Transition in the waiting-time distribution of price-change events in a global socioeconomic system

The goal of developing a firmer theoretical understanding of inhomogeneous temporal processes–in particular, the waiting times in some collective dynamical system–is attracting significant interest among physicists. Quantifying the deviations between the waiting-time distribution and the distribution generated by a random process may help unravel the feedback mechanisms that drive the underlying dynamics. We analyze the waiting-time distributions of high-frequency foreign exchange data for the best executable bid–ask prices across all major currencies. We find that the lognormal distribution yields a good overall fit for the waiting-time distribution between currency rate changes if both short and long waiting times are included. If we restrict our study to long waiting times, each currency pair’s distribution is consistent with a power-law tail with exponent near to 3.5. However, for short waiting times, the overall distribution resembles one generated by an archetypal complex systems model in which boundedly rational agents compete for limited resources. Our findings suggest that a gradual transition arises in trading behavior between a fast regime in which traders act in a boundedly rational way and a slower one in which traders’ decisions are driven by generic feedback mechanisms across multiple timescales and hence produce similar power-law tails irrespective of currency type.     

Power-law distributions in random multiplicative processes with non-Gaussian colored multipliers

One class of universal mechanisms that generate power-law probability distributions is that of random multiplicative processes. In this paper, we consider a multiplicative Langevin equation driven by non-Gaussian colored multipliers. We analytically derive a formula that relates the power-law exponent to the statistics of the multipliers and numerically confirm its validity using multiplicative noise generated by chaotic dynamical systems and by a two-valued Markov process. We also investigate the relationship between our treatment and the large deviation analysis of time series, and demonstrate the appearance of log-periodic fluctuations superimposed on the power-law distribution due to the non-Gaussian nature of the multipliers.     

Predicting the long tail of book sales: Unearthing the power-law exponent

The concept of the long tail has recently been used to explain the phenomenon in e-commerce where the total volume of sales of the items in the tail is comparable to that of the most popular items. In the case of online book sales, the proportion of tail sales has been estimated using regression techniques on the assumption that the data obeys a power-law distribution. Here we propose a different technique for estimation based on a generative model of book sales that results in an asymptotic power-law distribution of sales, but which does not suffer from the problems related to power-law regression techniques. We show that the proportion of tail sales predicted is very sensitive to the estimated power-law exponent. In particular, if we assume that the power-law exponent of the cumulative distribution is closer to 1.1 rather than to 1.2 (estimates published in 2003, calculated using regression by two groups of researchers), then our computations suggest that the tail sales of Amazon.com, rather than being 40% as estimated by Brynjolfsson, Hu and Smith in 2003, are actually closer to 20%, the proportion estimated by its CEO.     

Changes of firm size distribution: The case of Korea

In this paper, the distribution and inequality of firm sizes is evaluated for the Korean firms listed on the stock markets. Using the amount of sales, total assets, capital, and the number of employees, respectively, as a proxy for firm sizes, we find that the upper tail of the Korean firm size distribution can be described by power-law distributions rather than lognormal distributions. Then, we estimate the Zipf parameters of the firm sizes and assess the changes in the magnitude of the exponents. The results show that the calculated Zipf exponents over time increased prior to the financial crisis, but decreased after the crisis. This pattern implies that the degree of inequality in Korean firm sizes had severely deepened prior to the crisis, but lessened after the crisis. Overall, the distribution of Korean firm sizes changes over time, and Zipf’s law is not universal but does hold as a special case.     

