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Whether real-world complex networks are scale free or not has long been controversial. Recently, in Broido and Clauset [A. D. Broido, A. Clauset, Nat. Commun. 10, 1017 (2019)], it was claimed that the degree distributions of real-world networks are rarely power law under statistical tests. Here, we attempt to address this issue by defining a fundamental property possessed by each link, the degree–degree distance, the distribution of which also shows signs of being power law by our empirical study. Surprisingly, although full-range statistical tests show that degree distributions are not often power law in real-world networks, we find that in more than half of the cases the degree–degree distance distributions can still be described by power laws. To explain these findings, we introduce a bidirectional preferential selection model where the link configuration is a randomly weighted, two-way selection process. The model does not always produce solid power-law distributions but predicts that the degree–degree distance distribution exhibits stronger power-law behavior than the degree distribution of a finite-size network, especially when the network is dense. We test the strength of our model and its predictive power by examining how real-world networks evolve into an overly dense stage and how the corresponding distributions change. We propose that being scale free is a property of a complex network that should be determined by its underlying mechanism (e.g., preferential attachment) rather than by apparent distribution statistics of finite size. We thus conclude that the degree–degree distance distribution better represents the scale-free property of a complex network.

Understanding, quantifying and predicting market fluctuation has become increasingly important in recent decades. Volatility and volume are the two commonly used quantities to study the market dynamics and the relationship between these two has been modeled and debated for years with several hypothesis been put forward. Using empirical data, we investigate the causality and correlation between volume and volatility and find new ways in which they interact, particularly when the levels of both are high. We find that the volume-conditional volatility distribution scales with volume as a power-law function with an exponential cutoff. We exploit the characteristics of a volume-volatility scatterplot and find a strong correlation between logarithmic volume and a quantity we define as local maximum volatility (LMV), the highest volatility observed in a given range of volume. This supports our empirical analysis, showing that volume is an effective parameter for prediction of the maximum value of volatility for both same-day and near-future time periods. The joint conditional probability of volume and volatility also indicates if we invoke both quantities, the prediction of the largest next-day volatility will be better than invoking either one alone. This approach is thus a greatly improved method of risk assessment.

The linkages between crude oil spot and futures markets are widely studied in previous literatures. Most of these conventional methods are static, linear and parametric. In order to overcome this difficulty, we employ the nonparametric and nonlinear symmetric thermal optimal path (TOPS) method to determine the time-dependent lead–lag relationship between two time series. Based on the daily, weekly, monthly spot and futures returns for maturities of one, two, three and four months of the West Texas Intermediate (WTI) crude oil benchmark, we apply the TOPS method together with self-consistent test to investigate the time-varying lead–lag dependences between spot and futures markets from 1987 to 2017. We find that, for total three data sets with different frequencies, the TOPS paths indicate an alternate lead–lag structure instead of a dominance between spot and futures markets in most of the time periods. For the daily data, the lead–lag relationship between the two markets exists within one day except on some specific days. Moreover, without considering the units of time, on average, the lead–lag steps between the two markets are diminishing with the increasing frequency. Roughly speaking, these periods with strong lead–lag signals coincide with major influential changes in the oil and stock markets and the geopolitics, which can reflect real economic, financial and geopolitical regime changes. All in all, the results show that the lead–lag relationship between the spot and futures oil markets exists only temporarily.

We construct the empirical information network (EIN) of traders using the order flow data of the constituent stocks of SZSE 100 Index in 2013. A statistical validation method is applied to the edges of the network to filter out noises and uncover the intrinsic interaction behaviors of traders. We investigate the correlation between topological structures and statistical properties for their largest connected components. We find that the statistical validated network shows an assortative mixing pattern while the original network exhibits a disassortative mixing pattern. We consider two definitions of edge weight for comparison but there is no significant difference in a same network. We also analyze the mutual relationships among node degree, edge weight and node strength.

Being able to theoretically predict the outbreak threshold of an epidemic is essential in disease control. Real-world correlated networks are ubiquitous and their topological structure strongly affects the prediction accuracy of present-day theoretical methods. Quantifying their accuracy and fitness in predicting outbreak thresholds of correlated networks is thus essential. We use a susceptible–infected–removed (SIR) model to examine four widely-used theoretical methods – the heterogeneous mean-field ( H M F ), quenched mean-field ( Q M F ), dynamical message passing ( D M P ), and connectivity matrix ( C M ) methods – to predict the outbreak threshold of a correlated network. The potential topological structure of correlated network impacts on prediction accuracy of these four methods. We emphasize that the quantitative changes of degree correlation, degree distribution exponent and network size strongly affect the outbreak of SIR spreading dynamics, and compare the simulation results with the theoretical ones obtained from these four methods. The extensive experiments in synthetic networks show that (a) the increasing degree correlation coefficients reduce outbreak threshold and suppress outbreak size; (b) the increasing degree distribution exponents raise outbreak threshold but suppress outbreak size; (c) the increasing network sizes decrease outbreak threshold but do not affect outbreak size; (d) as for four theoretical methods, C M and D M P are more likely to precisely predict outbreak threshold because they to some extent incorporate network topology with dynamical correlations. The experimental results in 50 real-world networks also prove the above conclusions.

