Empirical analysis on temporal statistics of human correspondence patterns

Recently, extensive empirical evidence shows that the timing of human behaviors obeys non-Possion statistics with heavy-tailed interevent time distribution. In this paper, we empirically study the correspondence pattern of a great Chinese scientist, named Hsue-Shen Tsien. Both the interevent time distribution and response time distributions deviate from the Poisson statistics, showing an approximate power-law decaying. The two power-law exponents are more or less the same (about 2.1), which strongly support the hypothesis in [A. Vázquez, J.G. Oliveira, Z. Dezsö, K.-I. Goh, I. Kondor, A.-L. Barabási, Phys. Rev. E 73 (2006) 036127] that the response time distribution of the tasks could in fact drive the interevent time distribution, and both the two distributions should decay with the same exponent. Our result is against the claim in [A. Vázquez, J.G. Oliveira, Z. Dezsö, K.-I. Goh, I. Kondor, A.-L. Barabási, Phys. Rev. E 73 (2006) 036127], which suggests the human correspondence pattern belongs to a universality class with exponent 1.5.     

On the origin of bursts and heavy tails in animal dynamics

Over recent years there has been an accumulation of evidence that many animal behaviours are characterised by common scale-invariant patterns of switching between two contrasting activities over a period of time. This is evidenced in mammalian wake-sleep patterns, in the intermittent stop-start locomotion of Drosophila fruit flies, and in the Lévy walk movement patterns of a diverse range of animals in which straight-line movements are punctuated by occasional turns. Here it is shown that these dynamics can be modelled by a stochastic variant of Barabási’s model [A.-L. Barabási, The origin of bursts and heavy tails in human dynamics, Nature 435 (2005) 207-211] for bursts and heavy tails in human dynamics. The new model captures a tension between two competing and conflicting activities. The durations of one type of activity are distributed according to an inverse-square power-law, mirroring the ubiquity of inverse-square power-law scaling seen in empirical data. The durations of the second type of activity follow exponential distributions with characteristic timescales that depend on species and metabolic rates. This again is a common feature of animal behaviour. Bursty human dynamics, on the other hand, are characterised by power-law distributions with scaling exponents close to −1 and −3/2.     

Tuning degree distributions: Departing from scale-free networks

Scale-free networks are characterized by a degree distribution with power-law behavior. Although scale-free networks have been shown to arise in many areas, ranging from the World Wide Web to transportation or social networks, degree distributions of other observed networks often differ from the power-law type. Data based investigations require modifications of the typical scale-free network.We present an algorithm that generates networks in which the shape of the degree distribution is tunable by modifying the preferential attachment step of the Barabási-Albert construction algorithm. The shape of the distribution is represented by dispersion measures such as the variance and the skewness, both of which are highly correlated with the maximal degree of the network and, therefore, adequately represents the influence of superspreaders or hubs. By combining our algorithm with work of Holme and Kim, we show how to generate networks with a variety of degree distributions and clustering coefficients.     

Time evolution of the degree distribution of model A of random attachment growing networks

For random growing networks, Barabás and Albert proposed a kind of model in Barabás et al. [Physica A 272 (1999) 173], i.e. model A. In this paper, for model A, we give the differential format of master equation of degree distribution and obtain its analytical solution. The obtained result P(k, t) is the time evolution of degree distribution. P(k, t) is composed of two terms. At given finite time, one term decays exponentially, the other reflects size effect. At infinite time, the degree distribution is the same as that of Barabás and Albert. In this paper, we also discuss the normalization of degree distribution P(k, t) in detail.     

A continuous model of human dynamics

Ideas and tools from statistical physics have recently been applied to the investigation of human dynamics. The timing of human activities, in particular, has been studied both experimentally and analytically. Empirical data show that, in many different situations, the time interval separating two consecutive tasks executed by an individual follows a heavy-tailed probability distribution rather than Poisson statistics. To account for this data, human behaviour has been viewed as a decision-based queuing system where individuals select and execute tasks belonging to a finite list of items as an increasing function of a task priority parameter. It is then possible to obtain analytically the empirical result P(τ)∼1/τ, where P(τ) is the waiting time probability distribution.Here a continuous model of human dynamics is introduced using instead an infinite queuing list. In contrast with the results obtained by other models in the finite case we find a waiting time distribution explicitly depending on the priority distribution density function ρ. The power-law scaling P(τ)∼1/τ is then recovered when ρ is exponentially distributed.     

