基于随机矩阵理论的城市人群移动行为分析

移动通信应用为人类移动规律的研究提供了独特的数据来源. 本文通过城市手机用户的分布数据,研究城市移动人群的整体动力学行为. 借助随机矩阵理论的方法,通过比较移动人群数据与随机数据在互相关矩阵谱分布上的差异,发现移动人群数据互相关矩阵的相关系数均值、最大本征值及其对应的本征向量明显偏离于随机互相关矩阵的分布,指出这种差异体现了城市移动人群的整体行为特性,且这种差异在不同时间段也会有所不同. 研究结果体现出相关系数的均值和最大本征值的波动趋势,并指出本征向量成员权重的时空模式与城市移动人群整体行为特征的波动过 关键词： 随机矩阵理论 移动人群 宏观行为     

Identifying the node spreading influence with largest k-core values

Identifying the nodes with largest spreading influence of complex networks is one of the most promising domains. By taking into account the neighbors' k-core values, we present an improved neighbors' k-core (INK) method which is the sum of the neighbors' k-core values with a tunable parameter α to evaluate the node spreading influence with largest k-core values. Comparing with the Susceptible–Infected–Recovered (SIR) results for four real networks, the INK method could identify the node spreading influence with largest k-core values more accurately than the ones generated by the degree k, closeness C, betweenness B and coreness centrality method.     

Correlation and volatility in an Indian stock market: A random matrix approach

We examine the volatility of an Indian stock market in terms of correlation of stocks and quantify the volatility using the random matrix approach. First we discuss trends observed in the pattern of stock prices in the Bombay Stock Exchange for the three-year period 2000–2002. Random matrix analysis is then applied to study the relationship between the coupling of stocks and volatility. The study uses daily returns of 70 stocks for successive time windows of length 85 days for the year 2001. We compare the properties of matrix C of correlations between price fluctuations in time regimes characterized by different volatilities. Our analyses reveal that (i) the largest (deviating) eigenvalue of C correlates highly with the volatility of the index, (ii) there is a shift in the distribution of the components of the eigenvector corresponding to the largest eigenvalue across regimes of different volatilities, (iii) the inverse participation ratio for this eigenvector anti-correlates significantly with the market fluctuations and finally, (iv) this eigenvector of C can be used to set up a Correlation Index, CI whose temporal evolution is significantly correlated with the volatility of the overall market index.     

Impact of initial lattice structures on networks generated by traces of random walks

We show that the platform stage of network evolution plays a principal role in the topology of resulting networks generated by short-cuts stimulated by the movements of a random walker, the mechanism of which tends to produce power-law degree distributions. To examine the numerical results, we have developed a statistical method which relates the power-law exponent γ to random properties of the subgraph developed in the platform stage. As a result, we find that an important exponent in the network evolution is α, which characterizes the size of the subgraph in the form V∼t^α, where V and t denote the number of vertices in the subgraph and the time variable, respectively. 2D lattices can impose specific limitations on the walker’s diffusion, which keeps the value of α within a moderate range and provides typical properties of complex networks. 1D and 3D cases correspond to different ends of the spectrum for α, with 2D cases in between. Especially for 2D square lattices, a discontinuous change of the network structure is observed, which varies according to whether γ is greater or less than 2. For 1D cases, we show that emergence of nearly complete subgraphs is guaranteed by α<1/2, although the transient power-law is permitted at low increase rates of edges. Additionally, the model exhibits a spontaneous emergence of highly clustered structures regardless of its initial structure.     

BEHAVIOR OF DERIVATIVES OF EIGENVALUES AND EIGENVECTORS IN CURVE VEERING AND MODE LOCALIZATION AND THEIR RELATION TO CLOSE EIGENVALUES

The problem to measure the phenomena of eigenvalue curve veering and mode localization is addressed in this paper. The second derivative of an eigenvalue and the first derivative of an eigenvector are taken as the measures, numerically showing curve veering and mode localization. Based on the measurement, close eigenvalues, as a key factor for the occurrence of the phenomena, are defined. Two eigenvalues are considered to be close, if their difference is small enough to cause the occurrence of the phenomena. The curve veering and mode localization can be noted by comparison of the derivatives with a critical value and hence the associated eigenvalues are close. Weakly coupled springs are given as an example.     

Characterizing the dynamical importance of network nodes and links

The largest eigenvalue of the adjacency matrix of networks is a key quantity determining several important dynamical processes on complex networks. Based on this fact, we present a quantitative, objective characterization of the dynamical importance of network nodes and links in terms of their effect on the largest eigenvalue. We show how our characterization of the dynamical importance of nodes can be affected by degree-degree correlations and network community structure. We discuss how our characterization can be used to optimize techniques for controlling certain network dynamical processes and apply our results to real networks.     