To investigate the characteristics of extreme events in financial markets and the corresponding return intervals among these events, we use a Potts dynamic system to construct a random financial time series model of the attitudes of market traders. We use multiscale multifractal detrended cross-correlation analysis (MM-DCCA) and Lempel–Ziv complexity (LZC) perform numerical research of the return intervals for two significant China’s stock market indices and for the proposed model. The new MM-DCCA method is based on the Hurst surface and provides more interpretable cross-correlations of the dynamic mechanism between different return interval series. We scale the LZC method with different exponents to illustrate the complexity of return intervals in different scales. Empirical studies indicate that the proposed return intervals from the Potts system and the real stock market indices hold similar statistical properties.

Fundamental research in physics has long been a prerequisite for computer scientists and engineers to design innovative products, such as laptops and cell phones. Technological innovations and fundamental research are both part of the so-called STEM fields (Science, Technology, Engineering, and Mathematics), which are known to substantially contribute to economic growth. However, the questions still remain: how much contribution do these fields make to both wealth accumulation and inequality at different levels of analysis? First, analyzing the lists of world’s wealthiest individuals, the Zipf plot analysis demonstrates that STEM billionaires contribute more to wealth inequality than their non-STEM counterparts. Analyzing the companies in the S&P500, we find that STEM firms contribute more to wealth inequality and have larger growth rates on average than the non-STEM firms. Finally, we show that the more STEM graduates in a country, the larger its GDP growth rate. In combination, we demonstrate that STEM is a fractal mechanism that drives wealth accumulation—and the wealth inequality— at different scales of economy—from individual wealth to firm valuation to country GDP.

Flight delays have negative impacts on passengers, carriers, and airports. To reduce these unpopular influence, we need to find the statistical law of the collective behavior of passenger flights. We use a mean-field approach to analyze big data listing the departure and arrival records of all American domestic passenger flights in 20 years. We treat passenger flights as particle gases and define their dimensionless velocity, quasi-thermodynamic quantities — pressure, volume, temperature, and mole number, respectively. By introducing phenomenological parameters a and b to set up van der Waals-like state equations, we erect a universal gaseous constant R for actually operated passenger flights, their counterparts on schedule, and ”delayor gases” defined as the difference between them. We find that the attractive coefficient of ”delayor gases” positively correlates with the average delay per flight on airports. Rescaling state equations for passenger flights across all 20 years, we find a universal function. This is a significant step toward understanding flight delays and dealing with temporal big data with the tools of statistical physics.

Detection of power-law behavior and studies of scaling exponents uncover the characteristics of complexity in many real world phenomena. The complexity of financial markets has always presented challenging issues and provided interesting findings, such as the inverse cubic law in the tails of stock price fluctuation distributions. Motivated by the rise of novel digital assets based on blockchain technology, we study the distributions of cryptocurrency price fluctuations. We consider Bitcoin returns over various time intervals and from multiple digital exchanges, in order to investigate the existence of universal scaling behavior in the tails, and ascertain whether the scaling exponent supports the presence of a finite second moment. We provide empirical evidence on slowly decaying tails in the distributions of returns over multiple time intervals and different exchanges, corresponding to a power-law. We estimate the scaling exponent and find an asymptotic power-law behavior with 2 α 2 . 5 suggesting that Bitcoin returns, in addition to being more volatile, also exhibit heavier tails than stocks, which are known to be around 3. Our results also imply the existence of a finite second moment, thus providing a fundamental basis for the usage of standard financial theories and covariance-based techniques in risk management and portfolio optimization scenarios.

K-core and bootstrap percolation are widely studied models that have been used to represent and understand diverse deactivation and activation processes in natural and social systems. Since these models are considerably similar, it has been suggested in recent years that they could be complementary. In this manuscript we provide a rigorous analysis that shows that for any degree and threshold distributions heterogeneous bootstrap percolation can be mapped into heterogeneous k-core percolation and vice versa, if the functionality thresholds in both processes satisfy a complementary relation. Another interesting problem in bootstrap and k-core percolation is the fraction of nodes belonging to their giant connected components P∞b and P∞c, respectively. We solve this problem analytically for arbitrary randomly connected graphs and arbitrary threshold distributions, and we show that P∞b and P∞c are not complementary. Our theoretical results coincide with computer simulations in the limit of very large graphs. In bootstrap percolation, we show that when using the branching theory to compute the size of the giant component, we must consider two different types of links, which are related to distinct spanning branches of active nodes. Fil: Di Muro, Matias Alberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata. Instituto de Investigaciones Físicas de Mar del Plata. Universidad Nacional de Mar del Plata. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Físicas de Mar del Plata; Argentina Fil: Valdez, Lucas Daniel. Boston University; Estados Unidos Fil: Stanley, Harry Eugene. Boston University; Estados Unidos Fil: Buldyrev, Sergey V.. Yeshiva University; Estados Unidos Fil: Braunstein, Lidia Adriana. Politecnico di Milano; Italia

We study auto-correlations and cross-correlations of daily price changes and daily volume changes of thirteen global stock market indices, using multifractal detrended fluctuation analysis (MF-DFA) and multifractal detrended cross-correlation analysis (MF-DXA). We find rather distinct multifractal behavior of price and volume changes. Our results indicate that the time series of price changes are more complex than those of volume changes, and that large fluctuations dominate multifractal behavior of price changes, while small fluctuations dominate multifractal behavior of volume changes. We also find that there is an absence of correlations in price changes, there are anti-persistent long-term correlations in volume changes, and there are anti-persistent long-term cross-correlations between price and volume changes. Shuffling the series reveals that multifractality of both price changes and volume changes arises from a broad probability density function.

