S-curve networks and an approximate method for estimating degree distributions of complex networks

In the study of complex networks almost all theoretical models have the property of infinite growth,but the size of actual networks is finite.According to statistics from the China Internet IPv4(Internet Protocol version 4) addresses,this paper proposes a forecasting model by using S curve(logistic curve).The growing trend of IPv4 addresses in China is forecasted.There are some reference values for optimizing the distribution of IPv4 address resource and the development of IPv6.Based on the laws of IPv4 growth,that is,the bulk growth and the finitely growing limit,it proposes a finite network model with a bulk growth.The model is said to be an S-curve network.Analysis demonstrates that the analytic method based on uniform distributions(i.e.,Barab’asi-Albert method) is not suitable for the network.It develops an approximate method to predict the growth dynamics of the individual nodes,and uses this to calculate analytically the degree distribution and the scaling exponents.The analytical result agrees with the simulation well,obeying an approximately power-law form.This method can overcome a shortcoming of Baraba’si-Albert method commonly used in current network research.     

The critical properties of the agent-based model with environmental-economic interactions

The steady-state and nonequilibrium properties of the model of environmental-economic interactions are studied. The interacting heterogeneous agents are simulated on the platform of the emission dynamics of cellular automaton. The diffusive emissions are produced by the factory agents, and the local pollution is monitored by the randomly walking (mobile) sensors. When the threshold concentration is exceeded, a feedback signal is transmitted from the sensor to the nearest factory that affects its actual production rate. The model predicts the discontinuous phase transition between safe and catastrophic ecology. Right at the critical line, the broad-scale power-law distributions of emission rates have been identified. The power-law fluctuations are triggered by the screening effect of factories and by the time delay between the environment contamination and its detection. The system shows the typical signs of the self-organized critical systems, such as power-law distributions and scaling laws. The text was submitted by the authors in English.     

Exchange-Driven Growth with Birth Rate Less Than Death

We further study the kinetic behavior of the exchange-driven growth with birth and death for the case of birth rate kernel being less than that of death based on the mean-field theory. The symmetric exchange rate kernel is K(k,j)=K'(k,j)=Ikj^υ, and the birth and death rates are proportional to the aggregate's size. The long time asymptotic behavior of the aggregate size distribution a_k(t) is found to obey a much unusual scaling law with an exponentially growing scaling function Φ(x)=exp(x).     

Robust Characterization of Multidimensional Scaling Relations between Size Measures for Business Firms

Although the sizes of business firms have been a subject of intensive research, the definition of a “size” of a firm remains unclear. In this study, we empirically characterize in detail the scaling relations between size measures of business firms, analyzing them based on allometric scaling. Using a large dataset of Japanese firms that tracked approximately one million firms annually for two decades (1994–2015), we examined up to the trivariate relations between corporate size measures: annual sales, capital stock, total assets, and numbers of employees and trading partners. The data were examined using a multivariate generalization of a previously proposed method for analyzing bivariate scalings. We found that relations between measures other than the capital stock are marked by allometric scaling relations. Power–law exponents for scalings and distributions of multiple firm size measures were mostly robust throughout the years but had fluctuations that appeared to correlate with national economic conditions. We established theoretical relations between the exponents. We expect these results to allow direct estimation of the effects of using alternative size measures of business firms in regression analyses, to facilitate the modeling of firms, and to enhance the current theoretical understanding of complex systems.     

Kinetics of an n-Species Aggregation Chain Model with Complete Annihilation

The kinetic behavior of an n-species (n ≥ 3) aggregation-annihilation chain reaction model is studied. In this model, an irreversible aggregation reaction occurs between any two clusters of the same species, and an irreversible complete annihilation reaction occurs only between two species with adjacent number. Based on the rnean-field theory, we investigate the rate equations of the process with constant reaction rates to obtain the asymptotic solutions of the clustermass distributions for the system. The results show that the kinetic behavior of the system not only depends crucially on the ratio of the aggregation rate I to the annihilation rate J, but also has relation with the initial concentration of each species and the species number's odevity. We find that the cluster-mass distribution of each species obeys always a scaling law. The scaling exponents may strongly depend on the reaction rates for most cases, however, for the case in which the ratio of the aggregation rate to the annihilation rate is equal to a certain value, the scaling exponents are only dependent on the initial concentrations of the reactants.     

