the green s function and large time behavior of solutions for the one dimensional boltzmann equation we study the pointwise behavior of the green s function of the boltzmann equation our results reveal the particle and fluid aspects of the equation the particle aspect is represented by singular waves these waves are carried by transport equations and dominate the short time behavior of the solution we devise a picard type iteration for constructing the increasingly regular particlelike waves the fluidlike waves reveal the dissipative behavior of the type of navier stokes equations as usually seen by the chapman enskog expansion these waves are constructed as part of the long wave expansion in the spectrum of the fourier mode for the space variable the fluidlike waves represent the long time behavior of the solution as an application we obtain the pointwise description of the large time behavior of the convergence to the global maxwellian when the initial perturbation is not necessarily smooth in our analysis of the exchanges of the microscopic velocity decay and space decay we make essential use of the hard sphere models c 2004 wiley periodicals inc contents

continuum percolation in two dimensions monte carlo tests of scaling and universality for non interacting discs detailed results are reported for the connectivity properties of a system of discs of unit radius free to be situated anywhere within a square of area 2l2 ordinary lattke percolation would correspond to the discs being situated on the vertices of a j2l x j2l lattice computer simulations are carried out for a sequence of increasing system sizes ranging from l 20 to l 1000 for each value of l a large number of realisations are generated for 25 values of the disc concentration x we calculate a variety of estimates for the threshold parameter x as well as the critical exponents p y t and v our exponent estimates are in close agreement with accepted values for ordinary lattice percolation therefore this continuum system appears to be in the same universality class as lattice percolation

consensus dynamics on random rectangular graphs abstract a random rectangular graph rrg is a generalization of the random geometric graph rgg in which the nodes are embedded into a rectangle with side lengths a and b 1 a instead of on a unit square 0 1 2 two nodes are then connected if and only if they are separated at a euclidean distance smaller than or equal to a certain threshold radius r when a 1 the rrg is identical to the rgg here we apply the consensus dynamics model to the rrg our main result is a lower bound for the time of consensus i e the time at which the network reaches a global consensus state to prove this result we need first to find an upper bound for the algebraic connectivity of the rrg i e the second smallest eigenvalue of the combinatorial laplacian of the graph this bound is based on a tight lower bound found for the graph diameter our results prove that as the rectangle in which the nodes are embedded becomes more elongated the rrg becomes a large world i e the diameter grows to infinity and a poorly connected graph i e the algebraic connectivity decays to zero the main consequence of these findings is the proof that the time of consensus in rrgs grows to infinity as the rectangle becomes more elongated in closing consensus dynamics in rrgs strongly depend on the geometric characteristics of the embedding space and reaching the consensus state becomes more difficult as the rectangle is more elongated

geographic coarse graining analysis of the railway network of china abstract we investigate the detailed empirical analysis of the statistical properties of the railway network of china rnc in space l and space g constructed by geographic coarse graining process the rnc exhibits similar properties in the cumulative distributions of degree and strength in two spaces and it presents the hierarchical structure small world behavior and assortativity areciprocal connection both in space l and space g we also investigate the path length that every train runs the distribution of the railroad length per degree and the optimal distribution of stations

the vlasov poisson boltzmann system with angular cutoff for soft potentials abstract this paper is concerned with the cauchy problem of the vlasov poisson boltzmann system near a given global maxwellian with angular cutoff for a class of soft potentials in three space dimensions and the main purpose here is to derive the global solvability of such a problem without the neutral condition which was imposed in duan et al 2013 11 on the initial perturbation our analysis is based on a time weighted energy method and some delicate estimates on its solutions which show that the l 2 r x 3 r 3 norms of the solutions with higher order x derivatives enjoy sharper time decay rates

emergence of scale free behavior in networks from limited horizon linking and cost trade offs abstract we study network growth from a fixed set of initially isolated nodes placed at random on the surface of a sphere the growth mechanism we use adds edges to the network depending on strictly local gain and cost criteria only nodes that are not too far apart on the sphere may be considered for being joined by an edge given two such nodes the joining occurs only if the gain of doing it surpasses the cost our model is based on a multiplicative parameter that regulates in a function of node degrees the maximum geodesic distance that is allowed between nodes for them to be considered for joining for n nodes distributed uniformly on the sphere and for n within limits that depend on cost related parameters we have found that our growth mechanism gives rise to power law distributions of node degree that are invariant for constant n we also study connectivity and distance related properties of the networks

classical solutions to the boltzmann equation for molecules with an angular cutoff an important class of collision kernels in the boltzmann theory are governed by the inverse power law in which the intermolecular potential between two particles is an inverse power of their distance under the grad angular cutoff assumption global in time classical solutions near maxwellians are constructed in a periodic box for all soft potentials with 3 0