We investigate multifractal properties of daily price changes in currency rates using the multifractal detrended fluctuation analysis (MF-DFA). We analyze managed and independent floating currency rates in eight countries, and determine the changes in multifractal spectrum when transitioning between the two regimes. We find that after the transition from managed to independent float regime the changes in multifractal spectrum (position of maximum and width) indicate an increase in market efficiency. The observed changes are more pronounced for developed countries that have a well established trading market. After shuffling the series, we find that the multifractality is due to both probability density function and long term correlations for managed float regime, while for independent float regime multifractality is in most cases caused by broad probability density function.

Complex networks appear in almost every aspect of science and technology. Previous work in network theory has focused primarily on analyzing single networks that do not interact with other networks, despite the fact that many real-world networks interact with and depend on each other. Very recently an analytical framework for studying the percolation properties of interacting networks has been introduced. Here we review the analytical framework and the results for percolation laws for a Network Of Networks (NONs) formed by n interdependent random networks. The percolation properties of a network of networks differ greatly from those of single isolated networks. In particular, because the constituent networks of a NON are connected by node dependencies, a NON is subject to cascading failure. When there is strong interdependent coupling between networks, the percolation transition is discontinuous (first-order) phase transition, unlike the wellknown continuous second-order transition in single isolated networks. Moreover, although networks with broader degree distributions, e.g., scale-free networks, are more robust when analyzed as single networks, they become more vulnerable in a NON. We also review the effect of space embedding on network vulnerability. It is shown that for spatially embedded networks any finite fraction of dependency nodes will lead to abrupt transition. 2014 Elsevier Ltd. All rights reserved.

The properties of q-dependent cross-correlation matrices of stock market have been analyzed by using the random matrix theory and complex network. The correlation structures of the fluctuations at different magnitudes have unique properties. The cross-correlations among small fluctuations are much stronger than those among large fluctuations. The large and small fluctuations are dominated by different groups of stocks. We use complex network representation to study these q-dependent matrices and discover some new identities. By utilizing those q-dependent correlation-based networks, we are able to construct some portfolio by those most independent stocks which consistently perform the best. The optimal multifractal order for portfolio optimization is approximately $q=2$. These results have deepened our understanding about the collective behaviors of the complex financial system., 25 pages, 16 figures

To study the statistical structure of crosscorrel ations in empirical data, we generalize random matrix theory and propose a new method of cross- correlation analysis, known as autoregressive random matrix theory (ARRMT). ARRMT takes into account the influence of auto-correlations in the study of cross- correlations in multiple time series. We first analytically and numerically determine how auto-correlations affect the eigenva lue distribution of the correlation matrix. Then we introduce ARRMT with a detailed procedure of how to implement the method. Finally, we illustrate the method using two examples taken from inflation rates for air pressure data for 95 US cities. Published by AIP Publishing

Random walks and Sierpinski gaskets are used to introduce fractal dimensions, and diffusion limited aggregates finish this “dessert”, which is closely related to the critical exponents at the end of the previous chapter .

Cascading failures in complex systems have been studied extensively using two different models: $k$-core percolation and interdependent networks. We combine the two models into a general model, solve it analytically and validate our theoretical results through extensive simulations. We also study the complete phase diagram of the percolation transition as we tune the average local $k$-core threshold and the coupling between networks. We find that the phase diagram of the combined processes is very rich and includes novel features that do not appear in the models studying each of the processes separately. For example, the phase diagram consists of first and second-order transition regions separated by two tricritical lines that merge together and enclose a novel two-stage transition region. In the two-stage transition, the size of the giant component undergoes a first-order jump at a certain occupation probability followed by a continuous second-order transition at a lower occupation probability. Furthermore, at certain fixed interdependencies, the percolation transition changes from first-order $\rightarrow$ second-order $\rightarrow$ two-stage $\rightarrow$ first-order as the $k$-core threshold is increased. The analytic equations describing the phase boundaries of the two-stage transition region are set up and the critical exponents for each type of transition are derived analytically.

Made available in DSpace on 2018-12-11T16:46:06Z (GMT). No. of bitstreams: 0 Previous issue date: 2017-02-01 Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Using an agent-based model we examine the dynamics of stock price fluctuations and their rates of return in an artificial financial market composed of fundamentalist and chartist agents with and without confidence. We find that chartist agents who are confident generate higher price and rate of return volatilities than those who are not. We also find that kurtosis and skewness are lower in our simulation study of agents who are not confident. We show that the stock price and confidence index-both generated by our model-are cointegrated and that stock price affects confidence index but confidence index does not affect stock price. We next compare the results of our model with the S&P 500 index and its respective stock market confidence index using cointegration and Granger tests. As in our model, we find that stock prices drive their respective confidence indices, but that the opposite relationship, i.e., the assumption that confidence indices drive stock prices, is not significant. Department of Economics Sao Paulo State University (UNESP) Sao Paulo Metropolitan Company Federal Institute of Education Science and Technology Metropolitan College Boston University Center for Polymer Studies and Department of Physics Boston University Department of Economics Sao Paulo State University (UNESP) FAPESP: 2014/19534-8

We carry out a study of percolation behaviors of clustered networks with partial support‐dependence relations by adopting two different attacking strategies, attacking only one network and both networks, which help to further understand real coupled networks. For two different attacking strategies we find that the system changes from a second-order phase transition to a first-order phase transition as coupling strengthq increases. We also notice that the first-order region becomes smaller and the secondorder region becomes larger as average degree or clustering coefficient increases. And, as the average supported degree approaches infinity, coupled clustered networks become independent and only the second-order transition is observed, which is similar toq=0. Furthermore, we find that clustering coefficient has a significant impact on robustness of the system for strong coupling strength, but for weak coupling strength it has little influence, especially for attacking both networks. The study implies that we can obtain a more robust network by reducing clustering coefficient and increasing average degree for strong coupling strength. However, for weak coupling strength, a more robust network is obtained only by increasing average degree for the same support average degree.

