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

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Martin, Elliot

Shreim, Amer

Paczuski, Maya

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We define an activity dependent branching ratio that allows comparison of
different time series $X_{t}$. The branching ratio $b_x$ is defined as $b_x=
E[\xi_x/x]$. The random variable $\xi_x$ is the value of the next signal given
that the previous one is equal to $x$, so $\xi_x=\{X_{t+1}|X_t=x\}$. 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.

Comment: 7 pages, 11 figures

Comment: 7 pages, 11 figures

##### Keywords

Condensed Matter - Statistical Mechanics, Condensed Matter - Disordered Systems and Neural Networks, Physics - Physics and Society, Quantitative Finance - Statistical Finance