counting lattice animals a parallel attack a parallel algorithm for the enumeration of isolated connected clusters on a regular lattice is presented the algorithm has been implemented on 17 risc based workstations to calculate the perimeter polynomials for the plane triangular lattice up to clustersizes 21 new data for perimeter polynomials ds up tod21 total number of clustersgs up tog22 and coefficientsbr in the low density series expansion of the mean cluster size up tob21 are given

combinatorial approach to feynman path integration combinatorial relations can be used to convert the non relativistic time sliced feynman path integral into perturbation expansions these methods reveal that when the time interval is sliced into n increments each order of perturbation theory sustains an error o 1 square root n in this way the author provides exact path integral results for the following potentials delta function comb finite well tunnelling barrier and a generalized exponential cusp for the tunnelling barrier it is seen how the celebrated 1 reflection factor arises in the limit of infinite barrier height the one dimensional coulomb problem is solved as a limiting case of the exponential cusp in addition for power potentials the author indicates how this path integral approach yields sometimes divergent nevertheless asymptotic perturbation expansions

lattice animals a fast enumeration algorithm and new perimeter polynomials a fast computer algorithm for enumerating isolated connected clusters on a regular lattice and its fortran implementation are presented new perimeter polynomials are calculated for the square the triangular the simple cubic and the square lattice with next nearest neighbors

distribution of infected mass in disease spreading in scale free networks we use scale free networks to study properties of the infected mass m of the network during a spreading process as a function of the infection probability q and the structural scaling exponent we use the standard sir model and investigate in detail the distribution of m we find that for dense networks this function is bimodal while for sparse networks it is a smoothly decreasing function with the distinction between the two being a function of q we thus recover the full crossover transition from one case to the other this has a result that on the same network a disease may die out immediately or persist for a considerable time depending on the initial point where it was originated thus we show that the disease evolution is significantly influenced by the structure of the underlying population

spreading dynamics of forget remember mechanism we study extensively the forget remember mechanism frm for message spreading originally introduced in eur phys j b 62 247 2008 epjbfy1434 602810 1140 epjb e2008 00139 4 the freedom of specifying forget remember functions governing the frm can enrich the spreading dynamics to a very large extent the master equation is derived for describing the frm dynamics by applying the mean field techniques we have shown how the steady states can be reached under certain conditions which agrees well with the monte carlo simulations the distributions of forget and remember times can be explicitly given when the forget remember functions take linear or exponential forms which might shed some light on understanding the temporal nature of diseases like flu for time dependent frm there is an epidemic threshold related to the frm parameters we have proven that the mean field critical transmissibility for the sis model and the critical transmissibility for the sir model are the lower and the the upper bounds of the critical transmissibility for the frm model respectively

self avoiding walks on scale free networks several kinds of walks on complex networks are currently used to analyze search and navigation in different systems many analytical and computational results are known for random walks on such networks self avoiding walks saw s are expected to be more suitable than unrestricted random walks to explore various kinds of real life networks here we study long range properties of random saw s on scale free networks characterized by a degree distribution p k ensuremath sim k ensuremath ensuremath gamma in the limit of large networks system size n ensuremath rightarrow ensuremath infty the average number s n of saw s starting from a generic site increases as ensuremath mu n with ensuremath mu k 2 k ensuremath 1 for finite n s n is reduced due to the presence of loops in the network which causes the emergence of attrition of the paths for kinetic growth walks the average maximum length l increases as a power of the system size l ensuremath sim n ensuremath alpha with an exponent ensuremath alpha increasing as the parameter ensuremath gamma is raised we discuss the dependence of ensuremath alpha on the minimum allowed degree in the network a similar power law dependence is found for the mean self intersection length of nonreversal random walks simulation results support our approximate analytical calculations

spreading and shortest paths in systems with sparse long range connections spreading according to simple rules e g of fire or diseases and shortest path distances are studied on d dimensional systems with a small density p per site of long range connections small world lattices the volume v t covered by the spreading quantity on an infinite system is exactly calculated in all dimensions as a function of time t from this the average shortest path distance l r can be calculated as a function of euclidean distance r it is found that l r approximately r for r r c the characteristic length r c which governs the behavior of shortest path lengths diverges logarithmically with l for all p 0

