Damage spreading on the 3–12 lattice with competing Glauber and Kawasaki dynamics

The damage spreading of the Ising model on the 3–12 lattice with competing Glauber and Kawasaki dynamics is studied. The difference between the two kinds of nearest-neighboring spin interactions (interaction between two 12-gons, or interaction between a 12-gon and a triangle) are considered in the Hamiltonian. It is shown that the ratio of the interaction strengthF between the two kinds of interactions plays an important role in determining the critical temperature T_d of phase transition from frozen to chaotic. Two methods are used to introduce the bond dilution on the Ising model on the 3–12 lattice: regular and random. The maximum of the average damage spreading 〈D〉_max can approach values lower than 0.5 in both cases and the reason can be attributed to the ’survivors’ among the spins. We have also, for the first time, presented the phase diagram of the mixed G-K dynamics in the 3–12 lattice which shows what happens when going from pure Glauber to pure Kawasaki     

Time-dependent thermodynamic properties of the Ising model from damage spreading

The relationship between damage spreading and static thermodynamic properties in the Ising model developed by Coniglioet al. is here extended to include time-dependent thermodynamic quantities. We exploit this new result to measure the time-dependent spin correlation function from damage spreading in the Ising model with heat bath and Glauber dynamics. Until now, only static thermodynamic quantities have been correctly determined from damage spreading, and even then, only with heat bath dynamics. We also show that there are significant differences between the kinetics of damage spreading as found in heat bath and Glauber dynamics.     

Survivor statistics and damage spreading on social network with power-law degree distributions

Damage spreading(DS) of the random graph networks with power-law degree distributions is investigated using Glauber dynamics. Various subgraphs defined by the probability of acquaintance show distinct features in DS as measured by Hamming distance. A heuristic understanding of the long-time value of damage is achieved through an analysis of the survivor statistics. All survivors are dynamical, flipping in unison for the controlled sample and the damaged sample. Verification of these dynamic survivors is achieved through the introduction of a new measure of self-damage.     

Metastability and exponential approach to equilibrium for low-temperature stochastic Ising models

We present a proof of the exponential convergence to equilibrium of single-spin-flip stochastic dynamics for the two-dimensional Ising ferromagnet in the low-temperature case with not too small external magnetic fieldh uniformly in the volume and in the boundary conditions.     

Metastability and the Ising model

We discuss a recent theorem which establishes a precise connection between (i) the approximate degeneracy of the zero eigenvalue for the generator of the Glauber dynamics of the Ising model in a small nonzero field and below the critical temperature, (ii) the existence of a partition of the configuration space into a normal region and a metastable region. This enables us to demonstrate that the recent approach to metastability of Davies and Martin may be viewed as a simple (although in some ways fairly crude) approximation to the conventional approach. We also obtain what appear to be the first results concerning the stability of metastable states under small perturbations.     

Some new results on the two-dimensional kinetic Ising model in the phase coexistence region

We consider a Glauber dynamics reversible with respect to the two-dimensional Ising model in a finite square of sideL with open boundary conditions, in the absence of an external field and at large inverse temperature . We prove that the gap in the spectrum of the generator restricted to the invariant subspace of functions which are even under global spin flip is much larger than the true gap. As a consequence we are able to show that there exists a new time scalet _even, much smaller than the global relaxation timet _rel, such that, with large probability, any initial configuration first relaxes to one of the two phases in a time scale of ordert _even and only after a time scale of the order oft _rel does it reach the final equilibrium by jumping, via a large deviation, to the opposite phase. It also follows that, with large probability, the time spent by the system during the first jump from one phase to the opposite one is much shorter than the relaxation time.     

基于元胞自动机考虑传播延迟的复杂网络病毒传播研究

基于元胞自动机,研究传播延迟对复杂网络病毒传播动力学行为的影响,提出一种新的易染状态-感染状态-易染状态(SIS)传播模型.研究表明,传播延迟的存在显著降低了网络的传播临界值,增强了网络中病毒爆发的危险性.研究还发现,随着传播延迟的增大,病毒的感染程度以及传播速率都明显增大.此外,SIS传播模型不仅能够反映病毒的平均传播趋势,而且可以描述病毒随时间的动态演化过程以及病毒的爆发和消亡等概率事件,从而有效地克服了利用平均场方法构建的微分方程模型只能反映病毒平均传播趋势的局限性.同时,还给出有效控制网络中病毒传 关键词： 复杂网络 病毒传播 元胞自动机 传播延迟     

