Here we propose a new method, detrended cross-correlation analysis, which is a generalization of detrended fluctuation analysis and is based on detrended covariance. This method is designed to investigate power-law cross correlations between different simultaneously recorded time series in the presence of nonstationarity. We illustrate the method by selected examples from physics, physiology, and finance.

In finance, one usually deals not with prices but with growth rates R , defined as the difference in logarithm between two consecutive prices. Here we consider not the trading volume, but rather the volume growth rate R̃ , the difference in logarithm between two consecutive values of trading volume. To this end, we use several methods to analyze the properties of volume changes | R̃ |, and their relationship to price changes | R |. We analyze 14,981 daily recordings of the Standard and Poor's (S & P) 500 Index over the 59-year period 1950–2009, and find power-law cross-correlations between | R | and | R̃ | by using detrended cross-correlation analysis (DCCA). We introduce a joint stochastic process that models these cross-correlations. Motivated by the relationship between | R | and | R̃ |, we estimate the tail exponent α̃ of the probability density function P (| R̃ |) ∼ | R̃ | ^−1−α̃ for both the S & P 500 Index as well as the collection of 1819 constituents of the New York Stock Exchange Composite Index on 17 July 2009. As a new method to estimate α̃, we calculate the time intervals τ _q between events where R̃ > q . We demonstrate that τ̃ _q , the average of τ _q , obeys τ̃ _q ∼ q ^α̃ . We find α̃ ≈ 3. Furthermore, by aggregating all τ _q values of 28 global financial indices, we also observe an approximate inverse cubic law.

For stationary time series, the cross-covariance and the cross-correlation as functions of time lag n serve to quantify the similarity of two time series. The latter measure is also used to assess whether the cross-correlations are statistically significant. For nonstationary time series, the analogous measures are detrended cross-correlations analysis (DCCA) and the recently proposed detrended cross-correlation coefficient, $\rho${}DCCA(T,n), where T is the total length of the time series and n the window size. For $\rho${}DCCA(T,n), we numerically calculated the Cauchy inequality $-${}1$\le${}$\rho${}DCCA(T,n)$\le${}1. Here we derive $-${}1$\le${}$\rho${}DCCA(T,n)$\le${}1 for a standard variance-covariance approach and for a detrending approach. For overlapping windows, we find the range of $\rho${}DCCA within which the cross-correlations become statistically significant. For overlapping windows we numerically determine\char22{}and for nonoverlapping windows we derive\char22{}that the standard deviation of $\rho${}DCCA(T,n) tends with increasing T to 1/T. Using $\rho${}DCCA(T,n) we show that the Chinese financial market's tendency to follow the U.S. market is extremely weak. We also propose an additional statistical test that can be used to quantify the existence of cross-correlations between two power-law correlated time series.

Networks that fail can sometimes recover spontaneously—think of traffic jams suddenly easing or people waking from a coma. A model for such recoveries reveals spontaneous ‘phase flipping’ between high-activity and low-activity modes, in analogy with first-order phase transitions near a critical point. Much research has been carried out to explore the structural properties1,2,3,4,5,6,7,8,9,10 and vulnerability11,12,13,14,15,16,17,18,19 of complex networks. Of particular interest are abrupt dynamic events that cause networks to irreversibly fail13,14,15,16,17. However, in many real-world phenomena, such as brain seizures in neuroscience or sudden market crashes in finance, after an inactive period of time a significant part of the damaged network is capable of spontaneously becoming active again. The process often occurs repeatedly. To model this marked network recovery, we examine the effect of local node recoveries and stochastic contiguous spreading, and find that they can lead to the spontaneous emergence of macroscopic ‘phase-flipping’ phenomena. As the network is of finite size and is stochastic, the fraction of active nodes z switches back and forth between the two network collective modes characterized by high network activity and low network activity. Furthermore, the system exhibits a strong hysteresis behaviour analogous to phase transitions near a critical point. We present real-world network data exhibiting phase switching behaviour in accord with the predictions of the model.