topynate comments on Confound it! Correlation is (usually) not causation! But why not? - Less Wrong

44 Post author: gwern 09 July 2014 03:04AM

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Comment author: adam_strandberg 09 July 2014 04:14:07AM *  6 points [-]

(How many different DAGs are possible if you have 600 nodes? Apparently, >2^600.)

Naively, I would expect it to be closer to 600^600 (the number of possible directed graphs with 600 nodes).

And in fact, it is some complicated thing that seems to scale much more like n^n than like 2^n: http://en.wikipedia.org/wiki/Directed_acyclic_graph#Combinatorial_enumeration

Comment author: topynate 09 July 2014 03:31:08PM 8 points [-]

There's an asymptotic approximation in the OEIS: a(n) ~ n!2^(n(n-1)/2)/(M*p^n), with M and p constants. So log(a(n)) = O(n^2), as opposed to log(2^n) = O(n), log(n!) = O(n log(n)), log(n^n) = O(n log(n)).