corrected mean field model for random sequential adsorption on random geometric graphs a notorious problem in mathematics and physics is to create a solvable model for random sequential adsorption of non overlapping congruent spheres in the d dimensional euclidean space with d ge 2 spheres arrive sequentially at uniformly chosen locations in space and are accepted only when there is no overlap with previously deposited spheres due to spatial correlations characterizing the fraction of accepted spheres remains largely intractable we study this fraction by taking a novel approach that compares random sequential adsorption in euclidean space to the nearest neighbor blocking on a sequence of clustered random graphs this random network model can be thought of as a corrected mean field model for the interaction graph between the attempted spheres using functional limit theorems we characterize the fraction of accepted spheres and its fluctuations

random geometric graphs we analyze graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present the critical connectivity is found numerically by examining the size of the largest cluster we derive an analytical expression for the cluster coefficient which shows that the graphs are distinctly different from standard random graphs even for infinite dimensionality insights relevant for graph bipartitioning are included

random walks on fractals and stretched exponential relaxation stretched exponential relaxation exp t tau beta k is observed in a large variety of systems but has not been explained so far studying random walks on percolation clusters in curved spaces whose dimensions range from 2 to 7 we show that the relaxation is accurately a stretched exponential and is directly connected to the fractal nature of these clusters thus we find that in each dimension the decay exponent beta k is related to well known exponents of the percolation theory in the corresponding flat space we suggest that the stretched exponential behavior observed in many complex systems polymers colloids glasses is due to the fractal character of their configuration space

subgraphs and network motifs in geometric networks many real world networks describe systems in which interactions decay with the distance between nodes examples include systems constrained in real space such as transportation and communication networks as well as systems constrained in abstract spaces such as multivariate biological or economic data sets and models of social networks these networks often display network motifs subgraphs that recur in the network much more often than in randomized networks to understand the origin of the network motifs in these networks it is important to study the subgraphs and network motifs that arise solely from geometric constraints to address this we analyze geometric network models in which nodes are arranged on a lattice and edges are formed with a probability that decays with the distance between nodes we present analytical solutions for the numbers of all three and four node subgraphs in both directed and nondirected geometric networks we also analyze geometric networks with arbitrary degree sequences and models with a bias for directed edges in one direction scaling rules for scaling of subgraph numbers with system size lattice dimension and interaction range are given several invariant measures are found such as the ratio of feedback and feed forward loops which do not depend on system size dimension or connectivity function we find that network motifs in many real world networks including social networks and neuronal networks are not captured solely by these geometric models this is in line with recent evidence that biological network motifs were selected as basic circuit elements with defined information processing functions

extremal optimization for graph partitioning extremal optimization is a new general purpose method for approximating solutions to hard optimization problems we study the method in detail by way of the computationally hard np hard graph partitioning problem we discuss the scaling behavior of extremal optimization focusing on the convergence of the average run as a function of run time and system size the method has a single free parameter which we determine numerically and justify using a simple argument on random graphs our numerical results demonstrate that extremal optimization maintains consistent accuracy for increasing system sizes with an approximation error decreasing over run time roughly as a power law t 20 4 on geometrically structured graphs the scaling of results from the average run suggests that these are far from optimal with large fluctuations between individual trials but when only the best runs are considered results consistent with theoretical arguments are recovered

life lengths and association a dynamic approach we give two sets of conditions for the association of a system s component life lengths the conditions are dynamic in the sense that they are based on evaluating at any given time t the effect a failure would have on the system s future behaviour of basic importance is the concept weakened by failures introduced in the paper mathematically the paper is based on martingales in the case of jump processes or marked point processes

complex network based time series analysis abstract recent works show that complex network theory may be a powerful tool in time series analysis we propose in this paper a reliable procedure for constructing complex networks from the correlation matrix of a time series an original stock time series the corresponding return series and its amplitude series are considered the degree distribution of the original series can be well fitted with a power law while that of the return series can be well fitted with a gaussian function the degree distribution of the amplitude series contains two asymmetric gaussian branches reconstruction of networks from time series is a common problem in diverse research the proposed strategy may be a reasonable solution to this problem

motif distributions in phase space networks for characterizing experimental two phase flow patterns with chaotic features the dynamics of two phase flows have been a challenging problem in nonlinear dynamics and fluid mechanics we propose a method to characterize and distinguish patterns from inclined water oil flow experiments based on the concept of network motifs that have found great usage in network science and systems biology in particular we construct from measured time series phase space complex networks and then calculate the distribution of a set of distinct network motifs to gain insight we first test the approach using time series from classical chaotic systems and find a universal feature motif distributions from different chaotic systems are generally highly heterogeneous our main finding is that the distributions from experimental two phase flows tend to be heterogeneous as well suggesting the underlying chaotic nature of the flow patterns calculation of the maximal lyapunov exponent provides further support for this motif distributions can thus be a feasible tool to understand the dynamics of realistic two phase flow patterns

finding metastable states in real world time series with recurrence networks in the framework of time series analysis with recurrence networks we introduce a self adaptive method that determines the elusive recurrence threshold and identifies metastable states in complex real world time series as initial step we introduce a way to set the embedding parameters used to reconstruct the state space from the time series we set them as the ones giving the maximum shannon entropy of the diagonal line length distribution for the first simultaneous minima of recurrence rate and shannon entropy to identify metastable states as well as the transitions between them we use a soft partitioning algorithm for module finding which is specifically developed for the case in which a system shows metastability we illustrate our method with a complex time series example finally we show the robustness of our method for identifying metastable states our results suggest that our method is robust for identifying metastable states in complex time series even when introducing considerable levels of noise and missing data points

