Activity Dependent Branching Ratios in Stocks, Solar X-ray Flux, and the Bak-Tang-Wiesenfeld Sandpile Model

We define an activity dependent branching ratio that allows comparison of different time series $X_t$. The branching ratio $b_x$ is defined as $b_x= E[\xi_x/x]$\mathrm{.}Therandomvariable$\xi_x$ is the value of the next signal given that the previous one is equal to $x$, so ${\xi }_x=\{X_{t+1}\mathrm{\mid }{X}_t=x\}$. If $b_x>1$, the process is on average supercritical when the signal is equal to $x$, while if $b_x<1$, it is subcritical. For stock prices we find $b_x=1$ within statistical uncertainty, for all $x$, consistent with an ``efficient market hypothesis’'. For stock volumes, solar X-ray flux intensities, and the Bak-Tang-Wiesenfeld (BTW) sandpile model, $b_x$ is supercritical for small values of activity and subcritical for the largest ones, indicating a tendency to return to a typical value. For stock volumes this tendency has an approximate power law behavior. For solar X-ray flux and the BTW model, there is a broad regime of activity where $b_x\simeq 1$, which we interpret as an indicator of critical behavior. This is true despite different underlying probability distributions for $X_t$, and for ${\xi }_x$. For the BTW model the distribution of ${\xi }_x$ is Gaussian, for $x$ sufficiently larger than one, and its variance grows linearly with $x$. Hence, the activity in the BTW model obeys a central limit theorem when sampling over past histories. The broad region of activity where $b_x$ is close to one disappears once bulk dissipation is introduced in the BTW model – supporting our hypothesis that it is an indicator of criticality.