analyzing percolation of networks inspired by the 3x 1 problem abstract in this paper we investigate percolation in a sort of networks inspired by the observation of a collatz graph cg which is the network version of the famous 3 x 1 problem in mathematics the cg consists of positive integers that are connected based on the iteration relations actually we never mean to solve the 3 x 1 problem exactly but we observe it from the viewpoint of statistical physics we focus on the so called reduced collatz graph rcg that is a subgraph with all odd numbers since even numbers can be iterated into odd ones through 3 x 1 rules considering boundary conditions we obtain a special degree distribution of rcg for finite size set of odd integers and treat the infinite case as its limit with the percolation criterion through the approach of a generating function we determine the critical condition for the network ensemble of rcg furthermore we generalize the graph model with rcg type degree distributions beyond the 3 x 1 problem get the generic criterion of percolation and phase diagram for an ensemble of positive integer networks with rcg as its extreme case

topological properties of integer networks abstract inspired by pythagoras s belief that numbers represent the reality we study the topological properties of networks of composite numbers in which the vertices represent the numbers and two vertices are connected if and only if there exists a divisibility relation between them the network has a fairly large clustering coefficient c 0 34 which is insensitive to the size of the network the average distance between two nodes is shown to have an upper bound that is independent of the size of the network in contrast to the behavior in small world and ultra small world networks the out degree distribution is shown to follow a power law behavior of the form k 2 in addition these networks possess hierarchical structure as c k k 1 in accord with the observations of many real life networks

finite size effects for percolation on apollonian networks we study the percolation problem on the apollonian network model the apollonian networks display many interesting properties commonly observed in real network systems such as small world behavior scale free distribution and a hierarchical structure by taking advantage of the deterministic hierarchical construction of these networks we use the real space renormalization group technique to write exact iterative equations that relate percolation network properties at different scales more precisely our results indicate that the percolation probability and average mass of the percolating cluster approach the thermodynamic limit logarithmically we suggest that such ultraslow convergence might be a property of hierarchical networks since real complex systems are certainly finite and very commonly hierarchical we believe that taking into account finite size effects in real network systems is of fundamental importance

biased percolation on scale free networks biased degree dependent percolation was recently shown to provide strategies for turning robust networks fragile and vice versa here we present more detailed results for biased edge percolation on scale free net works we assume a network in which the probability for an edge between nodes i and j to be retained is proportional to kikjwith ki and kj the degrees of the nodes we discuss two methods of network recon struction sequential and simultaneous and investigate their properties by analytical and numerical means the system is examined away from the percolation transition where the size of the giant cluster is obtained and close to the transition where nonuniversal critical exponents are extracted using the generating functions method the theory is found to agree quite well with simulations by presenting an extension of the fortuin kasteleyn construction we find that biased percolation is well described by the q 1 limit of the q state potts model with inhomogeneous couplings

families and clustering in a natural numbers network we develop a network in which the natural numbers are the vertices the decomposition of natural numbers by prime numbers is used to establish the connections we perform data collapse and show that the degree distribution of these networks scales linearly with the number of vertices we explore the families of vertices in connection with prime numbers decomposition we compare the average distance of the network and the clustering coefficient with the distance and clustering coefficient of the corresponding random graph in case we set connections among vertices each time the numbers share a common prime number the network has properties similar to a random graph if the criterion for establishing links becomes more selective only prime numbers greater than p l are used to establish links where the network has high clustering coefficient

comment on families and clustering in a natural numbers network corso phys rev e 69 036106 2004 constructs a family of graphs from subsets of the natural numbers and numerically estimates diameter degree and clustering we give exact asymptotic formulas for these quantities and thereby argue that number theory is a more appropriate tool than simulation

improvement of the robustness on geographical networks by adding shortcuts abstract in a topological structure affected by geographical constraints on liking the connectivity is weakened by constructing local stubs with small cycles a something of randomness to bridge them is crucial for the robust network design in this paper we numerically investigate the effects of adding shortcuts on the robustness in geographical scale free network models under a similar degree distribution to the original one we show that a small fraction of shortcuts is highly contribute to improve the tolerance of connectivity especially for the intentional attacks on hubs the improvement is equivalent to the effect by fully rewirings without geographical constraints on linking even in the realistic internet topologies these effects are virtually examined

