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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 asymptotical
by C-x_C-f 2y 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_model#Properties_of_G(n,_p) https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_....
- bob1029 2y agoI think we are on the same page. https://en.wikipedia.org/wiki/Giant_component#Giant_component_in_Erdős–Rényi_model https://en.wikipedia.org/wiki/Giant_component#Giant_componen...
- C-x_C-f 2y agoAh 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 https://en.wikipedia.org/wiki/Lambert_W_function