Connectivity degrees in the threshold preferential attachment model

In this paper we present a study of the connectivity degrees of the threshold preferential attachment model, a generalization of the Barabási-Albert model to heterogeneous complex networks. The threshold model incorporates the states of the nodes in its preferential linking rule and assumes that the affinity between network nodes follows an inverse relationship with the distance between their states. We numerically analyze the connectivity degrees of the model, studying the influence of the main parameters on the distribution of connectivity degrees and its statistics, the average degree and highest degree of the network. We show that such statistics exhibit markedly different behaviors in the dependence on the model parameters, particularly as regards the interaction threshold. Nevertheless, we show that the two statistics converge in the limit of null threshold and often exhibit scaling that can be described by power laws of the model parameters.     

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.     

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.     

Strain Avalanches in Microsized Single Crystals:Avalanche Size Predicted by a Continuum Crystal Plasticity Model

Plastic deformation of small crystals occurs by power-law distributed strain avalanches whose universality is still debated.In this work we introduce a continuum crystal plasticity model for the deformation of microsized single crystals,which is able to reproduce the main experimental observations such as Row intermittency and statistics of strain avalanches.We report exact predictions for scaling exponents and scaling functions associated with random distribution of avalanche sizes.In this way,the developed model provides a routine for a quantitative characterization of the statistical aspects of strain avalanches in microsized single crystals.     

On the fractal characterization of Paretian Poisson processes

Paretian Poisson processes are Poisson processes which are defined on the positive half-line, have maximal points, and are quantified by power-law intensities. Paretian Poisson processes are elemental in statistical physics, and are the bedrock of a host of power-law statistics ranging from Pareto’s law to anomalous diffusion. In this paper we establish evenness-based fractal characterizations of Paretian Poisson processes. Considering an array of socioeconomic evenness-based measures of statistical heterogeneity, we show that: amongst the realm of Poisson processes which are defined on the positive half-line, and have maximal points, Paretian Poisson processes are the unique class of ‘fractal processes’ exhibiting scale-invariance. The results established in this paper are diametric to previous results asserting that the scale-invariance of Poisson processes–with respect to physical randomness-based measures of statistical heterogeneity–is characterized by exponential Poissonian intensities.     

Opinion evolution on a BA scaling network

In this paper, the dynamics of opinion formation is investigated based on a BA (Barabási–Albert) scale-free network, using a majority–minority rule governed by parameter q$q$. As the value of q$q$ is smoothly varied, a phase transition occurs between an ordered phase and a disordered one. By performing extensive Monte Carlo simulations, we show that the phase transition is dependent on the system size, as well as on m$m$, the number of edges added at each time step during the growth of the BA scaling network. Additionally, some theoretical analysis is given based on mean-field theory, by neglecting fluctuations and correlations. It is observed that the theoretical results coincide with results from simulations, especially for very large m$m$.     

Local affinity in heterogeneous growing networks

In this paper we present a study of the influence of local affinity in heterogeneous preferential attachment (PA) networks. Heterogeneous PA models are a generalization of the Barabási-Albert model to heterogeneous networks, where the affinity between nodes biases the attachment probability of links. Threshold models are a class of heterogeneous PA models where the affinity between nodes is inversely related to the distance between their states. We propose a generalization of threshold models where network nodes have individual affinity functions, which are then combined to yield the affinity of each potential interaction. We analyze the influence of the affinity functions in the topological properties averaged over a network ensemble. The network topology is evaluated through the distributions of connectivity degrees, clustering coefficients and geodesic distances. We show that the relaxation of the criterion of a single global affinity still leads to a reasonable power-law scaling in the connectivity and clustering distributions under a wide spectrum of assumptions. We also show that the richer behavior of the model often exhibits a better agreement with the empirical observations on real networks.     

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.     

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.     