随机稀释基底上刻蚀模型动力学标度行为的数值模拟研究

表面界面动力学粗化过程是凝聚态物理领域重要的研究内容,为研究基底不完整性对刻蚀模型动力学 标度行为的影响,本文采用Kinetic Monte Carlo(KMC)方法,分析研究了在随机稀释基底上刻蚀模型(Etching model)生长表面的动力学标度行为.研究发现:尽管随机稀释基底的不完整性会对刻蚀表面的动力学 行为产生显著的影响,导致刻蚀表面粗糙度指数和生长指数有明显的增加, 但其仍基本满足原有的动力学标度规律.此外,本文还对刻蚀表面动力学标度指数的有限尺寸效应进行了 分析讨论.     

Flat-head power-law, size-independent clustering, and scaling of coevolutionary scale-free networks

Scale-free topology and high clustering coexist in some real networks, and keep invariant for growing sizes of the systems. Previous models could hardly give out size-independent clustering with selforganized mechanism when succeeded in producing power-law degree distributions. Always ignored, some empirical statistic results display flat-head power-law behaviors. We modify our recent coevolutionary model to explain such phenomena with the inert property of nodes to retain small portion of unfavorable links in self-organized rewiring process. Flat-head power-law and size-independent clustering are induced as the new characteristics by this modification. In addition, a new scaling relation is found as the result of interplay between node state growth and adaptive variation of connections.     

Critical Behaviors in a Stochastic One-Dimensional Sand-Pile Model

A one-dimensional sand-pile model （Manna model）, which has a stochastic redistribution process, is studied both in discrete and continuous manners. The system evolves into a critical state after a transient period. A detailed analysis of the probability distribution of the avalanche size and duration is numerically investigated. Interestingly,contrary to the deterministic one-dimensional sand-pile model, where multifractal analysis works well, the analysis based on simple finite-size scaling is suited to fitting the data on the distribution of the avalanche size and duration. The exponents characterizing these probability distributions are measured. Scaling relations of these scaling exponents and their universality class are discussed.     

Critical Behaviors in a Stochastic One-Dimensional Sand-Pile Model

A one-dimensional sand-pile model (Manna model), which has a stochastic redistribution process, is studied both in discrete and continuous manners. The system evolves into a critical state after a transient period. A detailed analysis of the probability distribution of the avalanche size and duration is numerically investigated. Interestingly,contrary to the deterministic one-dimensional sand-pile model, where multifractal analysis works well, the analysis based on simple finite-size scaling is suited to fitting the data on the distribution of the avalanche size and duration. The exponents characterizing these probability distributions are measured. Scaling relations of these scaling exponents and their universality class are discussed.     

Modeling Evolution of Weighted Clique Networks

We propose a weighted clique network evolution model, which expands continuously by the addition of a new clique (maximal complete sub-graph) at each time step. And the cliques in the network overlap with each other. The structural expansion of the weighted clique network is combined with the edges' weight and vertices' strengths dynamical evolution. The model is based on a weight-driven dynamics and a weights' enhancement mechanism combining with the network growth. We study the network properties, which include the distribution of vertices' strength and the distribution of edges' weight, and find that both the distributions follow the scale-free distribution. At the same time, we also find that the relationship between strength and degree of a vertex are linear correlation during the growth of the network. On the basis of mean-field theory, we study the weighted network model and prove that both vertices' strength and edges' weight of this model follow the scale-free distribution. And we exploit an algorithm to forecast the network dynamics, which can be used to reckon the distributions and the corresponding scaling exponents. Furthermore, we observe that mean-field based theoretic results are
consistent with the statistical data of the model, which denotes the theoretical result in this paper is effective.     

Self-similar branching of aftershock sequences

In this paper we propose a branching aftershock sequence (BASS) model for seismicity. We suggest that the BASS model is a preferred alternative to the widely studied epidemic type aftershock sequence (ETAS) model. In the BASS model an initial, or seed, earthquake is specified. The subsequent earthquakes are obtained from the statistical distributions of magnitude, time, and location. The magnitude scaling is based on a combination of the Gutenberg-Richter scaling relation and the modified Båth’s law for the scaling relation of aftershocks relative to the magnitude of the seed earthquake. Omori’s law specifies the distribution of earthquake times, and a modified form of Omori’s law specifies the distribution of earthquake locations. Since the BASS model is specified by the four scaling relations, it is fully self-similar. This is not the case for ETAS. We also give a deterministic version of BASS and show that it satisfies Tokunaga side-branching statistics in a similar way to diffusion-limited aggregation (DLA).     

Universal distributions for growth processes in 1+1 dimensions and random matrices

We develop a scaling theory for Kardar-Parisi-Zhang growth in one dimension by a detailed study of the polynuclear growth model. In particular, we identify three universal distributions for shape fluctuations and their dependence on the macroscopic shape. These distribution functions are computed using the partition function of Gaussian random matrices in a cosine potential.     

