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Register a new userTowards uncovering the structure of power fluctuations of wind farms
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The structure of the turbulence-driven power fluctuations in a wind farm is fundamentally described from basic concepts. A derived tuning-free model, supported with experiments, reveals the underlying spectral content of the power fluctuations of a wind farm. It contains two power-law trends and oscillations in the relatively low- and high-frequency ranges. The former is mostly due to the turbulent interaction between the flow and the turbine properties; whereas the latter is due to the advection between turbine pairs. The spectral wind-farm scale power fluctuations $Phi_P$ exhibits a power-law decay proportional to $f^{-5/3-2}$ in the region corresponding to the turbulence inertial subrange and at relatively large scales, $Phi_Psim f^{-2}$. Due to the advection and turbulent diffusion of large-scale structures, a spectral oscillation exists with the product of a sinusoidal behavior and an exponential decay in the frequency domain.
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Many time series produced by complex systems are empirically found to follow power-law distributions with different exponents $alpha$. By permuting the independently drawn samples from a power-law distribution, we present non-trivial bounds on the memory strength (1st-order autocorrelation) as a function of $alpha$, which are markedly different from the ordinary $pm 1$ bounds for Gaussian or uniform distributions. When $1 < alpha leq 3$, as $alpha$ grows bigger, the upper bound increases from 0 to +1 while the lower bound remains 0; when $alpha > 3$, the upper bound remains +1 while the lower bound descends below 0. Theoretical bounds agree well with numerical simulations. Based on the posts on Twitter, ratings of MovieLens, calling records of the mobile operator Orange, and browsing behavior of Taobao, we find that empirical power-law distributed data produced by human activities obey such constraints. The present findings explain some observed constraints in bursty time series and scale-free networks, and challenge the validity of measures like autocorrelation and assortativity coefficient in heterogeneous systems.
In a network, we define shell $ell$ as the set of nodes at distance $ell$ with respect to a given node and define $r_ell$ as the fraction of nodes outside shell $ell$. In a transport process, information or disease usually diffuses from a random node and reach nodes shell after shell. Thus, understanding the shell structure is crucial for the study of the transport property of networks. For a randomly connected network with given degree distribution, we derive analytically the degree distribution and average degree of the nodes residing outside shell $ell$ as a function of $r_ell$. Further, we find that $r_ell$ follows an iterative functional form $r_ell=phi(r_{ell-1})$, where $phi$ is expressed in terms of the generating function of the original degree distribution of the network. Our results can explain the power-law distribution of the number of nodes $B_ell$ found in shells with $ell$ larger than the network diameter $d$, which is the average distance between all pairs of nodes. For real world networks the theoretical prediction of $r_ell$ deviates from the empirical $r_ell$. We introduce a network correlation function $c(r_ell)equiv r_{ell+1}/phi(r_ell)$ to characterize the correlations in the network, where $r_{ell+1}$ is the empirical value and $phi(r_ell)$ is the theoretical prediction. $c(r_ell)=1$ indicates perfect agreement between empirical results and theory. We apply $c(r_ell)$ to several model and real world networks. We find that the networks fall into two distinct classes: (i) a class of {it poorly-connected} networks with $c(r_ell)>1$, which have larger average distances compared with randomly connected networks with the same degree distributions; and (ii) a class of {it well-connected} networks with $c(r_ell)<1$.
The spectra of empirical correlation matrices, constructed from multivariate data, are widely used in many areas of sciences, engineering and social sciences as a tool to understand the information contained in typically large datasets. In the last two decades, random matrix theory-based tools such as the nearest neighbour eigenvalue spacing and eigenvector distributions have been employed to extract the significant modes of variability present in such empirical correlations. In this work, we present an alternative analysis in terms of the recently introduced spacing ratios, which does not require the cumbersome unfolding process. It is shown that the higher order spacing ratio distributions for the Wishart ensemble of random matrices, characterized by the Dyson index $beta$, is related to the first order spacing ratio distribution with a modified value of co-dimension $beta$. This scaling is demonstrated for Wishart ensemble and also for the spectra of empirical correlation matrices drawn from the observed stock market and atmospheric pressure data. Using a combination of analytical and numerics, such scalings in spacing distributions are also discussed.
To evaluate the performance of prediction of missing links, the known data are randomly divided into two parts, the training set and the probe set. We argue that this straightforward and standard method may lead to terrible bias, since in real biological and information networks, missing links are more likely to be links connecting low-degree nodes. We therefore study how to uncover missing links with low-degree nodes, namely links in the probe set are of lower degree products than a random sampling. Experimental analysis on ten local similarity indices and four disparate real networks reveals a surprising result that the Leicht-Holme-Newman index [E. A. Leicht, P. Holme, and M. E. J. Newman, Phys. Rev. E 73, 026120 (2006)] performs the best, although it was known to be one of the worst indices if the probe set is a random sampling of all links. We further propose an parameter-dependent index, which considerably improves the prediction accuracy. Finally, we show the relevance of the proposed index on three real sampling methods.
Starting from inhomogeneous time scaling and linear decorrelation between successive price returns, Baldovin and Stella recently proposed a way to build a model describing the time evolution of a financial index. We first make it fully explicit by using Student distributions instead of power law-truncated Levy distributions; we also show that the analytic tractability of the model extends to the larger class of symmetric generalized hyperbolic distributions and provide a full computation of their multivariate characteristic functions; more generally, the stochastic processes arising in this framework are representable as mixtures of Wiener processes. The Baldovin and Stella model, while mimicking well volatility relaxation phenomena such as the Omori law, fails to reproduce other stylized facts such as the leverage effect or some time reversal asymmetries. We discuss how to modify the dynamics of this process in order to reproduce real data more accurately.
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