PMC

Correlation between the country rank of the empirical power law exponent and the country rank based on average export ubiquity ( as reported in ). The dashed line indicates the linear least-squares fit and has slope 0.71 (, ).

Best fit power law exponents calculated for the empirical (filled triangles) and simulated (empty squares) patent distributions of each country as a function of . The linear least square fits are, respectively, (, ) and (, ).

(a) For Δt > 60 minutes, we note a decrease in the number of tails consistent with power law decay. We investigate whether the tails of the returns distributions are consistent with power law behavior or exponential decay using the Kolmogorov-Smirnov statistic, as described in the methods section. We first consider all trading days present in our dataset. At short time scales, we observe that the tails of most empirical distributions are consistent with power law behavior. As we increase the time lag, the number of tails consistent with power law behavior decreases and we see an increase in the number of tails of returns distributions that are consistent with exponential decay. We depict here the overall number of tails, both for the positive and negative returns distributions, for the 25 components of the DJIA. (b) We consider transaction days on which the prices of stocks have changed by more than 1%. We refer to this as a stress level of 1%. In this scenario, the number of tails consistent with power law decreases more sharply. Consistency with exponential decay appears when Δt is roughly 2 hours. (c) In a similar fashion, when we consider a stress level of 2%, we again observe a sharp decrease in the number of distributions consistent with power law behavior. We also find an increase in the number of tails consistent with exponential decay again when Δt is roughly 2 hours. (d) Under a stress level of 3%, the number of empirical distributions consistent with power law behavior decreases more quickly than in the other scenarios. The number of tails consistent with exponential decay peaks at a lower number than in other scenarios, but is again highest when Δt is roughly two hours, similar to other scenarios.

Empirical values (blue) of the pairs are compared to the piecewise regression lines of the PLL: for (red line), and for (green line), versus the regression line of the regular Power Law ( black). For all . For this trajectory , and .