical methods is established the condition of the mean square stability of analytical solution is obtained for a linear scalar system with multiplicative noise and the ms stability of the numerical scheme is proved

gibbs states and the set of solutions of random constraint satisfaction problems abstract an instance of a random constraint satisfaction problem defines a random subset the set of solutions of a large product space x n the set of assignments we consider two prototypical problem ensembles random k satisfiability and q coloring of random regular graphs and study the uniform measure with support on s as the number of constraints per variable increases this measure first decomposes into an exponential number of pure states clusters and subsequently condensates over the largest such states above the condensation point the mass carried by the n largest states follows a poisson dirichlet process for typical large instances the two transitions are sharp we determine their precise location further we provide a formal definition of each phase transition in terms of different notions of correlation between distinct variables in the problem the degree of correlation naturally affects the performances of many search sampling algorithms empirical evidence suggests that local monte carlo markov chain strategies are effective up to the clustering phase transition and belief propagation up to the condensation point finally refined message passing techniques such as survey propagation may also beat this threshold message passing algorithms phase transitions random graphs

sublinear time algorithms for monomer dimer systems on bounded degree graphs for a graph g let z g be the partition function of the monomer dimer system defined by k m k g k where m k g is the number of matchings of size k in g we consider graphs of bounded degree and develop a sublinear time algorithm for estimating logz g at an arbitrary value 0 within additive error en with high probability the query complexity of our algorithm does not depend on the size of g and is polynomial in 1 e and we also provide a lower bound quadratic in 1 e for this problem this is the first analysis of a sublinear time approximation algorithm for a p complete problem our approach is based on the correlation decay of the gibbs distribution associated with z g we show that our algorithm approximates the probability for a vertex to be covered by a matching sampled according to this gibbs distribution in a near optimal sublinear time we extend our results to approximate the average size and the entropy of such a matching within an additive error with high probability where again the query complexity is polynomial in 1 e and the lower bound is quadratic in 1 e our algorithms are simple to implement and of practical use when dealing with massive datasets our results extend to other systems where the correlation decay is known to hold as for the independent set problem up to the critical activity

the sharp threshold for bootstrap percolation in all dimensions in r neighbour bootstrap percolation on a graph g a typically random set a of initially infected vertices spreads by infecting at each time step vertices with at least r already infected neighbours this process may be viewed as a monotone version of the glauber dynamics of the ising model and has been extensively studied on the d dimensional grid n d the elements of the set a are usually chosen independently with some density p and the main question is to determine p c n d r the density at which percolation infection of the entire vertex set becomes likely in this paper we prove for every pair d ge r ge 2 that there is a constant l d r such that p c n d r l d r o 1 log r 1 n d r 1 as n o infty where log r denotes an r times iterated logarithm we thus prove the existence of a sharp threshold for percolation in any fixed number of dimensions moreover we determine l d r for every pair d r

on sums of independent random variables with unbounded variance and estimating the average degree in a graph we prove the following inequality for every positive integer n and every collection x 1 ldots x n of nonnegative independent random variables each with expectation 1 the probability that their sum remains below n 1 is at least alpha 0 our proof produces a value of alpha 1 13 simeq 0 077 but we conjecture that the inequality also holds with alpha 1 e simeq 0 368 as an example for the use of the new inequality we consider the problem of estimating the average degree of a graph by querying the degrees of some of its vertices we show the following threshold behavior approximation factors above 2 require far fewer queries than approximation factors below 2 the new inequality is used in order to get tight up to multiplicative constant factors relations between the number of queries and the quality of the approximation we show how the degree approximation algorithm can be used in order to quickly find those edges in a network that belong to many shortest paths

nonuniform mean square exponential dichotomies and mean square exponential stability abstract in this paper the existence conditions of nonuniform mean square exponential dichotomy nms ed for a linear stochastic differential equation sde are established the difference of the conditions for the existence of a nonuniform dichotomy between an sde and an ordinary differential equation ode is that the first one needs an additional assumption nonuniform lyapunov matrix to guarantee that the linear sde can be transformed into a decoupled one while the second does not therefore the first main novelty of our work is that we establish some preliminary results to tackle the stochasticity this paper is also concerned with the mean square exponential stability of nonlinear perturbation of a linear sde under the condition of nonuniform mean square exponential contraction nms ec for this purpose the concept of second moment regularity coefficient is introduced this concept is essential in determining the stability of the perturbed equation and hence we deduce the lower and upper bounds of this coefficient our results imply that the lower and upper bounds of the second moment regularity coefficient can be expressed solely by the drift term of the linear sde

minimizing energy below the glass thresholds focusing on the optimization version of the random k satisfiability problem the max k sat problem we study the performance of the finite energy version of the survey propagation algorithm we show that a simple linear time backtrack decimation strategy is sufficient to reach configurations well below the lower bound for the dynamic threshold energy and very close to the analytic prediction for the optimal ground states a comparative numerical study on one of the most efficient local search procedures is also given