epidemics and percolation in small world networks we study some simple models of disease transmission on small world networks in which either the probability of infection by a disease or the probability of its transmission is varied or both the resulting models display epidemic behavior when the infection or transmission probability rises above the threshold for site or bond percolation on the network and we give exact solutions for the position of this threshold in a variety of cases we confirm our analytic results by numerical simulation

the reflection of narrow slow quantum packets from mirrors we study the reflection of narrow in space quantum wavepackets from nonabsorptive mirrors totally or partially reflecting if the initial mean value of the momentum component perpendicular to the mirror surface is less than the momentum uncertainty then the mean value of this component gradually increases with time at the expense of shrinking the packet in the momentum space as a consequence very slow particles moving initially in the direction parallel to the mirror surface will be deflected to appreciable angles even when they pass at macroscopical distances from the mirror we give analytical expressions describing the asymptotical behaviour of wavefunctions and density matrices in the coordinate and momentum representations for arbitrary narrow initial packets we show that the asymptotical mean values and variances do not depend on the phases of the complex reflection and transmission coefficients moreover they are insensitive to the concrete form of the reflective potential in the case of totally reflecting mirrors for partially reflecting mirrors we introduce the concept of conditional wavefunctions and mean values the dependences of the asymptotical values of different quantities characterizing the packet e g the momentum transformation coefficient and the invariant uncertainty product on the parameters of the reflecting potential the height and width or characteristic length of the transition region are analysed in the examples of the potentials of epstein s type and their limit cases the ideal reflecting wall and the delta potential a possibility for the verification of the effect of quantum deflection in experiments with ultracold atoms is briefly discussed

finding rumor sources on random trees we consider the problem of detecting the source of a rumor which has spread in a network using only observations about which set of nodes are infected with the rumor and with no information as to when these nodes became infected in a recent work shah and zaman 2010 this rumor source detection problem was introduced and studied the authors proposed the graph score function rumor centrality as an estimator for detecting the source they establish it to be the maximum likelihood estimator with respect to the popular susceptible infected si model with exponential spreading times for regular trees they showed that as the size of the infected graph increases for a path graph 2 regular tree the probability of source detection goes to 0 and for d regular trees with d 3 the probability of detection say d remains bounded away from 0 and is less than 1 2 however their results stop short of providing insights for the performance of the rumor centrality estimator in more general settings such as irr

coupled effects of local movement and global interaction on contagion abstract by incorporating segregated spatial domain and individual based linkage into the sis susceptible infected susceptible model we propose a generalized epidemic model which can change from the territorial epidemic model to the networked epidemic model the role of the individual based linkage between different spatial domains is investigated as we adjust the timescale parameter from 0 to unity which represents the degree of activation of the individual based linkage three regions are found within the region of 0 0 02 the epidemic is determined by local movement and is sensitive to the timescale within the region of 0 02 0 5 the epidemic is insensitive to the timescale within the region of 0 5 1 the outbreak of the epidemic is determined by the structure of the individual based linkage as we keep an eye on the first region the role of activating the individual based linkage in the present model is similar to the role of the shortcuts in the two dimensional small world network only activating a small number of the individual based linkage can prompt the outbreak of the epidemic globally the role of narrowing segregated spatial domain and reducing mobility in epidemic control is checked these two measures are found to be conducive to curbing the spread of infectious disease only when the global interaction is suppressed a log log relation between the change in the number of infected individuals and the timescale is found by calculating the epidemic threshold and the mean first encounter time we heuristically analyze the microscopic characteristics of the propagation of the epidemic in the present model

dependency of percolation critical exponents on the exponent of power law size distribution abstract the standard percolation theory uses objects of the same size moreover it has long been observed that the percolation properties of the systems with a finite distribution of sizes are controlled by an effective size and consequently the universality of the percolation theory is still valid in this study the effect of power law size distribution on the critical exponents of the percolation theory of the two dimensional models is investigated two different object shapes i e stick shaped and square are considered these two shapes are the representative of the fractures in fracture reservoirs and the sandbodies in clastic reservoirs the finite size scaling arguments are used for the connectivity to determine the dependency of the critical exponents on the power law exponent in particular the deviations of percolation exponents from their universal values as well as the connectivity behavior of such systems are investigated numerically as a result this extends the applicability of the conventional percolation approach to study the connectivity of systems with a very broad size distribution

