power law

[ghost]

Power law was mentioned in lecture as something that occurs in nature, which was why the paper used degree-corrected SBM instead of just SBM. What is power law and why does it occur so often in nature?

[DSP137]

As far as I understand it, power law is a function between two variables where one varies as a power of the other. For example, if we have variables x and y, they follow a power law if there is some exponent n so that y=x^n. Why it occurs so often in nature is another matter entirely.

[dlee138]

The full form of the power law equation is f(x)=mx^n, where m and n are constants. If you look at this equation, you can tell it is scale invariant, because scaling x by a constant c will cause proportionate scaling of the function. This could be a possible reason as to why it applies to many phenomenon in nature, such as population growth, because the power law will retain its features even when its variables are multiplied by a common factor.

[ajulian3]

In regards to Neuronal Avalanches, power law distributions are applicable. Specifically, neuronal avalanches are a cascade of activity usually seen in acute cortical slices. This is the only area in which I've been able to find power law distributions present in statistical connectomics. What other neuronal processes do you think power laws will apply?

[mrjiaruiwang]

I think it stems from the robustness of systems. Power law systems have a small number of highly connected nodes and a large number of lowly connected nodes. Highly connected nodes tend to be more important, for example popular metabolites in cellular networks like ATP are very very important. Since biological systems undergo perturbation, often following some uniformly random damage model, the probability of damaging a highly-connected node is much lower in a power law system than in a random network.

[akim1]

Along this line, does anyone know if there is a way to show that there is thermodynamic basis for power laws? I feel like something like this might be a convincing argument that power law is an intrinsically natural process.

[adjordan]

I've been wondering this too. I know basic thermodynamics and statistical mechanics, but I'm not sure how it relates to the power law. If it's not thermodynamics, is there some other root that leads to the prevalence of the power law?