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Percolation Theory [pdf] (web.mit.edu)
nighthawk454 108 days ago [-]
This was a great YouTube video on the subject: https://www.youtube.com/watch?v=a-767WnbaCQ

A submission to 3blue1brown's SoME (summer of math explanation) competition

bob1029 108 days ago [-]
I was recently struggling with the best way to randomly construct a well-connected, recurrent topology of neurons until I encountered Percolation theory.

There is a basic natural log scaling rule that essentially guarantees that you will have a well-connected topology (even with random connections) as long as you ensure a minimum # of connections are assigned to each element.

The required fanout at each order of magnitude network size goes something like:

  10:              ~3 connections
  100:             ~5 connections
  1,000:           ~7 connections
  10,000:          ~10 connections
  100,000:         ~12 connections
  1,000,000:       ~14 connections 
  100,000,000,000: ~26 connections
I've been able to avoid a lot of complicated code by leveraging this.
C-x_C-f 108 days ago [-]
What do you mean by well-connected topology? If you mean that you can reach every neuron from any neuron then the number of connections you need is asymptotically n log n / 2 (not up to a constant factor or anything, just n log n / 2 on the nose, it's a sharp threshold), see [0]. In general when percolation is done on just n nodes without extra structure, it's called the Erdős–Rényi model [0], and most mathematicians even just call this "the" random graph model.

[0] https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_....

bob1029 108 days ago [-]
C-x_C-f 108 days ago [-]
Ah but that's a bit different. The giant component doesn't connect all the neurons, only a fraction. The wiki page doesn't say this but if you have c * n / 2 edges then the fraction of neurons in the giant component is 1 + W(-c * exp(-c))/c where W is the Lambert W function [0], also called the product logarithm. As c tends to infinity this fraction tends to 1 but it's never 1.

[0] https://en.wikipedia.org/wiki/Lambert_W_function

abetusk 108 days ago [-]
Christensen is one of the authors of "Complexity and Criticality", which I highly recommend to anyone interested in percolation theory, the Ising model, self organized criticality and other models [0].

[0] https://www.worldscientific.com/worldscibooks/10.1142/p365#t...

user070223 108 days ago [-]
Awesome primer on percolation https://www.youtube.com/watch?v=a-767WnbaCQ
esafak 108 days ago [-]
Relevant to the study of phase transitions in machine learning; e.g., https://openreview.net/forum?id=0pLCDJVVRD
mindcrime 107 days ago [-]
Thanks for sharing that! A high level (not at all fleshed out) notion that percolation could have some applications to AI/ML is what led me to lookint into this in the first place. :-)
esafak 107 days ago [-]
If you want to see the lay of the land read Statistical Mechanics of Deep Learning (https://ganguli-gang.stanford.edu/pdf/20.StatMechDeep.pdf)
mindcrime 107 days ago [-]
Righteous. Thanks!
physicsguy 108 days ago [-]
I loved studying this stuff at undergrad. Was one of my favourite courses.
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