The behaviors of fat-tailed distribution, linear long memory, and nonlinear long memory are considered as possible sources of apparent multifractality. Which behavior should be preserved in null models plays an important role in statistical tests of empirical multifractality. In this paper, we compare the performance of two null models on testing the existence of multifractality in fractional Brownian motions (fBm), Markov-switching multifractal (MSM) model, and financial returns. One null model is obtained by shuffling the original data, which keeps the distribution unchanged. The other null model is generated by the iterative amplitude adjusted Fourier transform (IAAFT) algorithm, which insures that the surrogate data and the original data sharing the same distribution and linear long memory behavior. We find that the tests based on the shuffle null model only reject the multifractality in fBm with and the tests based on the IAAFT null model reject the multifractality in fBms (except for ). And the multifractality in MSM and financial returns are significantly supported by the tests based on both null models. Our findings also shed light on the necessity of choosing suitable null models to test multifractality in other complex systems.

Two-dimensional (2D) multifractal detrended fluctuation analysis (MF-DFA) has been used to study monofractality and multifractality on 2D surfaces, but when it is used to calculate the generalized Hurst exponent in a fixed time scale, the presence of crossovers can bias the outcome. To solve this problem, multiscale multifractal analysis (MMA) was recent employed in a one-dimensional case. MMA produces a Hurst surface h(q,s) that provides a spectrum of local scaling exponents at different scale ranges such that the positions of the crossovers can be located. We apply this MMA method to a 2D surface and identify factors that influence the results. We generate several synthesized surfaces and find that crossovers are consistently present, which means that their fractal properties differ at different scales. We apply MMA to the surfaces, and the results allow us to observe these differences and accurately estimate the generalized Hurst exponents. We then study eight natural texture images and two real-world images and find (i) that the moving window length (WL) and the slide length (SL) are the key parameters in the MMA method, that the WL more strongly influences the Hurst surface than the SL, and that the combination of WL=4 and SL=4 is optimal for a 2D image; (ii) that the robustness of h(2,s) to four common noises is high at large scales but variable at small scales; and (iii) that the long-term correlations in the images weaken as the intensity of Gaussian noise and salt and pepper noise is increased. Our findings greatly improve the performance of the MMA method on 2D surfaces.

Increasing evidence shows that real-world systems interact with one another via dependency connectivities. Failing connectivities are the mechanism behind the breakdown of interacting complex systems, e.g., blackouts caused by the interdependence of power grids and communication networks. Previous research analyzing the robustness of interdependent networks has been limited to undirected networks. However, most real-world networks are directed, their in-degrees and out-degrees may be correlated, and they are often coupled to one another as interdependent directed networks. To understand the breakdown and robustness of interdependent directed networks, we develop a theoretical framework based on generating functions and percolation theory. We find that for interdependent Erdős-Renyi networks the directionality within each network increases their vulnerability and exhibits hybrid phase transitions. We also find that the percolation behavior of interdependent directed scale-free networks with and without degree correlations is so complex that two criteria are needed to quantify and compare their robustness: the percolation threshold and the integrated size of the giant component during an entire attack process. Interestingly, we find that the in-degree and out-degree correlations in each network layer increase the robustness of interdependent degree heterogeneous networks that most real networks are, but decrease the robustness of interdependent networks with homogeneous degree distribution and with strong coupling strengths. Moreover, by applying our theoretical analysis to real interdependent international trade networks, we find that the robustness of these real-world systems increases with the in-degree and out-degree correlations, confirming our theoretical analysis.

The self-consistent probabilistic approach has proven itself powerful in studying the percolation behavior of interdependent or multiplex networks without tracking the percolation process through each cascading step. In order to understand how directed dependency links impact criticality, we employ this approach to study the percolation properties of networks with both undirected connectivity links and directed dependency links. We find that when a random network with a given degree distribution undergoes a second-order phase transition, the critical point and the unstable regime surrounding the second-order phase transition regime are determined by the proportion of nodes that do not depend on any other nodes. Moreover, we also find that the triple point and the boundary between first- and second-order transitions are determined by the proportion of nodes that depend on no more than one node. This implies that it is maybe general for multiplex network systems, some important properties of phase transitions can be determined only by a few parameters. We illustrate our findings using Erdos-Renyi (ER) networks., Comment: 7 pager 5 figure

Being able to predict the occurrence of extreme returns is important in financial risk management. Using the distribution of recurrence intervals---the waiting time between consecutive extremes---we show that these extreme returns are predictable on the short term. Examining a range of different types of returns and thresholds we find that recurrence intervals follow a $q$-exponential distribution, which we then use to theoretically derive the hazard probability $W(\Delta t |t)$. Maximizing the usefulness of extreme forecasts to define an optimized hazard threshold, we indicates a financial extreme occurring within the next day when the hazard probability is greater than the optimized threshold. Both in-sample tests and out-of-sample predictions indicate that these forecasts are more accurate than a benchmark that ignores the predictive signals. This recurrence interval finding deepens our understanding of reoccurring extreme returns and can be applied to forecast extremes in risk management., Comment: 18 pages, 5 figues, 3 tables