networks based on collisions among mobile agents we investigate in detail a recent model of colliding mobile agents phys rev lett 96 088702 used as an alternative approach to construct evolving networks of interactions formed by the collisions governed by suitable dynamical rules the system of mobile agents evolves towards a quasi stationary state which is apart small fluctuations well characterized by the de nsity of the system and the residence time of the agents the residence time defines a collision rate and by varying the collision ra te the system percolates at a critical value with the emergence of a giant cluster whose critical exponents are the ones of t wodimensional percolation further the degree and clusteri ng coefficient distributions and the average path length sho w that the network associated with such a system presents non trivial features which depending on the collision rule enables one not only to recover the main properties of standard networks such as exponential random and scale free networks but also to obtain other topological structures namely we show a specific example where the obtained structure has topological fe atures which characterize accurately the structure and evolution of social networks in different contexts ranging from networks of acquaintances to networks of sexual contacts

box integrals by a box integral we mean here an expectation where email protected runs over the unit n cube with email protected and s fixed explicitly 0 email protected 0 1 r 1 q 1 2 r n q n 2 s 2dr 1 dr n the study of box integrals leads one naturally into several disparate fields of analysis while previous studies have focused upon symbolic evaluation and asymptotic analysis of special cases notably s 1 we work herein more generally in interdisciplinary fashion developing results such as 1 analytic continuation in complex s 2 relevant combinatorial identities 3 rapidly converging series 4 statistical inferences 5 connections to mathematical physics and 6 extreme precision quadrature techniques appropriate for these integrals these intuitions and results open up avenues of experimental mathematics with a view to new conjectures and theorems on integrals of this type

l2 stability theory of the boltzmann equation near a global maxwellian we present three a priori l2 stability estimates for classical solutions to the boltzmann equation with a cut off inverse power law potential when initial datum is a perturbation of a global maxwellian we show that l2 stability estimates of classical solutions depend on strichartz type estimates of perturbations and the non positive definiteness of the linearized collision operator several well known classical solutions to the boltzmann equation fit our l2 stability framework

random spherical graphs this work addresses a modification of the random geometric graph rgg model by considering a set of points uniformly and independently distributed on the surface of a d 1 sphere with radius r in a d dimensional euclidean space instead of on an unit hypercube 0 1 d then two vertices are connected by a link if their great circle distance is at most s in the case of d 3 the topological properties of the random spherical graphs rsgs generated by this model are studied as a function of s we obtain analytical expressions for the average degree degree distribution connectivity average path length diameter and clustering coefficient for rsgs by setting r sqrt pi 2 pi we also show the differences between the topological properties of rsgs and those of two dimensional rggs and random rectangular graphs rrgs surprisingly in terms of the average clustering coefficient rsgs look more similar to the analytical estimation for rggs than rggs themselves when their boundary effects are considered we support all our findings by computer simulations that corroborate the accuracy of the theoretical models proposed for rsgs

mapping and discrimination of networks in the complexity entropy plane complex networks are usually characterized in terms of their topological spatial or information theoretic properties and combinations of the associated metrics are used to discriminate networks into different classes or categories however even with the present variety of characteristics at hand it still remains a subject of current research to appropriately quantify a network s complexity and correspondingly discriminate between different types of complex networks like infrastructure or social networks on such a basis here we explore the possibility to classify complex networks by means of a statistical complexity measure that has formerly been successfully applied to distinguish different types of chaotic and stochastic time series it is composed of a network s averaged per node entropic measure characterizing the network s information content and the associated jenson shannon divergence as a measure of disequilibrium we study 29 real world networks and show that networks of the same category tend to cluster in distinct areas of the resulting complexity entropy plane we demonstrate that within our framework connectome networks exhibit among the highest complexity while e g transportation and infrastructure networks display significantly lower values furthermore we demonstrate the utility of our framework by applying it to families of random scale free and watts strogatz model networks we then show in a second application that the proposed framework is useful to objectively construct threshold based networks such as functional climate networks or recurrence networks by choosing the threshold such that the statistical network complexity is maximized

the vlasov poisson boltzmann system without angular cutoff this paper is concerned with the vlasov poisson boltzmann system for plasma particles of two species in three space dimensions the boltzmann collision kernel is assumed to be angular non cutoff with 3 2s and 1 2 s 1 where s are two parameters describing the kinetic and angular singularities respectively we establish the global existence and convergence rates of classical solutions to the cauchy problem when initial data is near maxwellians this extends the results in duan et al j diff eqs 252 12 6356 6386 2012 math models methods appl sci 23 6 927 2013 for the cutoff kernel with 2 1 to the case 3 2 as long as the angular singularity exists instead and is strong enough i e s is close to 1 the proof is based on the time weighted energy method building also upon the recent studies of the non cutoff boltzmann equation in gressman and strain j amer math soc 24 3 771 847 2011 and the vlasov poisson landau system in guo j amer math soc 25 759 812 2012

