critical state

The state between order and chaos / equilibrium and randomness, where a system is most adaptable and responsive to change. In this state, the system is at the edge of upheaval and can undergo phase transitions.

The closer to the critical point, the further information propagates - following a power law.

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The mere existence of power laws do not imply criticality.

E.g. successive fragmentation and combinations of exponential cruves produce power laws, but do not indivate criticality.

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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.