Complex systems are composed of mutually interacting components and the output values of these components usually exhibit long-range cross-correlations. Using wavelet analysis, we propose a method of characterizing the joint multifractal nature of these long-range cross correlations, a method we call multifractal cross wavelet analysis (MFXWT). We assess the performance of the MFXWT method by performing extensive numerical experiments on the dual binomial measures with multifractal cross correlations and the bivariate fractional Brownian motions (bFBMs) with monofractal cross correlations. For binomial multifractal measures, we find the empirical joint multifractality of MFXWT to be in approximate agreement with the theoretical formula. For bFBMs, MFXWT may provide spurious multifractality because of the wide spanning range of the multifractal spectrum. We also apply the MFXWT method to stock market indices, and in pairs of index returns and volatilities we find an intriguing joint multifractal behavior. The tests on surrogate series also reveal that the cross correlation behavior, particularly the cross correlation with zero lag, is the main origin of cross multifractality.

Employing detrended fluctuation analysis (DFA) and detrended cross-correlations analysis (DCCA), we analyze auto-correlations in the absolute returns for each of 30 Dow Jones Industrial Average (DJIA) constituents, S i , and cross-correlations in the absolute returns between the DJIA and each S i . We find that each DJIA member follows the DJIA in absolute returns, since the DCCA curve for each pair ( S i , DJIA i ) exhibits strong cross-correlations, with average DCCA exponent 〈 λ 〉 = 1.03 ± 0.04 . This value for 〈 λ 〉 implies that the power-law cross-correlations are of the 1 / f functional form. For the financial firms comprising the DJIA, we also find that the DFA and DCCA exponents controlling the duration of firm risk are somewhat larger than the corresponding values for the rest of the US financial industry.

The first return time (FRT) is the time it takes a random walker to first return to its original site, and the global first passage time (GFPT) is the first passage time for a random walker to move from a randomly selected site to a given site. We find that in finite networks the variance of FRT, Var(FRT), can be expressed Var(FRT)~$=2\langle$FRT$ \rangle \langle $GFPT$ \rangle -\langle $FRT$ \rangle^2-\langle $FRT$ \rangle$, where $\langle \cdot \rangle$ is the mean of the random variable. Therefore a method of calculating the variance of FRT on general finite networks is presented. We then calculate Var(FRT) and analyze the fluctuation of FRT on regular branched networks (i.e., Cayley tree) by using Var(FRT) and its variant as the metric. We find that the results differ from those in such other networks as Sierpinski gaskets, Vicsek fractals, T-graphs, pseudofractal scale-free webs, ($u,v$) flowers, and fractal and non-fractal scale-free trees., 7 pages,1 figure

In the threshold model of social contagions with non-redundant memory, researchers have overlooked the investigation on the limited imitation (LI) effect, which shows individual imitates the behavior adoption only for a certain range of ratio of his adopted informants. To understand such LI effect, we propose a social contagion model with a gate-like adoption probability consisting of the 'on' and 'off' thresholds. With extensive numerical simulations, we find that, given information transmission probability and a gate width (the 'off' threshold minus the 'on' threshold), there exists an optimal imitation capacity with the optimal 'on' threshold maximizing the final adoption size. And the large 'off' threshold serves for further enlarging the final adoption size at a given 'on' threshold. Besides, a cross phenomenon in phase transition is also uncovered: the increase of 'on' threshold causes the growth pattern of final behavior adoption size versus the information transmission probability change from the second-order to the first-order phase transition. We finally find that the above phenomena are qualitatively unaffected by the heterogeneity level of degree distribution. At last, an edge-based compartmental theory is conceived for theoretical analysis. And our suggested theory agrees well with the simulation results.

The static scaling law hypothesis for thermodynamic functions is formulated by the statement that the singular part of any thermodynamic potential is a generalized homogeneous function (GHF), where by definition a function f(x, y) is a GHF if there exist two numbers a, b such that for all positive values of λ, f (λax, λby) = λ f(x, y). We show that all Legendre transforms and all partial derivatives of a GHF are also GHF'S. Since every thermodynamic function is related to a given thermodynamic potential by some combination of Legendre transforms and partial derivatives, it follows that all thermodynamic functions are GHF'S. We then observe that every such function has a simple power law singularity at the critical point, and that the value of every critical-point exponent can be written down by inspection in terms of the two initial numbers a, b. Consideration of different paths of approach to the critical point lead to an important class of direct tests of the scaling hypothesis; among these are the familiar relations among certain combinations of critical-point exponents. Finally, the problems of extension to functions of more than two variables are discussed.

The size distribution of business firms is explained using number and size of firms' constituent components. It is a lognormal distribution multiplied by a stretching factor which can lead to a Pareto upper tail. This result is confirmed empirically.

We calculate the Shannon entropy of a time series by using the probability density functions of the characteristic sizes of the long-range correlated clusters introduced in [A. Carbone, G. Castelli, H.E. Stanley, Phys. Rev. E 69 (2004) 026105]. We define three different measures of the entropy related, respectively, to the length, the duration and the area of the clusters. For all the three cases, the entropy increases as the logarithm of a power of the size with exponents equal to the fractal dimension of the cluster length, duration and area, respectively. r 2007 Elsevier B.V. All rights reserved.

