Power-Law Tail in the Chinese Wealth Distribution

We analyse the data from the recently published lists of the richest Chinese from the year 2003 to 2005. The results confirm that in these years the wealth is distributed according to a power law with exponents between 1.758 and 2.285 in the high end. The power distribution is found to be quite robust although the persons in the list change drastically and the wealth increases rapidly. The relation between the wealth and the absolute change of wealth rejects the notion that the wealth evolution is a multiplicative stochastic process.     

Empirical regularities of opening call auction in Chinese stock market

We study the statistical regularities of an opening call auction using the ultra-high-frequency data of 22 liquid stocks traded on the Shenzhen Stock Exchange in 2003. The distribution of the relative price, defined as the relative difference between the order price in the opening call auction and the closing price on the last trading day, is asymmetric and that the distribution displays a sharp peak at the zero relative price and a relatively wide peak at the negative relative price. The detrended fluctuation analysis (DFA) method is adopted to investigate the long-term memory of relative order prices. We further study the statistical regularities of order sizes in the opening call auction, and observe a phenomenon of number preference, known as order size clustering. The probability density function (PDF) of order sizes could be well fitted by a q-Gamma function, and the long-term memory also exists in order sizes. In addition, both the average volume and the average number of orders decrease exponentially with the price level away from the best bid or ask price level in the limit-order book (LOB) established immediately after the opening call auction, and a price clustering phenomenon is observed.     

Empirical regularities of order placement in the Chinese stock market

Using ultra-high-frequency data extracted from the order flows of 23 stocks traded on the Shenzhen Stock Exchange, we study the empirical regularities of order placement in the opening call auction, cool period and continuous auction. The distributions of relative logarithmic prices against reference prices in the three time periods are qualitatively the same with quantitative discrepancies. The order placement behavior is asymmetric between buyers and sellers and between the inside-the-book orders and outside-the-book orders. In addition, the conditional distributions of relative prices in the continuous auction are independent of the bid-ask spread and volatility. These findings are crucial to build an empirical behavioral microscopic model based on order flows for Chinese stocks.     

Peculiar statistical properties of Chinese stock indices in bull and bear market phases

Chinese stock markets have experienced an extraordinary bull market since Jan 2006, which attracted global eyes. We investigate the statistical properties of the indices’ log-return r(t) for the bull market (Jan 2006-Oct 2007) and the previous bear market (Jan 2001-Dec 2005). Here we report three peculiar features of r(t): (i) the cumulative distribution function curve of r(t) in the bull market is similar to that in the bear market; (ii) the autocorrelation function of r(t) in the bull market has a stronger negative correlation and a shorter correlation time than that in the bear market; (iii) the bull market shows stronger long-term correlation than the bear market. This work has relevance to understanding novel statistical properties in economic systems.     

Evidence of Scaling in Chinese Income Distribution

With income data from Chinese household income projects in 1998-2002, we study the functional form of Chinese income distribution. The fitting results suggest a log-normal distribution plus a power-law tail. This distributional form has changed a lot from its appearance in the early stage of China＇s reform and turns out to be consistent with that of some complete market economies. The uncertainty and diversity of income growth rate aroused by marketing reform are the main causes of current Chinese income distribution.     

Stability of the BA Network: a New Approach to Rigorous Proof

We propose a new approach to rigorously prove the existence of the steady-state degree distribution for the BA network. The approach is based on a vector Markov chain of vertex numbers in the network evolving process. This framework provides a rigorous theoretical basis for the rate equation approach which has been widely applied to many problems in the field of complex networks, e.g., epidemic spreading and dynamic synchronization.     