exact solution of site and bond percolation on small world networks we study percolation on small world networks which has been proposed as a simple model of the propagation of disease the occupation probabilities of sites and bonds correspond to the susceptibility of individuals to the disease and the transmissibility of the disease respectively we give an exact solution of the model for both site and bond percolation including the position of the percolation transition at which epidemic behavior sets in the values of the critical exponents governing this transition the mean and variance of the distribution of cluster sizes disease outbreaks below the transition and the size of the giant component epidemic above the transition

intermittency from collatz s itineraries and complexity indicators we introduce a new type of discrete map on positive reals showing peculiar kinds of complexity they combine a large scale component collatz s itineraries on integers with a small scale component standard chaotic systems such as logistic maps or random noise usual characterizations for example by lyapunov exponents prove senseless or deceptive other indicators are used and in particular a new entropy is introduced to avoid difficulties regarding invariant measures all computations converge on the conclusion that the crucial point in qualifying the type of complexity and predictability does not consist in the alternative between deterministic perturbation or random noise but in the peculiar way the large scale component reacts on the small scale component altering its genericity possible connections with systems of physical interest are indicated

mixing patterns in networks we study assortative mixing in networks the tendency for vertices in networks to be connected to other vertices that are like or unlike them in some way we consider mixing according to discrete characteristics such as language or race in social networks and scalar characteristics such as age as a special example of the latter we consider mixing according to vertex degree i e according to the number of connections vertices have to other vertices do gregarious people tend to associate with other gregarious people we propose a number of measures of assortative mixing appropriate to the various mixing types and apply them to a variety of real world networks showing that assortative mixing is a pervasive phenomenon found in many networks we also propose several models of assortatively mixed networks both analytic ones based on generating function methods and numerical ones based on monte carlo graph generation techniques we use these models to probe the properties of networks as their level of assortativity is varied in the particular case of mixing by degree we find strong variation with assortativity in the connectivity of the network and in the resilience of the network to the removal of vertices

phase transition in a stochastic prime number generator we introduce a stochastic algorithm that acts as a prime number generator the dynamics of this algorithm gives rise to a continuous phase transition which separates a phase where the algorithm is able to reduce a whole set of integers into primes and a phase where the system reaches a frozen state with low prime density we present both numerical simulations and an analytical approach in terms of an annealed approximation by means of which the data are collapsed a critical slowing down phenomenon is also outlined

multiparticle random walks on a deformable medium multiparticle random walks on a deformable medium have been investigated in 2 1 dimensions the time evolution of the particle distribution is studied the results show that the randomly distributed particles in the beginning will be self organized into a cluster pattern in the intermediate stage and then return to the random distribution pattern in the late stage the dependence of the clustering degree on the stiffness parameter of medium ensuremath alpha stability parameter of systems ensuremath beta and average particle density ensuremath rho 0 is also investigated there exists an optimal clustering stability ensuremath beta p at which the system has the strongest clustering ability and corresponds to a maximum clustering coefficient ensuremath gamma p the dependence of the optimal clustering coefficient ensuremath gamma p on the stiffness ensuremath alpha and particle density ensuremath rho 0 is obtained and the landscape of the medium generated by particles is also investigated

maximal planar networks with large clustering coefficient and power law degree distribution in this article we propose a simple rule that generates scale free networks with very large clustering coefficient and very small average distance these networks are called random apollonian networks rans as they can be considered as a variation of apollonian networks we obtain the analytic results of power law exponent ensuremath gamma 3 and clustering coefficient c frac 46 3 ensuremath 36 phantom rule 0 2em 0ex mathrm ln phantom rule 0 2em 0ex frac 3 2 ensuremath approx 0 74 which agree with the simulation results very well we prove that the increasing tendency of average distance of rans is a little slower than the logarithm of the number of nodes in rans since most real life networks are both scale free and small world networks rans may perform well in mimicking the reality the rans possess hierarchical structure as c k ensuremath sim k ensuremath 1 that are in accord with the observations of many real life networks in addition we prove that rans are maximal planar networks which are of particular practicability for layout of printed circuits and so on the percolation and epidemic spreading process are also studied and the comparisons between rans and barab asi albert ba as well as newman watts nw networks are shown we find that when the network order n the total number of nodes is relatively small as n ensuremath sim 10 4 the performance of rans under intentional attack is not sensitive to n while that of ba networks is much affected by n and the diseases spread slower in rans than ba networks in the early stage of the suseptible infected process indicating that the large clustering coefficient may slow the spreading velocity especially in the outbreaks