The study of animal foraging behaviour is of practical ecological importance1, and exemplifies the wider scientific problem of optimizing search strategies2. Lévy flights are random walks, the step lengths of which come from probability distributions with heavy power-law tails3, 4, such that clusters of short steps are connected by rare long steps. Lévy flights display fractal properties, have no typical scale, and occur in physical3, 4, 5 and chemical6 systems. An attempt to demonstrate their existence in a natural biological system presented evidence that wandering albatrosses perform Lévy flights when searching for prey on the ocean surface7. This well known finding2, 4, 8, 9 was followed by similar inferences about the search strategies of deer10 and bumblebees10. These pioneering studies have triggered much theoretical work in physics (for example, refs 11, 12), as well as empirical ecological analyses regarding reindeer13, microzooplankton14, grey seals15, spider monkeys16 and fishing boats17. Here we analyse a new, high-resolution data set of wandering albatross flights, and find no evidence for Lévy flight behaviour. Instead we find that flight times are gamma distributed, with an exponential decay for the longest flights. We re-analyse the original albatross data7 using additional information, and conclude that the extremely long flights, essential for demonstrating Lévy flight behaviour, were spurious. Furthermore, we propose a widely applicable method to test for power-law distributions using likelihood18 and Akaike weights19, 20. We apply this to the four original deer and bumblebee data sets10, finding that none exhibits evidence of Lévy flights, and that the original graphical approach10 is insufficient. Such a graphical approach has been adopted to conclude Lévy flight movement for other organisms13, 14, 15, 16, 17, and to propose Lévy flight analysis as a potential real-time ecosystem monitoring tool17. Our results question the strength of the empirical evidence for biological Lévy flights.

We study the statistical properties of a recently proposed social networks measure of fragmentation F after removal of a fraction q of nodes or links from the network. The measure F is defined as the ratio of the number of pairs of nodes that are not connected in the fragmented network to the total number of pairs in the original fully connected network. We compare this measure with the one traditionally used in percolation theory, P ∞ , the fraction of nodes in the largest cluster relative to the total number of nodes. Using both analytical and numerical methods, we study Erdős–Renyi (ER) and scale-free (SF) networks under various node removal strategies. We find that for a network obtained after removal of a fraction q of nodes above criticality, P ∞ ≈ ( 1 - F ) 1 / 2 . For fixed P ∞ and close to criticality, we show that 1 - F better reflects the actual fragmentation. For a given P ∞ , 1 - F has a broad distribution and thus one can improve significantly the fragmentation of the network. We also study and compare the fragmentation measure F and the percolation measure P ∞ for a real national social network of workplaces linked by the households of the employees and find similar results.

Researchers have developed several theoretical methods for predicting epidemic thresholds, including the mean-field like (MFL) method, the quenched mean-field (QMF) method, and the dynamical message passing (DMP) method. When these methods are applied to predict epidemic threshold they often produce differing results and their relative levels of accuracy are still unknown. We systematically analyze these two issues---relationships among differing results and levels of accuracy---by studying the susceptible-infected-recovered (SIR) model on uncorrelated configuration networks and a group of 56 real-world networks. In uncorrelated configuration networks the MFL and DMP methods yield identical predictions that are larger and more accurate than the prediction generated by the QMF method. When compared to the 56 real-world networks, the epidemic threshold obtained by the DMP method is closer to the actual epidemic threshold because it incorporates full network topology information and some dynamical correlations. We find that in some scenarios---such as networks with positive degree-degree correlations, with an eigenvector localized on the high $k$-core nodes, or with a high level of clustering---the epidemic threshold predicted by the MFL method, which uses the degree distribution as the only input parameter, performs better than the other two methods. We also find that the performances of the three predictions are irregular versus modularity.

We use a key concept of the continuous-time random walk formalism, i.e., continuous and fluctuating interevent times in which mutual dependence is taken into account, to model market fluctuation data when traders experience excessive (or superthreshold) losses or excessive (or superthreshold) profits. We analytically derive a class of "superstatistics" that accurately model empirical market activity data supplied by Bogachev, Ludescher, Tsallis, and Bunde that exhibit transition thresholds. We measure the interevent times between excessive losses and excessive profits and use the mean interevent discrete (or step) time as a control variable to derive a universal description of empirical data collapse. Our dominant superstatistic value is a power-law corrected by the lower incomplete gamma function, which asymptotically tends toward robustness but initially gives an exponential. We find that the scaling shape exponent that drives our superstatistics subordinates itself and a "superscaling" configuration emerges. Thanks to the Weibull copula function, our approach reproduces the empirically proven dependence between successive interevent times. We also use the approach to calculate a dynamic risk function and hence the dynamic VaR, which is significant in financial risk analysis. Our results indicate that there is a functional (but not literal) balance between excessive profits and excessive losses that can be described using the same body of superstatistics but different calibration values and driving parameters. We also extend our original approach to cover empirical seismic activity data (e.g., given by Corral), the interevent times of which range from minutes to years. Superpositioned superstatistics is another class of superstatistics that protects power-law behavior both for short- and long-time behaviors. These behaviors describe well the collapse of seismic activity data and capture so-called volatility clustering phenomena.