Eigenvector approximation leading to exponential speedup of quantum eigenvalue calculation

We present an efficient method for preparing the initial state required by the eigenvalue approximation quantum algorithm of Abrams and Lloyd. Our method can be applied when solving continuous Hermitian eigenproblems, e.g., the Schr?dinger equation, on a discrete grid. We start with a classically obtained eigenvector for a problem discretized on a coarse grid, and we efficiently construct, quantum mechanically, an approximation of the same eigenvector on a fine grid. We use this approximation as the initial state for the eigenvalue estimation algorithm, and show the relationship between its success probability and the size of the coarse grid.     

Statistical properties of cross-correlation in the Korean stock market

We investigate the statistical properties of the cross-correlation matrix between individual stocks traded in the Korean stock market using the random matrix theory (RMT) and observe how these affect the portfolio weights in the Markowitz portfolio theory. We find that the distribution of the cross-correlation matrix is positively skewed and changes over time. We find that the eigenvalue distribution of original cross-correlation matrix deviates from the eigenvalues predicted by the RMT, and the largest eigenvalue is 52 times larger than the maximum value among the eigenvalues predicted by the RMT. The b₄₇₃\beta_{473} coefficient, which reflect the largest eigenvalue property, is 0.8, while one of the eigenvalues in the RMT is approximately zero. Notably, we show that the entropy function E(s)E(\sigma) with the portfolio risk σ for the original and filtered cross-correlation matrices are consistent with a power-law function, E(σ) ~ s^-g\sigma^{-\gamma}, with the exponent γ ~ 2.92 and those for Asian currency crisis decreases significantly.     

Inhomogenous Model of Crossing Loops and Multidegrees of Some Algebraic Varieties

We consider a quantum integrable inhomogeneous model based on the Brauer algebra B(1) and discuss the properties of its ground state eigenvector. In particular we derive various sum rules, and show how some of its entries are related to multidegrees of algebraic varieties.     

Topological properties of stock networks based on minimal spanning tree and random matrix theory in financial time series

We investigated the topological properties of stock networks constructed by a minimal spanning tree. We compared the original stock network with the estimated network; the original network is obtained by the actual stock returns, while the estimated network is the correlation matrix created by random matrix theory. We found that the consistency between the two networks increases as more eigenvalues are considered. In addition, we suggested that the largest eigenvalue has a significant influence on the formation of stock networks.     

Synchronizability on complex networks via pinning control

It is proved that the maximum eigenvalue sequence of the principal submatrices of coupling matrix is decreasing. The method of calculating the number of pinning nodes is given based on this theory. The findings reveal the relationship between the decreasing speed of maximum eigenvalue sequence of the principal submatrices for coupling matrix and the synchronizability on complex networks via pinning control. We discuss the synchronizability on some networks, such as scale-free networks and small-world networks. Numerical simulations show that different pinning strategies have different pinning synchronizability on the same complex network, and the consistence between the synchronizability with pinning control and one without pinning control in various complex networks.     

On purely gravito-magnetic vacuum space-times

It is shown that there are no purely magnetic, vacuum, spacetime metrics where any one of the following conditions holds: (a) the ratio of any two eigenvalues of the Weyl tensor is constant, (b) both of the Riemann principal null directions, defining the time-like blade, are nonrotating, (c) the shear tensor possesses an eigenvector v ^a which is defined by one of the space-like Riemann principal directions, (d) the vorticity is parallel to v ^a , where v ^a is defined by one of the space-like Riemann principal directions.This revised version was published online in April 2005. The publishing date was inserted.     

The Largest Eigenvalue of Real Symmetric,Hermitian and Hermitian Self-dual Random Matrix Models with Rank One External Source,Part I

We consider the limiting location and limiting distribution of the largest eigenvalue in real symmetric (β=1), Hermitian (β=2), and Hermitian self-dual (β=4) random matrix models with rank 1 external source. They are analyzed in a uniform way by a contour integral representation of the joint probability density function of eigenvalues. Assuming the “one-band” condition and certain regularities of the potential function, we obtain the limiting location of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is not the critical value, and further obtain the limiting distribution of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is greater than the critical value. When the nonzero eigenvalue of the external source matrix is less than or equal to the critical value, the limiting distribution of the largest eigenvalue will be analyzed in a subsequent paper. In this paper we also give a definition of the external source model for all β>0.     