magnetic models on apollonian networks thermodynamic and magnetic properties of ising models defined on the triangular apollonian network are investigated this and other similar networks are inspired by the problem of covering a euclidian domain with circles of maximal radii maps for the thermodynamic functions in two subsequent generations of the construction of the network are obtained by formulating the problem in terms of transfer matrices numerical iteration of this set of maps leads to very precise values for the thermodynamic properties of the model different choices for the coupling constants between only nearest neighbors along the lattice are taken into account for both ferromagnetic and antiferromagnetic constants long range magnetic ordering is obtained with exception of a size dependent effective critical behavior of the correlation length no evidence of asymptotic criticality was detected

free electron gas in the apollonian network multifractal energy spectrum and its thermodynamic fingerprints we study the free electron gas in an apollonian network within the tight binding framework the scale free and small world character of the underlying lattice is known to result in a quite structured energy spectrum with deltalike singularities gaps and minibands after an exact numerical diagonalization of the corresponding adjacency matrix of the network with a finite number of generations we employ a scaling analysis of the moments of the density of states to characterize its multifractality and report the associated singularity spectrum the fractal nature of the energy spectrum is also shown to be reflected in the thermodynamic behavior by logarithmic modulations on the temperature dependence of the specific heat the absence of chiral symmetry of the apollonian network leads to distinct thermodynamic behaviors due to electrons and holes thermal excitations

the influence of preferential attachment on evolving networks abstract with the consideration of local events and preferential attachment the extended model proposed by albert and barabasi is studied firstly the extended model is built only by local events it is found that without preferential attachment the networks connectivity distribution decays exponentially for large connectivity and the decaying exponent varies with different probabilities of the addition of new links and rewiring secondly the extended model with partial preferential attachment is studied when preferential attachment only presents in the addition of new links or new nodes there is scale free regime in the phase diagram while preferential attachment is brought into the rewiring there coexist two regimes i e the exponential regime and the scale free regime thirdly fitness is introduced to the extended model with complete preferential attachment it is obtained that the network s connectivity distribution is a result affected by many factors including the fitness the local events and preferential attachment

enumerations of lattice animals and trees we have developed an improved algorithm that allows us to enumerate the number of site animals on the square lattice up to size 46 we also calculate the number of lattice trees up to size 44 and the radius of gyration of both lattice animals and trees up to size 42 analysis of the resulting series yields an improved estimate 4 062570 8 for the growth constant of lattice animals and 0 3 795254 8 for the growth constant of trees and confirms to a very high degree of certainty that both the animal and tree generating functions have a logarithmic divergence analysis of the radius of gyration series yields the estimate 0 64115 5 for the size exponent

effect of dimensionality on the percolation thresholds of various d dimensional lattices we show analytically that the 0 1 1 1 and 2 1 pad e approximants of the mean cluster number s p for site and bond percolation on general d dimensional lattices are upper bounds on this quantity in any euclidean dimension d where p is the occupation probability these results lead to certain lower bounds on the percolation threshold p c that become progressively tighter as d increases and asymptotically exact as d becomes large these lower bound estimates depend on the structure of the d dimensional lattice and whether site or bond percolation is being considered we obtain explicit bounds on p c for both site and bond percolation on five different lattices d dimensional generalizations of the simple cubic body centered cubic and face centered cubic bravais lattices as well as the d dimensional generalizations of the diamond and kagom e or pyrochlore non bravais lattices these analytical estimates are used to assess available simulation results across dimensions up through d 13 in some cases it is noteworthy that the tightest lower bound provides reasonable estimates of p c in relatively low dimensions and becomes increasingly accurate as d grows we also derive high dimensional asymptotic expansions for p c for the 10 percolation problems and compare them to the bethe lattice approximation finally we remark on the radius of convergence of the series expansion of s in powers of p as the dimension grows