uniform framework for the recurrence network analysis of chaotic time series we propose a general method for the construction and analysis of unweighted e recurrence networks from chaotic time series the selection of the critical threshold e c in our scheme is done empirically and we show that its value is closely linked to the embedding dimension m in fact we are able to identify a small critical range e numerically that is approximately the same for the random and several standard chaotic time series for a fixed m this provides us a uniform framework for the nonsubjective comparison of the statistical measures of the recurrence networks constructed from various chaotic attractors we explicitly show that the degree distribution of the recurrence network constructed by our scheme is characteristic to the structure of the attractor and display statistical scale invariance with respect to increase in the number of nodes n we also present two practical applications of the scheme detection of transition between two dynamical regimes in a time delayed system and identification of the dimensionality of the underlying system from real world data with a limited number of points through recurrence network measures the merits limitations and the potential applications of the proposed method are also highlighted

connectivity distribution of spatial networks we study spatial networks constructed by randomly placing nodes on a manifold and joining two nodes with an edge whenever their distance is less than a certain cutoff we derive the general expression for the connectivity distribution of such networks as a functional of the distribution of the nodes we show that for regular spatial densities the corresponding spatial network has a connectivity distribution decreasing faster than an exponential in contrast we also show that scale free networks with a power law decreasing connectivity distribution are obtained when a certain information measure of the node distribution integral of higher powers of the distribution diverges we illustrate our results on a simple example for which we present simulation results finally we speculate on the role played by the limiting case p k proportional variant k 1 which appears empirically to be relevant to spatial networks of biological origin such as the ones constructed from gene expression data

enhancing robustness and immunization in geographical networks we find that different geographical structures of networks lead to varied percolation thresholds although these networks may have similar abstract topological structures thus strategies for enhancing robustness and immunization of a geographical network are proposed using the generating function formalism we obtain an explicit form of the percolation threshold q c for networks containing arbitrary order cycles for three cycles the dependence of q c on the clustering coefficients is ascertained the analysis substantiates the validity of the strategies with analytical evidence

topological properties and fractal analysis of a recurrence network constructed from fractional brownian motions many studies have shown that we can gain additional information on time series by investigating their accompanying complex networks in this work we investigate the fundamental topological and fractal properties of recurrence networks constructed from fractional brownian motions fbms first our results indicate that the constructed recurrence networks have exponential degree distributions the average degree exponent increases first and then decreases with the increase of hurst index h of the associated fbms the relationship between h and can be represented by a cubic polynomial function we next focus on the motif rank distribution of recurrence networks so that we can better understand networks at the local structure level we find the interesting superfamily phenomenon i e the recurrence networks with the same motif rank pattern being grouped into two superfamilies last we numerically analyze the fractal and multifractal properties of recurrence networks we find that the average fractal dimension db of recurrence networks decreases with the hurst index h of the associated fbms and their dependence approximately satisfies the linear formula db 2 h which means that the fractal dimension of the associated recurrence network is close to that of the graph of the fbm moreover our numerical results of multifractal analysis show that the multifractality exists in these recurrence networks and the multifractality of these networks becomes stronger at first and then weaker when the hurst index of the associated time series becomes larger from 0 4 to 0 95 in particular the recurrence network with the hurst index h 0 5 possesses the strongest multifractality in addition the dependence relationships of the average information dimension d 1 and the average correlation dimension d 2 on the hurst index h can also be fitted well with linear functions our results strongly suggest that the recurrence network inherits the basic characteristic and the fractal nature of the associated fbm series

nature s way of optimizing we propose a general purpose method for finding high quality solutions to hard optimization problems inspired by self organizing processes often found in nature the method called extremal optimization successively eliminates extremely undesirable components of sub optimal solutions drawing upon models used to simulate far from equilibrium dynamics it complements approximation methods inspired by equilibrium statistical physics such as simulated annealing with only one adjustable parameter its performance proves competitive with and often superior to more elaborate stochastic optimization procedures we demonstrate it here on two classic hard optimization problems graph partitioning and the traveling salesman problem 2000 elsevier science b v all rights reserved

complex network view of evolving manifolds we study complex networks formed by triangulations and higher dimensional simplicial complexes representing closed evolving manifolds in particular for triangulations the set of possible transformations of these networks is restricted by the condition that at each step all the faces must be triangles stochastic application of these operations leads to random networks with different architectures we perform extensive numerical simulations and explore the geometries of growing and equilibrium complex networks generated by these transformations and their local structural properties this characterization includes the hausdorff and spectral dimensions of the resulting networks their degree distributions and various structural correlations our results reveal a rich zoo of architectures and geometries of these networks some of which appear to be small worlds while others are finite dimensional with hausdorff dimension equal or higher than the original dimensionality of their simplices the range of spectral dimensions of the evolving triangulations turns out to be from about 1 4 to infinity our models include simplicial complexes representing manifolds with evolving topologies for example an h holed torus with a progressively growing number of holes this evolving graph demonstrates features of a small world network and has a particularly heavy tailed degree distribution