Human dynamics scaling characteristics for aerial inbound logistics operation

In recent years, the study of power-law scaling characteristics of real-life networks has attracted much interest from scholars; it deviates from the Poisson process. In this paper, we take the whole process of aerial inbound operation in a logistics company as the empirical object. The main aim of this work is to study the statistical scaling characteristics of the task-restricted work patterns. We found that the statistical variables have the scaling characteristics of unimodal distribution with a power-law tail in five statistical distributions — that is to say, there obviously exists a peak in each distribution, the shape of the left part closes to a Poisson distribution, and the right part has a heavy-tailed scaling statistics. Furthermore, to our surprise, there is only one distribution where the right parts can be approximated by the power-law form with exponent α=1.50. Others are bigger than 1.50 (three of four are about 2.50, one of four is about 3.00). We then obtain two inferences based on these empirical results: first, the human behaviors probably both close to the Poisson statistics and power-law distributions on certain levels, and the human-computer interaction behaviors may be the most common in the logistics operational areas, even in the whole task-restricted work pattern areas. Second, the hypothesis in Vázquez et al. (2006) [A. Vázquez, J. G. Oliveira, Z. Dezsö, K.-I. Goh, I. Kondor, A.-L. Barabási. Modeling burst and heavy tails in human dynamics, Phys. Rev. E 73 (2006) 036127] is probably not sufficient; it claimed that human dynamics can be classified as two discrete university classes. There may be a new human dynamics mechanism that is different from the classical Barabási models.     

An overview of shadowed fading wireless channels in terms of a cascaded approach

Wireless channels suffer from short term fading and shadowing simultaneously. While simple models of short term fading are based on the Nakagami-m distribution, short term fading can also be described as a cascading process allowing the modeling of wireless channels having worse fading than what exists in Nakagami-m channels. Shadowing, on the other hand, has been traditionally modeled as a lognormal process, making the analysis of shadowed fading channels cumbersome. Taking note of the fact that the lognormal density arises out of a multiplicative process, it was shown that shadowing can also be modeled as a cascading process. Utilizing such a vision of shadowing, this work provides an overview of a unified cascaded approach to model wireless channels when short term fading and shadowing are simultaneously present. The degradation in such shadowed fading channels is estimated in terms of error rates and outage probabilities. Results are compared to those of the exact model based on lognormal density as well as random number simulation. Analysis demonstrates that error rates and outage probabilities obtained using the exact model (lognormal model for shadowing) agree very well with those obtained through the composite cascaded model as well as random number simulations.     

Individual and group dynamics in purchasing activity

As a major part of the daily operation in an enterprise, purchasing frequency is in constant change. Recent approaches on the human dynamics can provide some new insights into the economic behavior of companies in the supply chain. This paper captures the attributes of creation times of purchase orders to an individual vendor, as well as to all vendors, and further investigates whether they have some kind of dynamics by applying logarithmic binning to the construction of distribution plots. It’s found that the former displays a power-law distribution with approximate exponent 2.0, while the latter is fitted by a mixture distribution with both power-law and exponential characteristics. Obviously, two distinctive characteristics are presented for the interval time distribution from the perspective of individual dynamics and group dynamics. Actually, this mixing feature can be attributed to the fitting deviations as they are negligible for individual dynamics, but those of different vendors are cumulated and then lead to an exponential factor for group dynamics. To better describe the mechanism generating the heterogeneity of the purchase order assignment process from the objective company to all its vendors, a model driven by product life cycle is introduced, and then the analytical distribution and the simulation result are obtained, which are in good agreement with the empirical data.     

Alternative interpretations of power-law distributions found in nature

We investigate two inherently different classes of probability density functions (pdfs) that share the common property of power law tails: the α-stable Le?vy process and the linear Markov diffusion process with additive and multiplicative Gaussian noise. Dynamical processes described by these distributions cannot be uniquely identified as belonging to one or the other class either by diverging variance due to power-law tails in the pdf or by the possible existence of skew. However, there are distinguishing features that may be found in sufficiently well sampled time series. We examine these features and discuss how they may guide the development of proper approximations to equations of motion underlying dynamical systems. An additional result of this research was the identification of a variable describing the relative importance of the multiplicative and independent additive noise forcing in our linear Markov process. The distribution of this variable is generally skewed, depending on the level of correlation between the additive and multiplicative noise.     

Self-similarity of rRNA secondary structures: A clue to RNA folding

In this paper, we analyze helices in the secondary structures of the 16S and 23S rRNAs from the statistical physics perspective. The results of the analysis lead to propose a possible mechanism of the RNA folding based on the premise that the structure of RNA may bear a trace of its folding. We show that the frequency distribution of the helix contact order approximately follows a power-law, which implies that helices of large contact orders should inevitably exist. Furthermore, the frequencies of helix contact orders can be characterized by the multifractal. Comprehending the multifractality and the power-law of the distribution of the helix contact orders, we suggest a nearest-preferred helix formation as a mechanism for RNA folding via a random binary multiplicative process. The proposed process was supported by reconstructing the multifractal spectrum based on the transfer matrix theory and the binary tree representation of helices in the secondary structures. This justifies, at least partially if not entirely, the relevance of the proposed process as the kinetics of RNA folding.     