Detecting unknown paths on complex networks through random walks

In this article, we investigate the problem of detecting unknown paths on complex networks through random walks. To detect a given path on a network a random walker should pass through the path from its initial node to its terminal node in turn. We calculate probability ?(t) that a random walker detects a given path on a connected network in t steps when it starts out from source node s. We propose an iteration formula for calculating ?(t). Generating function of ?(t) is also derived. Major factors affecting ?(t), such as walking time t, path length l, starting point of the walker, structure of the path, and topological structure of the underlying network are further discussed. Among these factors, two most outstanding ones are walking time t and path length l. On the one hand, ?(t) increases as t increases, and ?(∞)=1, which shows that the longer the walking time, the higher the chance of detecting a given path, and the walker will discover the path sooner or later so long as it keeps wandering on the network. On the other hand, ?(t) drops substantially as path length l increases, which shows that the longer the path, the lower the chance for the walker to find it in a given time. Apart from path length, path structure also has obvious effect on ?(t). Starting point of the walker has only minor influence on ?(t), but topological structure of the underlying network has strong influence on ?(t). Simulations confirm our analytic results.     

Numerical investigations of discrete scale invariance in fractals and multifractal measures

Fractals and multifractals and their associated scaling laws provide a quantification of the complexity of a variety of scale invariant complex systems. Here, we focus on lattice multifractals which exhibit complex exponents associated with observable log-periodicity. We perform detailed numerical analyses of lattice multifractals and explain the origin of three different scaling regions found in the moments. A novel numerical approach is proposed to extract the log-frequencies. In the non-lattice case, there is no visible log-periodicity, i.e., no preferred scaling ratio since the set of complex exponents spreads irregularly within the complex plane. A non-lattice multifractal can be approximated by a sequence of lattice multifractals so that the sets of complex exponents of the lattice sequence converge to the set of complex exponents of the non-lattice one. An algorithm for the construction of the lattice sequence is proposed explicitly.     

Testing a priority-based queue model with Linux command histories

We study human dynamics by analyzing Linux history files. The goodness-of-fit test shows that most of the collected datasets belong to the universality class suggested in the literature by a variable-length queuing process based on priority. In order to check the validity of this model, we design two tests based on mutual information between time intervals and a mathematical relationship known as the arcsine law. Since the previously suggested queuing process fails to pass these tests, the result suggests that the modelling of human dynamics should properly consider the statistical dependency in the temporal dimension.     

A delay fractioning approach to global synchronization of delayed complex networks with stochastic disturbances

In this Letter, the synchronization problem is investigated for a class of stochastic complex networks with time delays. By utilizing a new Lyapunov functional form based on the idea of ‘delay fractioning’, we employ the stochastic analysis techniques and the properties of Kronecker product to establish delay-dependent synchronization criteria that guarantee the globally asymptotically mean-square synchronization of the addressed delayed networks with stochastic disturbances. These sufficient conditions, which are formulated in terms of linear matrix inequalities (LMIs), can be solved efficiently by the LMI toolbox in Matlab. The main results are proved to be much less conservative and the conservatism could be reduced further as the number of delay fractioning gets bigger. A simulation example is exploited to demonstrate the advantage and applicability of the proposed result.     

Transient dynamics of sparsely connected Hopfield neural networks with arbitrary degree distributions

Using probabilistic approach, the transient dynamics of sparsely connected Hopfield neural networks is studied for arbitrary degree distributions. A recursive scheme is developed to determine the time evolution of overlap parameters. As illustrative examples, the explicit calculations of dynamics for networks with binomial, power-law, and uniform degree distribution are performed. The results are good agreement with the extensive numerical simulations. It indicates that with the same average degree, there is a gradual improvement of network performance with increasing sharpness of its degree distribution, and the most efficient degree distribution for global storage of patterns is the delta function.     

First passage time of multiple Brownian particles on networks with applications

In this article, we study some theoretical and technological problems with relation to multiple Brownian particles on networks. We are especially interested in the behavior of the first arriving Brownian particle when all the Brownian particles start out from the source s simultaneously and head to the destination h randomly. We analyze the first passage time (FPT) Y_sh(z) and the mean first passage time (MFPT) 〈Y_sh(z)〉 of multiple Brownian particles on complex networks. Equations of Y_sh(z) and 〈Y_sh(z)〉 are obtained. On a variety of commonly encountered networks, we observe first passage properties of multiple Brownian particles from different aspects. We find that 〈Y_sh(z)〉 drops substantially when particle number z increases at the first stage, and converges to d_sh, the distance between the source and the destination when z→∞. The distribution of FPT Prob{Y_sh(z)=t},t=0,1,2,… is also analyzed in these networks. The distribution curve peaks up towards t=d_sh when z increases. Consequently, if particle number z is set appropriately large, the first arriving Brownian particle will go along the shortest or near shortest paths between the source and the destination with high probability. Simulations confirm our analysis. Based on theoretical studies, we also investigate some practical problems using multiple Brownian particles, such as communication on P2P networks, optimal routing in small world networks, phenomenon of asymmetry in scale-free networks, information spreading in social networks, pervasion of viruses on the Internet, and so on. Our analytic and experimental results on multiple Brownian particles provide useful evidence for further understanding and properly tackling these problems.     

