3 ms·

> if you want a scale free process, it must be distributed as a power law. Could you elaborate a little, or maybe give an example?

sillysaurus3 9y ago
https://www.youtube.com/watch?v=fCn8zs912OE https://www.youtube.com/watch?v=fCn8zs912OE Well worth the 20 minutes.
Blackthorn 9y ago
This isn't a proof but if you want to understand it, these slides are alright: http://cs.brynmawr.edu/Courses/cs380/spring2013/section02/slides/10_ScaleFreeNetworks.pdf http://cs.brynmawr.edu/Courses/cs380/spring2013/section02/sl... I recommend reading them through, but slide 39 is the tl;dr.
antognini 9y ago
The links that others have provided are good, but the way I think about this is basically that the scale of a process basically sets the order of magnitude at which it occurs. We tend to think of processes that have a particular scale, which makes power laws somewhat unintuitive (even though there are also lots of scale-free processes in ordinary life, too). So, cars have a natural scale, since they have to fit people inside them. So the distribution of car sizes is pretty limited --- you aren't going to find cars that are microns long, and you won't find cars that are kilometers long. But if something can happen at any given scale, the distribution must operate very differently. The distribution can't pick out any favorite size to work at --- some fraction of events must happen at every possible scale. So, if we're looking at the distribution of stellar luminosities, then there will be some stars which are as bright as the Sun, some which are 1% as bright as the Sun, and some which are 100 times as bright as the Sun. And since the star formation process doesn't have any preferred scale, then the number of Solar luminosity stars to stars which are 1% as luminous as the Sun must be about the same as the number of stars which are 100 times as bright as the Sun to the number of Solar luminosity stars. it's perfectly fine for the distribution to prefer dimmer or brighter stars because this doesn't end up introducing a preferred scale --- it just tilts the slope of the distribution (which is a line on a log-log graph). I'm not a mathematician, so I'm not sure if these statements are true in any rigorous sense, but in practice scale-free processes tend to produce power laws in the universe.

sillysaurus3 9y ago

https://www.youtube.com/watch?v=fCn8zs912OE https://www.youtube.com/watch?v=fCn8zs912OE Well worth the 20 minutes.

Blackthorn 9y ago

This isn't a proof but if you want to understand it, these slides are alright: http://cs.brynmawr.edu/Courses/cs380/spring2013/section02/slides/10_ScaleFreeNetworks.pdf http://cs.brynmawr.edu/Courses/cs380/spring2013/section02/sl... I recommend reading them through, but slide 39 is the tl;dr.

antognini 9y ago

The links that others have provided are good, but the way I think about this is basically that the scale of a process basically sets the order of magnitude at which it occurs. We tend to think of processes that have a particular scale, which makes power laws somewhat unintuitive (even though there are also lots of scale-free processes in ordinary life, too). So, cars have a natural scale, since they have to fit people inside them. So the distribution of car sizes is pretty limited --- you aren't going to find cars that are microns long, and you won't find cars that are kilometers long. But if something can happen at any given scale, the distribution must operate very differently. The distribution can't pick out any favorite size to work at --- some fraction of events must happen at every possible scale. So, if we're looking at the distribution of stellar luminosities, then there will be some stars which are as bright as the Sun, some which are 1% as bright as the Sun, and some which are 100 times as bright as the Sun. And since the star formation process doesn't have any preferred scale, then the number of Solar luminosity stars to stars which are 1% as luminous as the Sun must be about the same as the number of stars which are 100 times as bright as the Sun to the number of Solar luminosity stars. it's perfectly fine for the distribution to prefer dimmer or brighter stars because this doesn't end up introducing a preferred scale --- it just tilts the slope of the distribution (which is a line on a log-log graph). I'm not a mathematician, so I'm not sure if these statements are true in any rigorous sense, but in practice scale-free processes tend to produce power laws in the universe.