solutions to homogeneous monge ampere equations of homothetic functions and their applications to production models in economics in economics homothetic functions are production functions whose marginal technical rate of substitution is homogeneous of degree zero mathematically a homothetic function is a function of the form f x f h x1 xn where f is a monotonically increasing function and h is a homogeneous function of any degree d 0 in this paper we classify homothetic functions satisfying the homogeneous monge ampere equation several applications to production models in economics will also be given

strong convergence of euler type methods for nonlinear stochastic differential equations traditional finite time convergence theory for numerical methods applied to stochastic differential equations sdes requires a global lipschitz assumption on the drift and diffusion coefficients in practice many important sde models satisfy only a local lipschitz property and since brownian paths can make arbitrarily large excursions the global lipschitz based theory is not directly relevant in this work we prove strong convergence results under less restrictive conditions first we give a convergence result for euler maruyama requiring only that the sde is locally lipschitz and that the pth moments of the exact and numerical solution are bounded for some p 2 as an application of this general theory we show that an implicit variant of euler maruyama converges if the diffusion coefficient is globally lipschitz but the drift coefficient satisfies only a one sided lipschitz condition this is achieved by showing that the implicit method has bounded moments and may be viewed as an euler maruyama approximation to a perturbed sde of the same form second we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial

belief propagation and lp relaxation for weighted matching in general graphs loopy belief propagation has been employed in a wide variety of applications with great empirical success but it comes with few theoretical guarantees in this paper we analyze the performance of the max product form of belief propagation for the weighted matching problem on general graphs we show that the performance of max product is exactly characterized by the natural linear programming lp relaxation of the problem in particular we first show that if the lp relaxation has no fractional optima then max product always converges to the correct answer this establishes the extension of the recent result by bayati shah and sharma which considered bipartite graphs to general graphs perhaps more interestingly we also establish a tight converse namely that the presence of any fractional lp optimum implies that max product will fail to yield useful estimates on some of the edges we extend our results to the weighted b matching and r edge cover problems we also demonstrate how to simplify the max product message update equations for weighted matching making it easily deployable in distributed settings like wireless or sensor networks

belief propagation an asymptotically optimal algorithm for the random assignment problem the random assignment problem asks for the minimum cost perfect matching in the complete n n bipartite graph knn with i i d edge weights say uniform on 0 1 in a remarkable work by aldous aldous d 2001 the 2 limit in the random assignment problem rsa18 381 418 the optimal cost was shown to converge to 2 as n as conjectured by mezard and parisi mezard m g parisi 1987 on the solution of the random link matching problem j phys 48 1451 1459 through the so called cavity method the latter also suggested a nonrigorous decentralized strategy for finding the optimum which turned out to be an instance of the belief propagation bp heuristic discussed by pearl pearl j 1988 probabilistic reasoning in intelligent systems networks of plausible inference morgan kaufmann san francisco in this paper we use the objective method to analyze the performance of bp as the size of the underlying graph becomes large specifically we establish that the dynamic of bp on knn converges in distribution as n to an appropriately defined dynamic on the poisson weighted infinite tree and we then prove correlation decay for this limiting dynamic as a consequence we obtain that bp finds an asymptotically correct assignment in o n2 time only this contrasts with both the worst case upper bound for convergence of bp derived by bayati et al bayati m d shah m sharma 2008 max product for maximum weight matching convergence correctness and lp duality ieee trans inform theory54 3 1241 1251 and the best known computational cost of n3 achieved by edmonds and karp s algorithm edmonds j r karp 1972 theoretical improvements in algorithmic efficiency for network flow problems j acm19 248 264

the asymptotic k sat threshold abstract since the early 2000s physicists have developed an ingenious but non rigorous formalism called the cavity method to put forward precise conjectures on phase transitions in random problems mezard et al 2002 37 the cavity method predicts that the satisfiability threshold in the random k sat problem is r k sat 2 k ln 2 1 2 1 ln 2 e k with lim k e k 0 mertens et al 2006 35 this paper contains a proof of the conjecture

instability of one step replica symmetry broken phase in satisfiability problems we reconsider the one step replica symmetry breaking 1rsb solutions of two random combinatorial problems k xorsat and k sat we present a general method for establishing the stability of these solutions with respect to further steps of replica symmetry breaking our approach extends the ideas of montanari and ricci tersenghi 2003 eur phys j b 33 339 to more general combinatorial problems it turns out that 1rsb is always unstable at sufficiently small clause density or high energy in particular the recent 1rsb solution to 3 sat is unstable at zero energy for m with m 4 153 on the other hand the sat unsat phase transition seems to be correctly described within 1rsb