Simulation study on nanocluster growth deposited on a substrate

We present a model equation that describes nucleation and growth of hemispherical nanoclusters or islands deposited on a substrate for the small surface coverage case. The model is formulated in terms of a set of rate equations for the island sizes, combined with the time-dependent behavior of supersaturation and island nucleation rate. As an example to demonstrate the usefulness of the model, we study effects of the deposition rate of adatoms on the nanocluster growth. Large-scale computer simulation results show that the broadness of island size distribution is a decreasing function of the deposition rate for small rates, and bimodal distributions are obtained for large rates.     

分形基底上受限固-固模型动力学性质的数值模拟研究

为了探讨非完整基底结构对生长表面动力学行为的影响,本文在具有相同分形维数而不同谱维数的谢尔宾斯基箭头和蟹状分形基底上对受限固-固(restricted solid-on-solid,RSOS)模型的生长过程进行了大量的数值模拟研究.通过计算表面宽度和饱和表面极值高度的统计行为对生长表面的动力学行为进行了分析.结果表明,分形基底结构对生长表面的动力学行为具有显著的影响.尽管在两种基底上受限固-固模型的表面宽度均表现出很好的动力学标度行为,仍然满足Family-Vicsek标度规律,但由此计算得到的动力学标度指数并不相同.饱和生长表面的极值高度并不能满足三种常用的极值统计分布,即Weibull,Gumbel和Frechet分布,而是能很好地符合Asym2Sig分布.     

Energy barrier distributions in magnetic systems from the Tln(t/τ0) scaling

A method based in the Tln(t/τ₀) scaling is proposed to obtain the energy barrier distribution of a randomly oriented assembly of magnetic entities. It essentially consists on performing the logarithmic time derivative of the master relaxation curve. The method has been applied to two small particle systems and the obtained distributions are in good agreement with the ones used to fit the master curves to the magnetization decay of a system following the Arrhenius law. The effect of the specific shape of the distribution functions on the low temperature behaviour of the relaxation rate and the possibility to erroneously attribute this behaviour to non-thermal processes is also discussed.     

Reaction kinetic model of height selection in heteroepitaxial growth of quantum dots

A reaction kinetic model is proposed for height selection of heteroepitaxially growing nanometer-thick quantum dots. The model describes the growth by a set of rate equations for the combined size and height distributions of the dots. In addition to nucleation and growth, the model includes a coarse-grained conversion rate incorporating kinetics of height changes. With suitably chosen rate coefficients the model reproduces qualitatively the experimentally observed height-selected size distributions and their evolution. The results support the view that the height selection and the form of the size distribution both result from the oscillating energy barrier for the transformation of dots of different heights, and this transformation barrier is considerably larger in magnitude than oscillations in the electronic energy due to quantum well states in the dot.     

Scaling in the distribution of intertrade durations of Chinese stocks

The distribution of intertrade durations, defined as the waiting times between two consecutive transactions, is investigated based upon the limit order book data of 23 liquid Chinese stocks listed on the Shenzhen Stock Exchange in the whole year 2003. A scaling pattern is observed in the distributions of intertrade durations, where the empirical density functions of the normalized intertrade durations of all 23 stocks collapse onto a single curve. The scaling pattern is also observed in the intertrade duration distributions for filled and partially filled trades and in the conditional distributions. The ensemble distributions for all stocks are modeled by the Weibull and the Tsallis q-exponential distributions. Maximum likelihood estimation shows that the Weibull distribution outperforms the q-exponential for not-too-large intertrade durations which account for more than 98.5% of the data. Alternatively, nonlinear least-squares estimation selects the q-exponential as a better model, in which the optimization is conducted on the distance between empirical and theoretical values of the logarithmic probability densities. The distribution of intertrade durations is Weibull followed by a power-law tail with an asymptotic tail exponent close to 3.     

Allometry and growth: A unified view

Allometry is crucial in biology; scaling relations are implied in laws of growth of living systems. The self similarity of Gompertzian growths of biological organisms plays a key role, in this regard, in biological similitude. The origin of allometric relationships and values of the scaling exponents is a source of debate, as well as the origin of the range of biological scales. But, besides biology, scaling and growth characterize many different complex systems. Encompassing these aspects in a unified view is an interesting target. In this paper we propose a coarse but significant model that is applied to astrophysical structures and biological organisms. The inclusion of such different systems suggests applications to fields where allometry is emerging, such as economics, urban planning and the social sciences.