Epidemic spreading on networks with vaccination

In this paper, a new susceptible-infected-susceptible (SIS) model on complex networks with imperfect vaccination is proposed. Two types of epidemic spreading patterns (the recovered individuals have or have not immunity) on scale-free networks are discussed. Both theoretical and numerical analyses are presented. The epidemic thresholds related to the vaccination rate, the vaccination-invalid rate and the vaccination success rate on scale-free networks are demonstrated, showing different results from the reported observations. This reveals that whether or not the epidemic can spread over a network under vaccination control is determined not only by the network structure but also by the medicine's effective duration. Moreover, for a given infective rate, the proportion of individuals to vaccinate can be calculated theoretically for the case that the recovered nodes have immunity. Finally, simulated results are presented to show how to control the disease prevalence.     

Epidemic spreading on weighted complex networks

Nowadays, the emergence of online services provides various multi-relation information to support the comprehensive understanding of the epidemic spreading process. In this Letter, we consider the edge weights to represent such multi-role relations. In addition, we perform detailed analysis of two representative metrics, outbreak threshold and epidemic prevalence, on SIS and SIR models. Both theoretical and simulation results find good agreements with each other. Furthermore, experiments show that, on fully mixed networks, the weight distribution on edges would not affect the epidemic results once the average weight of whole network is fixed. This work may shed some light on the in-depth understanding of epidemic spreading on multi-relation and weighted networks.     

复杂网络中考虑不完全免疫的病毒传播研究

复杂网络中不完全免疫包括免疫失败和免疫失效两种情况,本文研究两者同时存在对网络病毒传播行为的影响,基于平均场理论,提出一种新的传播模型.理论分析表明,免疫失败和免疫失效同时存在显著降低了网络的传播临界值,增强了病毒的感染程度.根据传播临界值与免疫节点密度、免疫成功率以及免疫失效率之间的关系,给出有效控制网络病毒传播的策略.通过数值仿真进行验证。     

Metastability in Glauber Dynamics in the Low-Temperature Limit: Beyond Exponential Asymptotics

We consider Glauber dynamics of classical spin systems of Ising type in the limit when the temperature tends to zero in finite volume. We show that information on the structure of the most profound minima and the connecting saddle points of the Hamiltonian can be translated into sharp estimates on the distribution of the times of metastable transitions between such minima as well as the low lying spectrum of the generator. In contrast with earlier results on such problems, where only the asymptotics of the exponential rates is obtained, we compute the precise pre-factors up to multiplicative errors that tend to 1 as T0. As an example we treat the nearest neighbor Ising model on the 2 and 3 dimensional square lattice. Our results improve considerably earlier estimates obtained by Neves–Schonmann,⁽¹⁾ Ben Arous–Cerf,⁽²⁾ and Alonso–Cerf.⁽³⁾ Our results employ the methods introduced by Bovier, Eckhoff, Gayrard, and Klein in refs. 4 and 5.     

Epidemic spreading on uncorrelated heterogenous networks with non-uniform transmission

We investigate the combinational effects of non-uniform transmission of edges and the network structure in susceptible-infected-removed models of the epidemic spreading. Here the degree-degree correlation is neglected and the transmission of individual edge depends on the degree of infected node. With the additional assumption that we can compartment a independent factor λ₀ from the transmission, we analyzed the effects of degree distribution P(k) and transmission distribution λ(k) on the threshold of λ₀.     

SIHR rumor spreading model in social networks

There are significant differences between rumor spreading and epidemic spreading in social networks, especially with consideration of the mutual effect of forgetting and remembering mechanisms. In this paper, a new rumor spreading model, Susceptible-Infected-Hibernator-Removed (SIHR) model, is developed. The model extends the classical Susceptible-Infected-Removed (SIR) rumor spreading model by adding a direct link from ignorants to stiflers and a new kind of people-Hibernators. We derive mean-field equations that describe the dynamics of the SIHR model in social networks. Then a steady-state analysis is conducted to investigate the final size of the rumor spreading under various spreading rate, stifling rate, forgetting rate, and average degree of the network. We discuss the spreading threshold and find the relationship between the final size of the rumor and two probabilities. Also Runge-Kutta method is used for numerical simulation which shows that the direct link from the ignorants to the stiflers advances the rumor terminal time and reduces the maximum rumor influence. Moreover, the forgetting and remembering mechanisms of hibernators postpone the rumor terminal time and reduce the maximum rumor influence.     