Polydispersity effects on the magnetization of diluted ferrofluids: a lognormal analysis

Based on a lognormal particle size distribution, this paper makes a model analysis on the polydispersity effects on the magnetization behaviour of diluted ferrofluids. Using a modified Langevin relationship for the lognormal dispersion, it first performs reduced calculations without material parameters. From the results, it is extrapolated that for the ferrofluid of lognormal polydispersion, in comparison with the corresponding monodispersion, the saturation magnetization is enhanced higher by the particle size distribution. It also indicates that in an equivalent magnetic field, the lognormally polydispersed ferrofluid is magnetically saturated faster than the corresponding monodispersion. Along the theoretical extrapolations, the polydispersity effects are evaluated for a typical ferrofluid of magnetite, with a dispersity of σ =0.20. The results indicate that the lognormal polydispersity leads to a slight increase of the saturation magnetization, but a noticeable increase of the speed to reach the saturation value in an equivalent magnetic field.     

Stochastic resonance of a periodically driven neuron under non-Gaussian noise

We investigate the first-passage-time statistics of the integrate-fire neuron model driven by a sub-threshold harmonic signal superposed with a non-Gaussian noise. Here, we considered the noise as the result of a random multiplicative process displaced from the origin by an additive term. Such a mechanism generates a power-law distributed noise whose characteristic decay exponent can be finely tuned. We performed numerical simulations to analyze the influence of the noise non-Gaussian character on the stochastic resonance condition. We found that when the noise deviates from Gaussian statistics, the resonance condition occurs at weaker noise intensities, achieving a minimum at a finite value of the distribution function decay exponent. We discuss the possible relevance of this feature to the efficiency of the firing dynamics of biological neurons, as the present result indicates that neurons would require a lower noise level to detect a sub-threshold signal when its statistics departs from Gaussian.     

A Langevin approach to the Log–Gauss–Pareto composite statistical structure

The distribution of wealth in human populations displays a Log–Gauss–Pareto composite statistical structure: its density is Log–Gauss in its central body, and follows power-law decay in its tails. This composite statistical structure is further observed in other complex systems, and on a logarithmic scale it displays a Gauss-Exponential structure: its density is Gauss in its central body, and follows exponential decay in its tails. In this paper we establish an equilibrium Langevin explanation for this statistical phenomenon, and show that: (i) the stationary distributions of Langevin dynamics with sigmoidal force functions display a Gauss-Exponential composite statistical structure; (ii) the stationary distributions of geometric Langevin dynamics with sigmoidal force functions display a Log–Gauss–Pareto composite statistical structure. This equilibrium Langevin explanation is universal — as it is invariant with respect to the specific details of the sigmoidal force functions applied, and as it is invariant with respect to the specific statistics of the underlying noise.     

Statistical properties of daily ensemble variables in the Chinese stock markets

We study dynamical behavior of the Chinese stock markets by investigating the statistical properties of daily ensemble return and variety defined, respectively, as the mean and the standard deviation of the ensemble daily price return of a portfolio of stocks traded in China's stock markets on a given day. The distribution of the daily ensemble return has an exponential form in the center and power-law tails, while the variety distribution is lognormal in the bulk followed by a power-law tail for large variety. Based on detrended fluctuation analysis, R/S analysis and modified R/S analysis, we find evidence of long memory in the ensemble return and strong evidence of long memory in the evolution of variety.     

幂律指数在1与3之间的一类无标度网络

借助排队系统中顾客批量到达的概念，提出节点批量到达的Poisson网络模型.节点按照到达率为λ的Poisson过程批量到达系统.模型1，批量按照到达批次的幂律非线性增长，其幂律指数为θ(0≤θ<+∞).BA模型是在θ＝0时的特例.利用Poisson过程理论和连续化方法进行分析，发现这个网络稳态平均度分布是幂律分布，而且幂律指数在1和3之间.模型2，批量按照节点到达批次的对数非线性增长，得出当批量增长较缓慢时，稳态度分布幂律指数为3.因此，节点批量到达的Poisson网络模型不仅是BA模型的推广，也为许多幂律指数在1和2之间的现实网络提供了理论依据.     