Spectral properties of Google matrix of Wikipedia and other networks

We study the properties of eigenvalues and eigenvectors of the Google matrix of the Wikipedia articles hyperlink network and other real networks. With the help of the Arnoldi method, we analyze the distribution of eigenvalues in the complex plane and show that eigenstates with significant eigenvalue modulus are located on well defined network communities. We also show that the correlator between PageRank and CheiRank vectors distinguishes different organizations of information flow on BBC and Le Monde web sites.     

Analysis of cross-correlations between financial markets after the 2008 crisis

We analyze the cross-correlation matrix C of the index returns of the main financial markets after the 2008 crisis using methods of random matrix theory. We test the eigenvalues of C for universal properties of random matrices and find that the majority of the cross-correlation coefficients arise from randomness. We show that the eigenvector of the largest deviating eigenvalue of C represents a global market itself. We reveal that high volatility of financial markets is observed at the same times with high correlations between them which lowers the risk diversification potential even if one constructs a widely internationally diversified portfolio of stocks. We identify and compare the connection and cluster structure of markets before and after the crisis using minimal spanning and ultrametric hierarchical trees. We find that after the crisis, the co-movement degree of the markets increases. We also highlight the key financial markets of pre and post crisis using main centrality measures and analyze the changes. We repeat the study using rank correlation and compare the differences. Further implications are discussed.     

Approximate vacuums in (:φ2m :g(x))2 field theories and perturbation series

A class of perturbation problems is considered, in which the Rayleigh-Schrödinger perturbation series for the ground state eigenvalue and eigenvector are presumed to diverge. This class includes the (:^2m :g(x))₂, (m=2, 3) quantum field theory models and the quantum mechanical anharmonic oscillator. It is shown that, using matrix elements and vectors which occur in the series coefficients, one may construct convergent approximants to the eigenvalue and eigenvector. Results of a calculation of the ground state energy of thex ⁴ anharmonic oscillator are given.Supported in part by the National Research Council of Canada.     

Improved steered response power method for sound source localization based on principal eigenvector

Sound source localization is essential in many microphone arrays application, ranging from teleconferencing systems to artificial perception in a reverberant noisy environment. The steered response power (SRP) using the phase transform (SRP-PHAT) source localization algorithm has been proved robust, however, the performance of the SRP-PHAT algorithm degrades in highly reverberant noisy environment. Though the SRP-based maximum likelihood localizers are more robust than SRP-PHAT, they have the drawback of requiring noise variance to be estimated in a silent room. This paper presents an improved SRP-PHAT algorithm based on principal eigenvector. Sound source location is estimated from the principal eigenvector computed from the frequency-domain correlation matrix. Using both simulated and real data, we show that the proposed algorithm achieves higher source localization accuracy compared to the SRP-PHAT algorithm.     

k-Core organization of complex networks

We analytically describe the architecture of randomly damaged uncorrelated networks as a set of successively enclosed substructures--k-cores. The k-core is the largest subgraph where vertices have at least k interconnections. We find the structure of k-cores, their sizes, and their birthpoints--the bootstrap percolation thresholds. We show that in networks with a finite mean number zeta2 of the second-nearest neighbors, the emergence of a k-core is a hybrid phase transition. In contrast, if zeta2 diverges, the networks contain an infinite sequence of k-cores which are ultrarobust against random damage.     

Generalized Hellmann-Feynman Theorem for Ensemble Average Expressed in the Formalism of Pure State Expectation

We show that the generalized Hellmann-Feynman theorem (GHFT) for ensemble average can be expressed in the formalism of pure state expectation. We employ the thermo field dynamics proposed by Umezawa and Takahash to realize this goal, the pure state |0(β)〉 is defined in doubled eigenvector space of the Hamiltonian. Relation between the GHFT and entropy-variation in the context of |0(β)〉 is also derived.     

Predicting criticality and dynamic range in complex networks: effects of topology

The collective dynamics of a network of coupled excitable systems in response to an external stimulus depends on the topology of the connections in the network. Here we develop a general theoretical approach to study the effects of network topology on dynamic range, which quantifies the range of stimulus intensities resulting in distinguishable network responses. We find that the largest eigenvalue of the weighted network adjacency matrix governs the network dynamic range. When the largest eigenvalue is exactly one, the system is in a critical state and its dynamic range is maximized. Further, we examine higher order behavior of the steady state system, which predicts that networks with more homogeneous degree distributions should have higher dynamic range. Our analysis, confirmed by numerical simulations, generalizes previous studies in terms of the largest eigenvalue of the adjacency matrix.