Analysis of the spatial and temporal distributions between successive earthquakes: Nonextensive statistical mechanics viewpoint

The spatial and temporal distributions between successive earthquakes are treated in the framework of nonextensive statistical mechanics. We find temporal distributions exhibit the power law behavior; q-exponential with q>1. It means the earthquakes are strongly correlated in time. The spatial distributions obey the q-exponential form with q<1. We also examine the dependence of the q exponent on magnitude range, covering period, time interval and size of the region where data are gathered. The conjecture of Abe et al. [S. Abe, N. Suzuki, Physica A 350 (2005) 588] has been examined for different categories of data. The results show a strange relation between q values of the spatial and temporal distributions.     

Universal power laws in the threshold network model: A theoretical analysis based on extreme value theory

We theoretically and numerically investigated the threshold network model with a generic weight function where there were a large number of nodes and a high threshold. Our analysis was based on extreme value theory, which gave us a theoretical understanding of the distribution of independent and identically distributed random variables within a sufficiently high range. Specifically, the distribution could be generally expressed by a generalized Pareto distribution, which enabled us to formulate the generic weight distribution function. By using the theorem, we obtained the exact expressions of degree distribution and clustering coefficient which behaved as universal power laws within certain ranges of degrees. We also compared the theoretical predictions with numerical results and found that they were extremely consistent.     

Phase transitions in social networks

We study a model of network with clustering and desired node degree. The original purpose of the model was to describe optimal structures of scientific collaboration in the European Union. The model belongs to the family of exponential random graphs. We show by numerical simulations and analytical considerations how a very simple Hamiltonian can lead to surprisingly complicated and eventful phase diagram.     

Possible canonical distributions for finite systems with nonadditive energy

It is shown that a small system in thermodynamic equilibrium with a finite thermostat can have a q-exponential probability distribution which closely depends on the energy nonextensivity and the particle number of the thermostat. The distribution function will reduce to the exponential one at the thermodynamic limit. However, the nonextensivity of the system should not be neglected.     

Boltzmann power laws

In this paper we propose a model based on the Boltzmann distribution as a mechanism for generating power laws, Boltzmann Power Laws (BPL). Some of these power laws are studied and compared to popular power laws such as ‘1/f’ noise and self-organized criticality (SOC). We will show how, in some cases, these BPLs reproduce behaviors similar to the finite size scaling (FSS) scenario, which is typical of SOC.     

A complex network model based on the Gnutella protocol

Gnutella is one of the basic protocols for P2P software. In this paper, a novel network model based on Gnutella is introduced. The mechanism of this network is based on resource occupancy and search activities of peers. As for the structure, the power-law exponent of in-degree γ_in≈4.2, the length of the average shortest path 〈l〉=57.74, and the diameter of the network is 156; these topological properties of the proposed structure differ from known results.     

Sensitivity of Exponents of Three-Power Laws to Hybrid Ratio in Weighted HUHPM

The sensitivity of exponents of three-power laws for node degree, node strength and edged weight to hybrid ratio are studied analytically and numerically in the weighted harmonious unifying hybrid preferential model （HUHPM）, which is extended from an-weighted hybrid preferential attachment model we proposed previously [Chin. Phys. Lett. 22 （2005）719]. Our weighted HUHPMs plus the Barrat-Barthelemy-Vespignani model and the traffic-driven evolution model, respectively, are taken as two typical examples for demonstration and application of the HUHPM.