Real-world attacks can be interpreted as the result of competitive interactions between networks, ranging from predator-prey networks to networks of countries under economic sanctions. Although the purpose of an attack is to damage a target network, it also curtails the ability of the attacker, which must choose the duration and magnitude of an attack to avoid negative impacts on its own functioning. Nevertheless, despite the large number of studies on interconnected networks, the consequences of initiating an attack have never been studied. Here, we address this issue by introducing a model of network competition where a resilient network is willing to partially weaken its own resilience in order to more severely damage a less resilient competitor. The attacking network can take over the competitor nodes after their long inactivity. However, due to a feedback mechanism the takeovers weaken the resilience of the attacking network. We define a conservation law that relates the feedback mechanism to the resilience dynamics for two competing networks. Within this formalism, we determine the cost and optimal duration of an attack, allowing a network to evaluate the risk of initiating hostilities., 8 two-column pages, 6 figures, supplementary material; accepted for publication in Journal of the Royal Society Interface

The stability of networks is greatly influenced by their degree distributions and in particular by their breadth. Networks with broader degree distributions are usually more robust to random failures but less robust to localized attacks. To better understand the effect of the breadth of the degree distribution we study two models in which the breadth is controlled and compare their robustness against localized attacks (LA) and random attacks (RA). We study analytically and by numerical simulations the cases where the degrees in the networks follow a bi-Poisson distribution, P(k)=αe^{-λ_{1}}λ_{1}^{k}/k!+(1-α)e^{-λ_{2}}λ_{2}^{k}/k!,α∈[0,1], and a Gaussian distribution, P(k)=Aexp(-(k-μ)^{2}/2σ^{2}), with a normalization constant A where k≥0. In the bi-Poisson distribution the breadth is controlled by the values of α, λ_{1}, and λ_{2}, while in the Gaussian distribution it is controlled by the standard deviation, σ. We find that only when α=0 or α=1, i.e., degrees obeying a pure Poisson distribution, are LA and RA the same. In all other cases networks are more vulnerable under LA than under RA. For a Gaussian distribution with an average degree μ fixed, we find that when σ^{2} is smaller than μ the network is more vulnerable against random attack. When σ^{2} is larger than μ, however, the network becomes more vulnerable against localized attack. Similar qualitative results are also shown for interdependent networks.

Being able to forcast extreme volatility is a central issue in financial risk management. We present a large volatility predicting method based on the distribution of recurrence intervals between volatilities exceeding a certain threshold $Q$ for a fixed expected recurrence time $\tau_Q$. We find that the recurrence intervals are well approximated by the $q$-exponential distribution for all stocks and all $\tau_Q$ values. Thus a analytical formula for determining the hazard probability $W(\Delta t |t)$ that a volatility above $Q$ will occur within a short interval $\Delta t$ if the last volatility exceeding $Q$ happened $t$ periods ago can be directly derived from the $q$-exponential distribution, which is found to be in good agreement with the empirical hazard probability from real stock data. Using these results, we adopt a decision-making algorithm for triggering the alarm of the occurrence of the next volatility above $Q$ based on the hazard probability. Using a "receiver operator characteristic" (ROC) analysis, we find that this predicting method efficiently forecasts the occurrance of large volatility events in real stock data. Our analysis may help us better understand reoccurring large volatilities and more accurately quantify financial risks in stock markets., Comment: 13 pages and 7 figures

Estimating the critical points at which complex systems abruptly flip from one state to another is one of the remaining challenges in network science. Due to lack of knowledge about the underlying stochastic processes controlling critical transitions, it is widely considered difficult to determine the location of critical points for real-world networks, and it is even more difficult to predict the time at which these potentially catastrophic failures occur. We analyse a class of decaying dynamic networks experiencing persistent failures in which the magnitude of the overall failure is quantified by the probability that a potentially permanent internal failure will occur. When the fraction of active neighbours is reduced to a critical threshold, cascading failures can trigger a total network failure. For this class of network we find that the time to network failure, which is equivalent to network lifetime, is inversely dependent upon the magnitude of the failure and logarithmically dependent on the threshold. We analyse how permanent failures affect network robustness using network lifetime as a measure. These findings provide new methodological insight into system dynamics and, in particular, of the dynamic processes of networks. We illustrate the network model by selected examples from biology, and social science.

When common factors strongly influence two power-law cross-correlated time series recorded in complex natural or social systems, using classic detrended cross-correlation analysis (DCCA) without considering these common factors will bias the results. We use detrended partial cross-correlation analysis (DPXA) to uncover the intrinsic power-law cross-correlations between two simultaneously recorded time series in the presence of nonstationarity after removing the effects of other time series acting as common forces. The DPXA method is a generalization of the detrended cross-correlation analysis that takes into account partial correlation analysis. We demonstrate the method by using bivariate fractional Brownian motions contaminated with a fractional Brownian motion. We find that the DPXA is able to recover the analytical cross Hurst indices, and thus the multi-scale DPXA coefficients are a viable alternative to the conventional cross-correlation coefficient. We demonstrate the advantage of the DPXA coefficients over the DCCA coefficients by analyzing contaminated bivariate fractional Brownian motions. We calculate the DPXA coefficients and use them to extract the intrinsic cross-correlation between crude oil and gold futures by taking into consideration the impact of the US dollar index. We develop the multifractal DPXA (MF-DPXA) method in order to generalize the DPXA method and investigate multifractal time series. We analyze multifractal binomial measures masked with strong white noises and find that the MF-DPXA method quantifies the hidden multifractal nature while the MF-DCCA method fails., 7 Latex pages including 3 figures

We analyze the European transition economics and show that many time series of major indices exhibit (i) power-law correlations in their values, (ii) power-law correlations in their magnitudes and (iii) an asymmetric probability distribution. Applying the phase randomization procedure to these time series, we show that magnitude correlations completely vanish. We propose a stochastic model that can generate time series with features (i), (ii) and (iii), and we show by means of numerical simulations that this model is capable of reproducing these three features found in the empirical data.

