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> The fraction turned out to be approximately 69%, making the graphs neither common nor rare.

The wording kinda bothers me... Either 31% or 69% is exceedingly common.

Rare would be asymptotically few, or constant but smaller than e.g. 1 in 2^256.

I guess the article covers it's working definition of common, ever so briefly:

> that if you randomly select a graph from a large bucket of possibilities, you can practically guarantee it to be an optimal expander.

So it's not a reliable property, either way.

The article reads as written by someone who just learned about graphs, it focuses so much on the bet and so less on explaining Ramanujan expanders

Might be fruitful to apply this on p2p mesh networks. I suppose you should be able to make a model describing how the relationship between the fraction of byzantine nodes affects the probability distribution of connectedness. Then you could figure out what algorithm parameters would put you within desired bounds for tolerated ratios of byzantine.

Coming to a leetcode interview soon near you!