gradient mechanism in a communication network we study the efficiency of the gradient mechanism of message transfer in a two dimensional communication network of regular nodes and randomly distributed hubs each hub on the network is assigned some randomly chosen capacity and hubs with lower capacities are connected to the hubs with maximum capacity the average travel times of single messages traveling on the lattice decrease rapidly as the number of hubs increase the functional dependence of the average travel times on the hub density shows q exponential behavior with a power law tail we also study the relaxation behavior of the network when a large number of messages are created simultaneously at random locations and travel on the network toward their designated destinations for this situation in the absence of the gradient mechanism the network can show congestion effects due to the formation of transport traps we show that if hubs of high betweenness centrality are connected by the gradient mechanism efficient decongestion can be achieved the gradient mechanism is less prone to the formation of traps than other decongestion schemes we also study the spatial configurations of transport traps and propose minimal strategies for their elimination

a revisiting of the l 2 stability theory of the boltzmann equation near global maxwellians we study the l 2 stability theory of the boltzmann equation near a global maxwellian when an initial datum is a perturbation of a global maxwellian we show that the l 2 distance between two classical solutions can be controlled by the initial data in a lipschitz manner which illustrates the lipschitz continuity of the solution operator for the boltzmann equation in l 2 topology our local in time l 2 stability results cover cutoff very soft potentials as well as non cutoff hard and soft potentials these cases were not treated in the previous work ha et al in arch ration mech anal 197 657 688 2010 thus our results together with the results in ha et al 2010 complete the l 2 stability theory for the boltzmann equation near a global maxwellian for this l 2 stability estimate we use the coercivity estimate of the linearized collision operator the smallness of perturbation in a mixed lebesgue norm and strichartz type estimates of perturbation we also show that for all classical solutions available in the literature the lipschitz constant can be chosen as independent of time to obtain the uniform l 2 stability of the boltzmann equation

spatially embedded random networks many real world networks analyzed in modern network theory have a natural spatial element e g the internet social networks neural networks etc yet aside from a comparatively small number of somewhat specialized and domain specific studies the spatial element is mostly ignored and in particular its relation to network structure disregarded in this paper we introduce a model framework to analyze the mediation of network structure by spatial embedding specifically we model connectivity as dependent on the distance between network nodes our spatially embedded random networks construction is not primarily intended as an accurate model of any specific class of real world networks but rather to gain intuition for the effects of spatial embedding on network structure nevertheless we are able to demonstrate in a quite general setting some constraints of spatial embedding on connectivity such as the effects of spatial symmetry conditions for scale free degree distributions and the existence of small world spatial networks we also derive some standard structural statistics for spatially embedded networks and illustrate the application of our model framework with concrete examples

on the continuous fermat weber problem we give the firstexact algorithmic study of facility location problems that deal with finding a median for acontinuum of demand points in particular we consider versions of the continuousk median fermat weber problem where the goal is to select one or more center points that minimize the average distance to a set of points in a demandregion in such problems the average is computed as an integral over the relevant region versus the usual discrete sum of distances the resulting facility location problems are inherently geometric requiring analysis techniques of computational geometry we provide polynomial time algorithms for various versions of thel1 1 median fermat weber problem we also consider the multiple center version of thel1k median problem which we prove is np hard for largek

recurrence plots of discrete time gaussian stochastic processes abstract we investigate the statistical properties of recurrence plots rps of data generated by discrete time stationary gaussian random processes we analytically derive the theoretical values of the probabilities of occurrence of recurrence points and consecutive recurrence points forming diagonals in the rp with an embedding dimension equal to 1 these results allow us to obtain theoretical values of three measures i the recurrence rate r e c ii the percent determinism d e t and iii rp based estimation of the e entropy e in the sense of correlation entropy we apply these results to two gaussian processes namely first order autoregressive processes and fractional gaussian noise for these processes we simulate a number of realizations and compare the rp based estimations of the three selected measures to their theoretical values these comparisons provide useful information on the quality of the estimations such as the minimum required data length and threshold radius used to construct the rp

uncertainty principle and kinetic equations abstract a large number of mathematical studies on the boltzmann equation are based on the grad s angular cutoff assumption however for particle interaction with inverse power law potentials the associated cross sections have a non integrable singularity corresponding to the grazing collisions smoothing properties of solutions are then expected on the other hand the uncertainty principle established by heisenberg in 1927 has been developed so far in various situations and it has been applied to the study of the existence and smoothness of solutions to partial differential equations this paper is the first one to apply this celebrated principle to the study of the singularity in the cross sections for kinetic equations precisely we will first prove a generalized version of the uncertainty principle and then apply it to justify rigorously the smoothing properties of solutions to some kinetic equations in particular we give some estimates on the regularity of solutions in sobolev spaces w r t all variables for both linearized and nonlinear space inhomogeneous boltzmann equations without angular cutoff as well as the linearized space inhomogeneous landau equation