Dynamical behaviors of inter-out-of-equilibrium state intervals in Korean futures exchange markets

A recently discovered feature of financial markets, the two-phase phenomenon, is utilized to categorize a financial time series into two phases, namely equilibrium and out-of-equilibrium states. For out-of-equilibrium states, we analyze the time intervals at which the state is revisited. The power-law distribution of inter-out-of-equilibrium state intervals is shown and we present an analogy with discrete-time heat bath dynamics, similar to random Ising systems. In the mean-field approximation, this model reduces to a one-dimensional multiplicative process. By varying global and local model parameters, the relevance between volatilities in financial markets and the interaction strengths between agents in the Ising model are investigated and discussed.     

Surname distribution in population genetics and in statistical physics

Surnames tend to behave like neutral genes, and their distribution has attracted a growing attention from genetists and physicists. We review the century-long history of surname studies and discuss the most recent developments. Isonymy has been regarded as a tool for the measurement of consanguinity of individuals and populations and has been applied to the analysis of migrations. The analogy between patrilineal surname transmission and the propagation of Y chromosomes has been exploited for the genetic characterization of families, communities and control groups. Surname distribution is the result of a stochastic dynamics, which has been studied either as a Yule process or as a branching phenomenon: both approaches predict the asymptotic power-law behavior which has been observed in many empirical researches. Models of neutral evolution based on the theory of disordered systems have suggested the application of field-theoretical techniques, and in particular the Renormalization Group, to describe the dynamics leading to scale-invariant distributions and to compute the related (critical) exponents.     

Equilibrium of a mesoscopic system with conformation dependent damping: An alternative approach

We look at the dynamics of a Brownian particle with internal degrees of freedom and conformation dependent damping. Inhomogeneous damping apparently makes the problem a stochastic process with multiplicative noise. We derive the equilibrium distribution of such a system on the basis of a single postulate that the stochastic forces on the system produces no drift. Based on this postulate, we generalize the expression of the stochastic force for the equilibrium of such systems. The equilibrium probability distribution obtained deviates from the exact canonical form, although, the equipartition of energy remains intact when the internal degrees of freedom are integrated out. We also show a crucial local balance of the rate of average energy inflow and outflow as a consequence of the equilibrium probability distribution.     

Emergence of power-law in a market with mixed models

We investigate the problem of wealth distribution from the viewpoint of asset exchange. Robust nature of Pareto's law across economies, ideologies and nations suggests that this could be an outcome of trading strategies. However, the simple asset exchange models fail to reproduce this feature. A Yardsale (YS) model in which amount put on the bet is a fraction of minimum of the two players leads to condensation of wealth in hands of some agent while theft and fraud (TF) model in which the amount to be exchanged is a fraction of loser's wealth leads to an exponential distribution of wealth. We show that if we allow few agents to follow a different model than others, i.e., there are some agents following TF model while rest follow YS model, it leads to distribution with power-law tails. Similar effect is observed when one carries out transactions for a fraction of one's wealth using TF model and for the rest YS model is used. We also observe a power-law tail in wealth distribution if we allow the agents to follow either of the models with some probability.     

Transition state theory: A generalization to nonequilibrium systems with power-law distributions

Transition state theory (TST) is generalized to nonequilibrium systems with power-law distributions. The stochastic dynamics that gives rise to the power-law distributions for the reaction coordinate and momentum is modeled by Langevin equations and corresponding Fokker-Planck equations. It is considered that a system far away from equilibrium does not have to relax to a thermal equilibrium state with Boltzmann-Gibbs distribution, but asymptotically approaches a nonequilibrium stationary state with a power-law distribution. Thus, we obtain a possible generalization of TST rates to nonequilibrium systems with power-law distributions. Furthermore, we derive the generalized TST rate constants for one-dimensional and n-dimensional Hamiltonian systems away from equilibrium, and obtain a generalized Arrhenius rate for systems with power-law distributions.