Percolation Theory [pdf]

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

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

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