degree correlations in random geometric graphs spatially embedded networks are important in several disciplines the prototypical spatial network we assume is the random geometric graph of which many properties are known here we present new results for the two point degree correlation function in terms of the clustering coefficient of the graphs for two dimensional space in particular with extensions to arbitrary finite dimensions

new heuristic percolation criterion for continuum systems a heuristic criterion is introduced and applied for derivation of percolation thresholds in continuum systems the criterion is comprehensive in the sense that it applies to systems with diverse bonding criteria as well as to systems with a correlated distribution of sites its usefulness is manifested by the excellent quantitative agreement between the values obtained and available monte carlo data

stochastic comparison of random vectors with a common copula we consider two random vectorsx andy such that the components of x are dominated in the convex order by the corresponding components of y we want to find conditions under which this implies that any positive linear combination of the components of x is dominated in the convex order by the same positive linear combination of the components of y this problem has a motivation in the comparison of portfolios in terms of risk the conditions for the above dominance will concern the dependence structure of the two random vectorsx andy namely the two random vectors will have a common copula and will be conditionally increasing this new concept of dependence is strictly related to the idea of conditionally increasing in sequence but in addition it is invariant under permutation we will actually prove that under the above conditions x will be dominated byy in the directionally convex order which yields as a corollary the dominance for positive linear combinations this result will be applied to a portfolio optimization problem

precise determination of the critical threshold and exponents in a three dimensional continuum percolation model we present a large scale computer simulation of the prototypical three dimensional continuum percolation model consisting of a distribution of overlapping spatially uncorrelated spheres by using simulations of up to particles and studying the finite size scaling of various effective percolation thresholds we obtain a value of this value is significantly smaller than the values obtained for simulations that have been carried out using smaller systems employing this value of and systems of size l 160 relative to a sphere of unit radius we also obtain estimates of the critical exponents and for the continuum system and show that the values are different than those obtained using previous values of

initial boundary value problem for one dimensional wave solutions of the boltzmann equation tai ping liu and his collaborator shih hsien yu solve the initial boundary value problem for the boltzmann equation because of the inclusion of microscopic velocity in the kinetic theory the boltzmann equation can model richer and physically more realistic boundary effects more so than the fluid equations such as the navier stokes this motivates our study the reflection of fluid waves off the boundary is indicated in the figure for the case only one of the euler wave velocities is positive

graph bipartitioning and spin glasses on a random network of fixed finite valence the authors study the problem of bipartitioning a random graph of fixed finite valence using a mean field replica symmetric theory of an ising ferromagnet with zero magnetisation constraint the thermodynamics is determined by the probability distribution of an auxiliary field the expression for the ground state energy agrees with that proposed by mezard and parisi 1986 using a cavity field method but their expression for the fraction of crazy spins is reinterpreted

ambiguities in recurrence based complex network representations of time series recently different approaches have been proposed for studying basic properties of time series from a complex network perspective in this work the corresponding potentials and limitations of networks based on recurrences in phase space are investigated in some detail we discuss the main requirements that permit a feasible system theoretic interpretation of network topology in terms of dynamically invariant phase space properties possible artifacts induced by disregarding these requirements are pointed out and systematically studied finally a rigorous interpretation of the clustering coefficient and the betweenness centrality in terms of invariant objects is proposed

multivariate stochastic orderings and positive dependence in reliability theory stochastic orderings are introduced for vectors of lifetimes based on hazard rates and cumulative hazard functions relations are established between those orderings and other ones such as the totally positive ordering of karlin and rinott 1980 positive dependence notions in reliability theory are recast in terms of the new stochastic orderings relationships are also established between the notion of multivariate total positivity and the notion of hazard increase upon failure

topological versus spectral properties of random geometric graphs in this work we perform a detailed statistical analysis of topological and spectral properties of random geometric graphs rggs a graph model used to study the structure and dynamics of complex systems embedded in a two dimensional space rggs g n l consist of n vertices uniformly and independently distributed on the unit square where two vertices are connected by an edge if their euclidian distance is less than or equal to the connection radius l 0 sqrt 2 to evaluate the topological properties of rggs we chose two well known topological indices the randic index r g and the harmonic index h g we characterize the spectral and eigenvector properties of the corresponding randomly weighted adjacency matrices by the use of random matrix theory measures the ratio between consecutive eigenvalue spacings the inverse participation ratios and the information or shannon entropies s g first we review the scaling properties of the averaged measures topological and spectral on rggs then we show that i the averaged scaled indices r g and h g are highly correlated with the average number of nonisolated vertices v g and ii surprisingly the averaged scaled shannon entropy s g is also highly correlated with v g therefore we suggest that very reliable predictions of eigenvector properties of rggs could be made by computing topological indices