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

dynamics of epidemics on random networks this paper examines how diseases on random networks spread in time the disease is described by a probability distribution function for the number of infected and recovered individuals and the probability distribution is described by a generating function the time development of the disease is obtained by iterating the generating function in cases where the disease can expand to an epidemic the probability distribution function is the sum of two parts one that is static at long times and another whose mean grows exponentially the time development of the mean number of infected individuals is obtained analytically when epidemics occur the probability distributions are very broad and the uncertainty in the number of infected individuals at any given time is typically larger than the mean number of infected individuals

phase transition in the countdown problem we present a combinatorial decision problem inspired by the celebrated quiz show called countdown t hat involves the computation of a given target number t from a set of k randomly chosen integers along with a set of arithmetic operations we find that the probability of winning the game evidences a threshold phenomenon that can be understood in the terms of an algorithmic phase transition as a function of the set size k numerical simulations show that such probability sharply transitions from zero to one at some critical value of the control parameter hence separating the algorithm s parameter space in different phases we also find that the system is maximally efficient close to the critical point we derive analytical expressions that match the numerical results for finite size and permit us to extrapolate the behavior in the thermodynamic limit

analytical approach to directed sandpile models on the apollonian network we investigate a set of directed sandpile models on the apollonian network which are inspired by the work of dhar and ramaswamy phys rev lett 63 1659 1989 on euclidian lattices they are characterized by a single parameter q which restricts the number of neighbors receiving grains from a toppling node due to the geometry of the network two and three point correlation functions are amenable to exact treatment leading to analytical results for avalanche distributions in the limit of an infinite system for q 1 2 the exact recurrence expressions for the correlation functions are numerically iterated to obtain results for finite size systems when larger values of q are considered finally a detailed description of the local flux properties is provided by a multifractal scaling analysis

multifractality of prime numbers abstract the multifractal formalism is applied to prime numbers the spectrum of critical indices is found to be contained in the interval min 1 where min tends to 1 for increasing sets of numbers besides the scaling of moments with respect to the length of intervals the scaling with respect to the sizes of subsets of natural numbers is also considered we have found the cusps in the plots of the functions and we claim that they are not caused by numerical roundings but they are a real effect besides the computer method some analytical calculations are presented

a generalization of the collatz problem building cycles and a stochastic approach the 3x 1 2 problem is generalized into the n furcation problem l i x m i n where i 0 1 n 1 it is shown that under some constraints on l i and m i the main bijection property between the k less significant digits of the seed written in base n and the sequence of generalized parities of the k first iterates is preserved this property is used to investigate a stochastic treatment of ensemble of large value seeds the bijection property predicts stochasticity for a number of iterations equals to the number of significant digits of the seed in fact the stochastic approach is valid for much larger numbers a property which is more easily shown by using increasing sequences than decreasing ones finally we extend the stochastic approach to cases where the bijection theorem does not hold introducing the matrix giving the probability that a j number where j is the last significant digit of this number written in base n gives a i number iterate

quantum treatment of a class of time dependent potentials the time dependent potential v x f t is studied by path integrals it is shown that the problem can be mapped into the static form of the potential plus a linear term with a time dependent coefficient after the presentation of the general formulation some exactly solvable examples are discussed a perturbative treatment is also suggested

the susceptible infected recovered model on a euclidean network we consider a susceptible infected recovered sir epidemic model on a euclidean network in one dimension in which nodes at a distance l are connected with probability p l l in addition to nearest neighbors the topology of the network changes as is varied and its effect on the sir model is studied r t the recovered fraction of population up to time t and the total duration of the epidemic are calculated for different values of the infection probabilities q and a threshold behavior is observed for all up to 2 0 above the threshold value q qc the saturation value rsat attains a finite value both rsat and show scaling behavior in a finite system of size n and qc is constant for 0 qc the results are compared to static percolation phenomena and also to mean field results for finite systems discussions on the properties of the euclidean network are made in the light of the present results

3x 1 search programs abstract algorithms for computing peaks of certain statistics related to the 3x 1 problem are described along with data on such peaks up to 56 trillion 5 6 10 13 the data result from several years of computation the design of the algorithms used illustrates several techniques for program optimization