A neighbourhood evolving network model

Many social, technological, biological and economical systems are best described by evolved network models. In this short Letter, we propose and study a new evolving network model. The model is based on the new concept of neighbourhood connectivity, which exists in many physical complex networks. The statistical properties and dynamics of the proposed model is analytically studied and compared with those of Barabási–Albert scale-free model. Numerical simulations indicate that this network model yields a transition between power-law and exponential scaling, while the Barabási–Albert scale-free model is only one of its special (limiting) cases. Particularly, this model can be used to enhance the evolving mechanism of complex networks in the real world, such as some social networks development.     

一种新型电力网络局域世界演化模型

现实世界中的许多系统都可以用复杂网络来描述,电力系统是人类创造的最为复杂的网络系统之一.当前经典的网络模型与实际电力网络存在较大差异.从电力网络本身的演化机理入手,提出并研究了一种可以模拟电力网络演化规律的新型局域世界网络演化模型.理论分析表明该模型的度分布具有幂尾特性,且幂律指数在3—∞之间可调.最后通过对中国北方电网和美国西部电网的仿真以及和无标度网络、随机网络的对比,验证了该模型可以很好地反映电力网络的演化规律,并且进一步证实了电力网络既不是无标度网络,也不是完全的随机网络. 关键词： 电力网络 演化模型 局域世界 幂律分布     

Correlated dynamics in human printing behavior

Arrival times of requests to print in a student laboratory were analyzed. Inter-arrival times between subsequent requests follow a universal scaling law relating time intervals and the size of the request, indicating a scale invariant dynamics with respect to the size. The cumulative distribution of file sizes is well-described by a modified power-law often seen in non-equilibrium critical systems. For each user, waiting times between their individual requests show long range dependence and are broadly distributed from seconds to weeks. All results are incompatible with Poisson models, and may provide evidence of critical dynamics associated with voluntary thought processes in the brain.     

Weblog patterns and human dynamics with decreasing interest

In order to describe the phenomenon that people’s interest in doing something always keep high in the beginning while gradually decreases until reaching the balance, a model which describes the attenuation of interest is proposed to reflect the fact that people’s interest becomes more stable after a long time. We give a rigorous analysis on this model by non-homogeneous Poisson processes. Our analysis indicates that the interval distribution of arrival-time is a mixed distribution with exponential and power-law feature, which is a power law with an exponential cutoff. After that, we collect blogs in ScienceNet.cn and carry on empirical study on the interarrival time distribution. The empirical results agree well with the theoretical analysis, obeying a special power law with the exponential cutoff, that is, a special kind of Gamma distribution. These empirical results verify the model by providing an evidence for a new class of phenomena in human dynamics. It can be concluded that besides power-law distributions, there are other distributions in human dynamics. These findings demonstrate the variety of human behavior dynamics.     

A probabilistic walk up power laws

We establish a path leading from Pareto’s law to anomalous diffusion, and present along the way a panoramic overview of power-law statistics. Pareto’s law is shown to universally emerge from “Central Limit Theorems” for rank distributions and exceedances, and is further shown to be a finite-dimensional projection of an infinite-dimensional underlying object — Pareto’s Poisson process  . The fundamental importance and centrality of Pareto’s Poisson process is described, and we demonstrate how this process universally generates an array of anomalous diffusion statistics characterized by intrinsic power-law structures: sub-diffusion and super-diffusion, Lévy laws and the “Noah effect”, long-range dependence and the “Joseph effect”, 1/f$1/f$ noises, and anomalous relaxation.     

Emergent organization of oscillator clusters in coupled self-regulatory chaotic maps

Here we introduce a model of parametrically coupled chaotic maps on a one-dimensional lattice. In this model, each element has its internal self-regulatory dynamics, whereby at fixed intervals of time the nonlinearity parameter at each site is adjusted by feedback from its past evolution. Additionally, the maps are coupled sequentially and unidirectionally, to their nearest neighbor, through the difference of their parametric variations. Interestingly we find that this model asymptotically yields clusters of superstable oscillators with different periods. We observe that the sizes of these oscillator clusters have a power-law distribution. Moreover, we find that the transient dynamics gives rise to a 1/f power spectrum. All these characteristics indicate self-organization and emergent scaling behavior in this system. We also interpret the power-law characteristics of the proposed system from an ecological point of view.