Activity-dependent branching ratios in stocks, solar x-ray flux, and the Bak-Tang-Wiesenfeld sandpile model

Abstract

We define an activity dependent branching ratio that allows comparison of different time series Xt. The branching ratio bx is defined as bx = E[ξx/x]. The random variable ξx is the value of the next signal given that the previous one is equal to x, so ξx = {Xt+1|Xt = x}. If bx > 1, the process is on average supercritical when the signal is equal to x, while if bx < 1, it is subcritical. For stock prices we find bx = 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, bx 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 bx ≃ 1, which we interpret as an indicator of critical behavior. This is true despite different underlying probability distributions for Xt, and for ξx. For the BTW model the distribution of ξx 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 bx is close to one disappears once bulk dissipation is introduced in the BTW model -supporting our hypothesis that it is an indicator of criticality.

Spike-Timing-Dependent Plasticity With Axonal Delay Tunes Networks of Izhikevich Neurons to the Edge of Synchronization Transition With Scale-Free Avalanches

“…Another important quantity to characterize critical dynamics is activity-dependent branching ratio (Martin et al, 2010). Essentially, this function gives the (relative) expectation value of the timeseries in the next time step for a given amount of activity at the present time step.…”

“…More precisely, it is defined as, b ( M ) = E {ξ M / M }. The variable ξ M is the value of the next signal given that the present one is equal to M , so ξ M = { M ( t + dt )| M ( t ) = M } (Martin et al, 2010). Since a critical system is on the edge and is inherently unpredictable, b ( M ) ≈ 1, ∀ M .…”

“…Therefore, one expects b ( M ) < 1 to generally indicate subcritical behavior, while b ( M ) > 1 to indicate supercritical behavior. In fact, b ( M ) has been used to ascertain criticality in a wide range of systems including sandpile models of SOC or solar flares (Martin et al, 2010) as well as neural networks (Larremore et al, 2014; Moosavi and Montakhab, 2014; Moosavi et al, 2017).…”

Refractory period in network models of excitable nodes: self-sustaining stable dynamics, extended scaling region and oscillatory behavior

“…Condition (ii) has to do with lack of characteristic scale for a critical systems in the thermodynamic limit. We note that b(M) has been used to ascertain criticality in a wide range of systems including sandpile models of SOC or solar flares 44 as well as neural networks 33 .…”

Mean-field behavior as a result of noisy local dynamics in self-organized criticality: Neuroscience implications

“…We report the exponents obtained from the main panel in Table I. ries analysis of branching ratios has been proposed which assigns a branching ratio to a given activity and is a reliable method of distinguishing criticality [41]. Activity dependent branching ratio (b z ) is defined for a time series {Z t } as the expectation value of ξ z /z, i.e.…”