The Spectral Gap of the 2-D Stochastic Ising Model with Mixed Boundary Conditions

We establish upper bounds for the spectral gap of the stochastic Ising model at low temperatures in an l×l box with boundary conditions which are not purely plus or minus; specifically, we assume the magnitude of the sum of the boundary spins over each interval of length l in the boundary is bounded by l, where <1. We show that for any such boundary condition, when the temperature is sufficiently low (depending on ), the spectral gap decreases exponentially in l.     

An extended chain Ising model and its Glauber dynamics

It was first proposed that an extended chain Ising (ECI) model contains the Ising chain model, single spin double-well potentials and a pure phonon heat bath of a specific energy exchange with the spins. The extension method is easy to apply to high dimensional cases. Then the single spin-flip probability (rate) of the ECI model is deduced based on the Boltzmann principle and general statistical principles of independent events and the model is simplified to an extended chain Glauber-Ising (ECGI) model. Moreover, the relaxation dynamics of the ECGI model were simulated by the Monte Carlo method and a comparison with the predictions of the special chain Glauber-Ising (SCGI) model was presented. It was found that the results of the two models are consistent with each other when the Ising chain length is large enough and temperature is relative low, which is the most valuable case of the model applications. These show that the ECI model will provide a firm physical base for the widely used single spin-flip rate proposed by Glauber and a possible route to obtain the single spin-flip rate of other form and even the multi-spin-flip rate.     

The Spectral Gap of the 2-D Stochastic Ising Model with Nearly Single-Spin Boundary Conditions

We establish upper bounds for the spectral gap of the stochastic Ising model at low temperature in an N×N box, with boundary conditions which are plus except for small regions at the corners which are either free or minus. The spectral gap decreases exponentially in the size of the corner regions, when these regions are of size at least of order logN. This means that removing as few as O(logN) plus spins from the corners produces a spectral gap far smaller than the order N ^–2 gap believed to hold under the all-plus boundary condition. Our results are valid at all subcritical temperatures.     

Zero-temperature dynamics for the ferromagnetic Ising model on random graphs

We consider Glauber dynamics at zero temperature for the ferromagnetic Ising model on the usual random graph model on N vertices, with on average γ edges incident to each vertex, in the limit as N→∞. Based on numerical simulations, Svenson (Phys. Rev. E 64 (2001) 036122) reported that the dynamics fails to reach a global energy minimum for a range of values of γ. The present paper provides a mathematically rigorous proof that this failure to find the global minimum in fact happens for all γ>0. A lower bound on the residual energy is also given.     

Some New Results on the Kinetic Ising Model in a Pure Phase

We consider a general class of Glauber dynamics reversible with respect to the standard Ising model in ^d with zero external field and inverse temperature strictly larger than the critical value _c in dimension 2 or the so called slab threshold in dimension d 3. We first prove that the inverse spectral gap in a large cube of side N with plus boundary conditions is, apart from logarithmic corrections, larger than N in d = 2 while the logarithmic Sobolev constant is instead larger than N ² in any dimension. Such a result substantially improves over all the previous existing bounds and agrees with a similar computations obtained in the framework of a one dimensional toy model based on mean curvature motion. The proof, based on a suggestion made by H. T. Yau some years ago, explicitly constructs a subtle test function which forces a large droplet of the minus phase inside the plus phase. The relevant bounds for general d 2 are then obtained via a careful use of the recent –approach to the Wulff construction. Finally we prove that in d = 2 the probability that two independent initial configurations, distributed according to the infinite volume plus phase and evolving under any coupling, agree at the origin at time t is bounded from below by a stretched exponential , again apart from logarithmic corrections. Such a result should be considered as a first step toward a rigorous proof that, as conjectured by Fisher and Huse some years ago, the equilibrium time auto-correlation of the spin at the origin decays as a stretched exponential in d = 2.     

复杂网络上灾害蔓延动力学研究

针对关键生命线系统，如电网、供水网、供气网、交通网、通信网等的一些共性特征，建立一个普适性的灾害蔓延动力学模型. 这个模型考虑网络结点的自修复功能、灾害蔓延机制和内部随机噪声，并研究自修复因子、延迟时间因子和噪声强度三个重要特征参数对三种网络(随机网络、无标度网络和小世界网络)结点修复率和崩溃结点数的影响. 模拟结果与这些实际生命线系统的特征一致，表明所建立的模型可以有效地模拟生命线系统的灾害演化动力学. 关键词： 复杂网络 生命线系统 灾害蔓延