scale-free

Transclude of The-Cortex-and-The-Critical-Point---Understanding-The-Power-of-Emergence#^af35d7

Scale-free avalanche properties.

Two avalanches of sizes $S_1,S_2$, have the probabilities $P(S)=k{S}^{-\tau }$ occuring, where $k$ is a constant and $\tau$ controls the avalanche size distribution.
The ratio between the two sizes is $a=\frac{S_2}{S_1}$. What is the ratio between their probabilities?
Since $S_2=a{S}_1$, we can wirte the ratio as $\frac{P(S_2)}{P(S_1)}=\frac{ka{S}_1^{-\tau }}{k{S}_1^{-\tau }}=a^{-\tau }$.
→ The ratio is preserved, regardless of avalanche size (10/100, 10k/100k) → scale-free
Another scale-free property for e.g. for neuronal avalanches is the number of neurons evolved in the avalanche over time. The “avalanche shape” at different scales is the same but scaled by a constant.
Such scale-free properties are indicative of criticality.

Link to original

(See also the power law note)