For both stock and currency markets, we study the return intervals τ between the daily volatilities of the price changes that are above a certain threshold q . We find that the distribution function P q (τ) scales with the mean return interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}{\bar {{\tau}}}\end{equation*}\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}P_{q}({\tau})={\bar {{\tau}}}^{-1}f({\tau}/{\bar {{\tau}}})\end{equation*}\end{document} . The scaling function f ( x ) is similar in form for all seven stocks and for all seven currency databases analyzed, and f ( x ) is consistent with a power-law form, f ( x ) ∼ x -γ with γ ≈ 2. We also quantify how the conditional distribution P q (τ|τ 0 ) depends on the previous return interval τ 0 and find that small (or large) return intervals are more likely to be followed by small (or large) return intervals. This “clustering” of the volatility return intervals is a previously unrecognized phenomenon that we relate to the long-term correlations known to be present in the volatility.

We review the analysis of the length of the optimal path l opt in random networks with disorder (i.e., random weights on the links). In the case of strong disorder, in which the maximal weight along the path dominates the sum, we find that l opt increases dramatically compared to the known small-world result for the minimum distance l min : for Erdős–Renyi (ER) networks l opt ∼ N 1 / 3 , while for scale-free (SF) networks, with degree distribution P ( k ) ∼ k - λ , we find that l opt scales as N ( λ - 3 ) / ( λ - 1 ) for 3 λ 4 and as N 1 / 3 for λ ⩾ 4 . Thus, for these networks, the small-world nature is destroyed. For 2 λ 3 , our numerical results suggest that l opt scales as ln λ - 1 N . We also find numerically that for weak disorder l opt ∼ ln N for ER models as well as for SF networks. We also study the transition between the strong and weak disorder regimes in the scaling properties of the average optimal path l opt in ER and SF networks.

We study the transition between the strong and weak disorder regimes in the scaling properties of the average optimal path l opt in a disordered Erdős–Renyi (ER) random network and scale-free (SF) network. Each link i is associated with a weight τ i ≡ exp ( ar i ) , where r i is a random number taken from a uniform distribution between 0 and 1 and the parameter a controls the strength of the disorder. We find that for any finite a, there is a crossover network size N * ( a ) such that for N ⪡ N * ( a ) the scaling behavior of l opt is in the strong disorder regime, while for N ⪢ N * ( a ) the scaling behavior is in the weak disorder regime. We derive the scaling relation between N * ( a ) and a with the help of simulations and also present an analytic derivation of the relation.

We report some recent attempts to understand the dynamics of complex physiologic fluctuations by adapting and extending concepts and methods developed very recently in statistical physics. We first review recent progress using wavelet-based multifractal analysis, magnitude and sign decomposition analysis and a new segmentation algorithm to quantify multiscale features of heartbeat interval series. We then investigate how heartbeat dynamics change with circadian influences and under pathologic conditions, and we discuss their possible relation to the underlaying cardiac control mechanisms. The analytic tools we discuss may be used on a wider range of physiologic signals.

We model the power-law stability in distribution of returns for S&P500 index by the GARCH process which we use to account for the long memory in the variance correlations. Precisely, we analyze the distributions corresponding to temporal aggregation of the GARCH process, i.e., the sum of n GARCH variables. The stability in the power-law tails is controlled by the GARCH parameters. We model the crossover behavior in magnitude correlations of returns by the so-called two-FIARCH process. Besides detrended fluctuation analysis, we employ the method proposed by Geweke and Porter-Hudak to estimate the fractional parameter in magnitude correlations.

We discuss a family of clusters C corresponding to the region whose boundary is formed by a fractional Brownian path y(i) and by the moving average function y n (i)≡ 1 n ∑ k=0 n−1 y(i−k) . Our model generates fractal directed patterns showing spatio-temporal complexity, and we demonstrate that the cluster area, length and duration exhibit the characteristic scaling behavior of SOC clusters. The function Cn(i) acts as a magnifying lens, zooming in (or out) the ‘avalanches’ formed by the cluster construction rule, where the magnifying power of the zoom is set by the value of the amplitude window n. On the basis of the construction rule of the clusters C n (i)≡y(i)− y n (i) and of the relationship among the exponents, we hypothesize that our model might be considered to be a generalized stochastic directed model, including the Dhar–Ramaswamy (DR) model and the stochastic models as particular cases. As in the DR model, the growth and annihilation of our clusters are obtained from the set of intersections of two random walk paths, and we argue that our model is a variant of the directed self-organized criticality scheme of the DR model.

We study the optimal distance l opt in random networks in the presence of disorder implemented by assigning random weights to the links. The optimal distance between two nodes is the length of the path for which the sum of weights along the path (“cost”) is a minimum. We study the case of strong disorder for which the distribution of weights is so broad that its sum along any path is dominated by the largest link weight in the path. We find that in random graphs, l opt scales as N 1/3 , where N is the number of nodes in the network. Thus, l opt increases dramatically compared to the known small-world result for the minimum distance l min , which scales as log N . We also study, theoretically and by simulations, scale-free networks characterized by a power law distribution for the number of links, P ( k )∼ k − λ , and find that l opt scales as N 1/3 for λ >4 and as N ( λ −3)/( λ −1) for 3 λ λ opt scales logarithmically with N .