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

statistical properties of an iterated arithmetic mapping we study the 3x 1 2 problem from a probabilistic viewpoint and show a forgetting mechanism for the last k binary digits of the seed after k iterations the problem is subsequently generalized to a trifurcation process the lx m 3 problem finally the sequence of a set of seeds is empirically shown to be equivalent to a random walk of the variable log sub 2 x or log sub 3 x though computer simulations

stabilization of a steady state in network oscillators by using diffusive connections with two long time delays the present study shows that diffusive connections with two long time delays can induce the stabilization of a steady state in network oscillators a linear stability analysis shows that if the two delay times retain a proportional relation with a certain bias the stabilization can be achieved independent of the delay times furthermore a simple systematic procedure for designing the coupling strength and the delay times in the connections is proposed the procedure has the following two advantages one can employ time delays as long as one wants and the stabilization can be achieved independently of its network topology our analytical results are applied to the well known double scroll circuit model on a small world network

exact propagators for atom laser interactions a class of exact propagators describing the interaction of an n level atom with a set of on resonance lasers is obtained by means of the laplace transform method state selective mirrors are described in the limit of strong lasers for the two level case the transient effects arising as a result of the interaction between both a semi infinite beam and a wavepacket with the on resonance laser are examined

ising model on the apollonian network with node dependent interactions this work considers an ising model on the apollonian network where the exchange constant j i j approximately 1 k i k j mu between two neighboring spins i j is a function of the degree k of both spins using the exact geometrical construction rule for the network the thermodynamical and magnetic properties are evaluated by iterating a system of discrete maps that allows for very precise results in the thermodynamic limit the results can be compared to the predictions of a general framework for spin models on scale free networks where the node distribution p k approximately k gamma with node dependent interacting constants we observe that by increasing mu the critical behavior of the model changes from a phase transition at t infinity for a uniform system mu 0 to a t 0 phase transition when mu 1 in the thermodynamic limit the system shows no true critical behavior at a finite temperature for the whole mu or 0 interval the magnetization and magnetic susceptibility are found to present noncritical scaling properties

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

phase transition in the number partitioning problem universit at magdeburg institut fu r theoretische physik universit atsplatz 2 d 39106 magdeburg germany february 1 2008 number partitioning is an np complete problem of com binatorial optimization a statistical mechanics analysis re veals the existence of a phase transition that separates theeasy from the hard to solve instances and that re ects thepseudo polynomiality of number partitioning the phase di agram and the value of the typical ground state energy arecalculated 64 60 cn 02 60 pn 02 70 lq 89 80 h

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

delta function perturbations and neumann boundary conditions by path integration delta function perturbations and neumann boundary conditions are incorporated into the path integral formalism the starting point is consideration of the path integral representation for the one dimensional dirac particle together with a relativistic point interaction the nonrelativistic limit yields either the usual delta function or a delta function perturbation making their strengths infinitely repulsive one respectively obtains either dirichlet or neumann boundary conditions in the path integral

simulations of lattice animals and trees the scaling behaviour of randomly branched polymers in a good solvent is studied in two to nine dimensions using as microscopic models lattice animals and lattice trees on simple hypercubic lattices as a stochastic sampling method we use a biased sequential sampling algorithm with re sampling similar to the pruned enriched rosenbluth method perm used extensively for linear polymers essentially we start simulating percolation clusters either site or bond re weigh them according to the animal tree ensemble and prune or branch the further growth according to a heuristic fitness function in contrast to previous applications of perm this fitness function is not the weight with which the actual configuration would contribute to the partition sum but is closely related to it we obtain high statistics of animals with up to several thousand sites in all dimension 2 d 9 in addition to the partition sum number of different animals we estimate gyration radii and numbers of perimeter sites in all dimensions we verify the parisi sourlas prediction and we verify all exactly known critical exponents in dimensions 2 3 4 and 8 in addition we present the hitherto most precise estimates for growth constants in d 3 for clusters with one site attached to an attractive surface we verify for d 3 the superuniversality of the cross over exponent at the adsorption transition predicted by janssen and lyssy but not for d 2 there we find 0 480 4 instead of the conjectured 1 2 finally we discuss the collapse of animals and trees arguing that our present version of the algorithm is also efficient for some of the models studied in this context but showing that it is not very efficient for the classical model for collapsing animals