spatial network surrogates for disentangling complex system structure from spatial embedding of nodes networks with nodes embedded in a metric space have gained increasing interest in recent years the effects of spatial embedding on the networks structural characteristics however are rarely taken into account when studying their macroscopic properties here we propose a hierarchy of null models to generate random surrogates from a given spatially embedded network that can preserve certain global and local statistics associated with the nodes embedding in a metric space comparing the original network s and the resulting surrogates global characteristics allows one to quantify to what extent these characteristics are already predetermined by the spatial embedding of the nodes and links we apply our framework to various real world spatial networks and show that the proposed models capture macroscopic properties of the networks under study much better than standard random network models that do not account for the nodes spatial embedding depending on the actual performance of the proposed null models the networks are categorized into different classes since many real world complex networks are in fact spatial networks the proposed approach is relevant for disentangling the underlying complex system structure from spatial embedding of nodes in many fields ranging from social systems over infrastructure and neurophysiology to climatology

time dependent limited penetrable visibility graph analysis of nonstationary time series abstract recent years have witnessed the development of visibility graph theory which allows us to analyze a time series from the perspective of complex network we in this paper develop a novel time dependent limited penetrable visibility graph tdlpvg two examples using nonstationary time series from rr intervals and gas liquid flows are provided to demonstrate the effectiveness of our approach the results of the first example suggest that our tdlpvg method allows characterizing the time varying behaviors and classifying heart states of healthy congestive heart failure and atrial fibrillation from rr interval time series for the second example we infer tdlpvgs from gas liquid flow signals and interestingly find that the deviation of node degree of tdlpvgs enables to effectively uncover the time varying dynamical flow behaviors of gas liquid slug and bubble flow patterns all these results render our tdlpvg method particularly powerful for characterizing the time varying features underlying realistic complex systems from time series

signatures of chaotic and stochastic dynamics uncovered with recurrence networks an old and important problem in the field of nonlinear time series analysis entails the distinction between chaotic and stochastic dynamics recently recurrence networks have been proposed as a t

random geometric graphs in high dimension many machine learning algorithms used for dimensional reduction and manifold learning leverage on the computation of the nearest neighbors to each point of a data set to perform their tasks these proximity relations define a so called geometric graph where two nodes are linked if they are sufficiently close to each other random geometric graphs where the positions of nodes are randomly generated in a subset of r d offer a null model to study typical properties of data sets and of machine learning algorithms up to now most of the literature focused on the characterization of low dimensional random geometric graphs whereas typical data sets of interest in machine learning live in high dimensional spaces d 10 2 in this work we consider the infinite dimensions limit of hard and soft random geometric graphs and we show how to compute the average number of subgraphs of given finite size k e g the average number of k cliques this analysis highlights that local observables display different behaviors depending on the chosen ensemble soft random geometric graphs with continuous activation functions converge to the naive infinite dimensional limit provided by erdos renyi graphs whereas hard random geometric graphs can show systematic deviations from it we present numerical evidence that our analytical results exact in infinite dimensions provide a good approximation also for dimension d 10

positive dependence properties of conditionally independent random lifetimes conditions under which conditionally independent random variables are positive dependent are described in this paper for example it is shown that if the conditionally independent random variables increase in the hazard rate sense in the condition th en they are wbf weakened by failures it is also shown that if the conditionally independent random variables increase in the likelihood ratio sense in the condition then they are mtp2 multivariate totally positive of order 2 the results are given for the cases in which the condition is either a random variable or a random vector some applications in the areas of imperfect repair budget allocation random environments and discrete time filtering illustrate the theory

uncovering dynamic behaviors underlying experimental oil water two phase flow based on dynamic segmentation algorithm abstract characterizing complex dynamic behaviors arising from various inclined oil water two phase flow patterns is a challenging problem in the fields of nonlinear dynamics and fluid mechanics we systematically carried out inclined oil water two phase flow experiments for measuring the time series conductance fluctuating signals of different flow patterns we using the dynamic segmentation algorithm incorporating with phase space reconstruction analyze the measured experimental signals to uncover the dynamic behaviors underlying different flow patterns specifically given a time series from a two phase flow we move a sliding pointer over the time series and for each position of the pointer we calculate the dynamic difference measure of the phase space orbits generated from the segment to the left and to the right of the pointer a number of experimental signals under different flow conditions are investigated in order to reveal the dynamical characteristics of inclined oil water flows the results indicate that the heterogeneity of dynamic difference measure series is sensitive to the transition among different flow patterns and the standard deviation of dynamic difference measure series can yield quantitative insights into the nonlinear dynamics of the two phase flow these properties render the dynamic segmentation algorithm based approach particularly useful for uncovering the dynamic behaviors of inclined oil water two phase flows

boltzmann equation micro macro decompositions and positivity of shock profiles we introduce an elementary energy method for the boltzmann equation based on a decomposition of the equation into macroscopic and microscopic components the decomposition is useful for the study of time asymptotic stability of nonlinear waves the wave location is determined by the macroscopic equation the microscopic component has an equilibrating property the coupling of macroscopic and microscopic components gives rise naturally to the dissipations similar to those obtained by the chapman enskog expansion our main result is the establishment of the positivity of shock profiles for the boltzmann equation this is shown by the time asymptotic approach and the maximal principle for the collision operator

the boltzmann equation without angular cutoff in the whole space i global existence for soft potential it is known that the singularity in the non cutoff cross section of the boltzmann equation leads to the gain of regularity and gain of weight in the velocity variable by defining and analyzing a non isotropy norm which precisely captures the dissipation in the linearized collision operator we first give a new and precise coercivity estimate for the non cutoff boltzmann equation for general physical cross sections then the cauchy problem for the boltzmann equation is considered in the framework of small perturbation of an equilibrium state in this part for the soft potential case in the sense that there is no positive power gain of weight in the coercivity estimate on the linearized operator we derive some new functional estimates on the nonlinear collision operator together with the coercivity estimates we prove the global existence of classical solutions for the boltzmann equation in weighted sobolev spaces

