Examples of the cross-correlation functions between two time series <i>x</i> and <i>y</i> (left column); their Fourier transforms (right column).

<p>(a) Impulse response function corresponding to the fundamental solution of Eq. (5); (b) impulse response function of the same equation with the variables <i>x</i> and <i>y</i> being interchanged. A symmetric shape of results in the zero imaginary part of (c), while its small shift (d) results in the qualitatively similar behavior of the imaginary part as for the impulse response. The Fourier transform of the cross-correlation function which decays to zero with different speed for negative and positive lag values (e) also demonstrates the similar features.</p

Historical dynamics of the (top-bottom) market volatility , maximum value of the average cross-correlation , peak value of the real and imaginary parts of the average susceptibility .

<p>The historical dynamics is calculated for the different SMA windows: , (a) and , (b) days. Filled areas under the panel mark the periods where it is not significantly bigger than 0.5. The distance between two labeled dates is 500 trading days and the highlighted periods correspond to the major financial crises described in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0105874#pone-0105874-g002" target="_blank">Fig. 2</a>.</p

Histograms of correlation coefficients (top) and their Fisher transforms (bottom) observed on Jun 15, 2011 for lags (a) and 80 (b) days, and randomly shuffled returns (c).

<p>Stock and market volatilities are calculated using an SMA with the window days. The correlations between them are calculated using an SMA with the window days. The red curves denote fitted normal distribution. In the case of large correlations, the Fisher transform makes the highly skewed distribution approximately Gaussian (a).</p

List of the companies which stock prices are used for the calculations in the paper.

<p>Sectors are defined as basic materials (Bas), conglomerate (Cng), consumer goods (Con), financial (Fin), healthcare (Hea), industrial goods (Ind), services (Ser), technology (Tec) and utilities (Uti).</p><p>List of the companies which stock prices are used for the calculations in the paper.</p

Cross-correlation function between stock and market volatility (blue solid line) averaged over stocks for two different dates (Jun 15, 2011 and Sep 9, 2011) near the European sovereign debt crisis.

<p>The cross-correlations possess asymmetry with respect to zero lag (): (a) changes in individual stock risks on average precede changes in the market risk with lag of 14 days; (b) individual stock risks on average are prone to follow the market risk. Stock and market volatilities are calculated using an SMA with the window days. The cross-correlations between them are calculated using an SMA with the window days. Highlighted ranges with a blue background around zero lag ( days) are further used for the calculation of the susceptibilities depicted in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0105874#pone-0105874-g004" target="_blank">Fig. 4</a>. Grey solid line corresponds to when the underlying stock returns are randomly shuffled. The corresponding 95% confidence intervals for the mean correlations are denoted with dotted lines.</p

Susceptibilities for the averaged cross-correlation functions depicted in Fig. 1.

<p>The peaks of the imaginary parts hint at the causal relationships between the individual and collective risks: (a) individual stock risks on average tend to influence overall market risk; (b) market risk tends to influence risks of separate stocks. The susceptibilities are calculated using the discrete Fourier transform for the range of days around zero lag (61 days in total) which is highlighted with a blue background in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0105874#pone-0105874-g001" target="_blank">Fig. 1</a>.</p

Part of Fig. 7 enlarged for the historical period of the high correlation between individual and systemic risk for different SMA window sizes used for calculation of the volatilities <i>T</i> (a), cross-correlations <i>M</i> (b) and the range of the Fourier transform used for calculation of the average susceptibility (c).

<p>Part of Fig. 7 enlarged for the historical period of the high correlation between individual and systemic risk for different SMA window sizes used for calculation of the volatilities <i>T</i> (a), cross-correlations <i>M</i> (b) and the range of the Fourier transform used for calculation of the average susceptibility (c).</p

Average cross-correlation functions and corresponding susceptibilities calculated using different SMA window sizes for volatility (<i>T</i> ) and cross-correlations (<i>M</i> ): Jun 15, 2011 (a), (d); Sep 9, 2011 (b), (e); randomly shuffled returns (c), (f).

<p>The susceptibilities are calculated using the discrete Fourier transform for the range of days around zero lag (61 days in total) which is highlighted with a blue background. Bigger values of increase spurious correlations (c) due to smoothing effects.</p

Historical dynamics of the US stock market volatility .

<p>It is represented by the SMA standard deviation of returns of the portfolio consisting of 71 US stocks (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0105874#pone-0105874-t001" target="_blank">Table 1</a>), calculated using windows of (green line), 90 (blue line) and 180 (red line) days. The distance between two labeled dates is 500 trading days. Market crashes correspond to abrupt jumps of the volatility. Use of the bigger values of <i>T</i> leads to smoothing of small crashes, while the biggest ones are still clearly seen. Main financial crises are highlighted with a light green background: (1) Asian and Russian crisis of 1997–1998, (2) dot-com bubble, (3) US stock market downturn of 2002, (4) US housing bubble, (5) bankruptcy of Lehman Brothers followed by the global financial crisis, (6) European sovereign debt crisis.</p

Histogram of number of atoms per unit cell for 37,941 organic compounds from four experimental organic chemistry journals contained within the COD database.

<p>Blue solid line denotes log-normal fit with a median value exp(<i>μ</i>) of 222.04 atoms and a standard deviation <i>σ</i> of 0.64 ln(atoms).</p