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

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

are randomly grown graphs really random we analyze a minimal model of a growing network at each time step a new vertex is added then with probability ensuremath delta two vertices are chosen uniformly at random and joined by an undirected edge this process is repeated for t time steps in the limit of large t the resulting graph displays surprisingly rich characteristics in particular a giant component emerges in an infinite order phase transition at ensuremath delta 1 8 at the transition the average component size jumps discontinuously but remains finite in contrast a static random graph with the same degree distribution exhibits a second order phase transition at ensuremath delta 1 4 and the average component size diverges there these dramatic differences between grown and static random graphs stem from a positive correlation between the degrees of connected vertices in the grown graph older vertices tend to have higher degree and to link with other high degree vertices merely by virtue of their age we conclude that grown graphs however randomly they are constructed are fundamentally different from their static random graph counterparts

analytical framework for recurrence network analysis of time series recurrence networks are a powerful nonlinear tool for time series analysis of complex dynamical systems while there are already many successful applications ranging from medicine to paleoclimatology a solid theoretical foundation of the method has still been missing so far here we interpret an ensuremath varepsilon recurrence network as a discrete subnetwork of a continuous graph with uncountably many vertices and edges corresponding to the system s attractor this step allows us to show that various statistical measures commonly used in complex network analysis can be seen as discrete estimators of newly defined continuous measures of certain complex geometric properties of the attractor on the scale given by ensuremath varepsilon in particular we introduce local measures such as the ensuremath varepsilon clustering coefficient mesoscopic measures such as ensuremath varepsilon motif density path based measures such as ensuremath varepsilon betweennesses and global measures such as ensuremath varepsilon efficiency this new analytical basis for the so far heuristically motivated network measures also provides an objective criterion for the choice of ensuremath varepsilon via a percolation threshold and it shows that estimation can be improved by so called node splitting invariant versions of the measures we finally illustrate the framework for a number of archetypical chaotic attractors such as those of the bernoulli and logistic maps periodic and two dimensional quasiperiodic motions and for hyperballs and hypercubes by deriving analytical expressions for the novel measures and comparing them with data from numerical experiments more generally the theoretical framework put forward in this work describes random geometric graphs and other networks with spatial constraints which appear frequently in disciplines ranging from biology to climate science

efficient measurement of the percolation threshold for fully penetrable discs we study the percolation threshold for fully penetrable discs by measuring the average location of the frontier for a statistically inhomogeneous distribution of fully penetrable discs we use two different algorithms to efficiently simulate the frontier including the continuum analogue of an algorithm previously used for gradient percolation on a square lattice we find that c 0 676 339 0 000 004 thus providing an extra significant digit of accuracy to this constant some figures in this article appear in colour in the electronic version see www iop org

fractal and complex network analyses of protein molecular dynamics based on protein molecular dynamics we investigate the fractal properties of energy pressure and volume time series using the multifractal detrended fluctuation analysis mf dfa and the topological and fractal properties of their converted horizontal visibility graphs hvgs the energy parameters of protein dynamics we considered are bonded potential angle potential dihedral potential improper potential kinetic energy van der waals potential electrostatic potential total energy and potential energy the shape of the h q h q curves from mf dfa indicates that these time series are multifractal the numerical values of the exponent h 2 h 2 of mf dfa show that the series of total energy and potential energy are non stationary and anti persistent the other time series are stationary and persistent apart from series of pressure with h 0 5h 0 5 indicating the absence of long range correlation the degree distributions of their converted hvgs show that these networks are exponential the results of fractal analysis show that fractality exists in these converted hvgs for each energy pressure or volume parameter it is found that the values of h 2 h 2 of mf dfa on the time series exponent of the exponential degree distribution and fractal dimension dbdb of their converted hvgs do not change much for different proteins indicating some universality we also found that after taking average over all proteins there is a linear relationship between h 2 h 2 from mf dfa on time series and db db of the converted hvgs for different energy pressure and volume

extremal optimization of graph partitioning at the percolation threshold the benefits of a recently proposed method to approximate hard optimization problems are demonstrated on the graph partitioning problem the performance of this new method called extremal optimization eo is compared with simulated annealing sa in extensive numerical simulations while generally a complex np hard problem the optimization of the graph partitions is particularly difficult for sparse graphs with average connectivities near the percolation threshold at this threshold the relative error of sa for large graphs is found to diverge relative to eo at equalized runtime on the other hand eo based on the extremal dynamics of self organized critical systems reproduces known results about optimal partitions at this critical point quite well