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Had a thought that sqrt(n) to n^2 via n is a vastly more important phenomenon than 0 to 1. sqrt(n) = random walk n = muddling through (filtered random walk) n^2 = network effect This is what things like product-market fit really are.
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Quick math question: if you constrain a random walk so that the system does not respond to impulses in all but a tiny pizza slice, the distance from origin should grow in the slice direction linearly in number of steps, right? n instead of sqrt(n)?
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This seems quite true. But as a headline, "Sqrt[n] -> n^2" won't really fly off the shelves. Needs the Gladwell "catchy phrase for people who don't really want to bother with the math" treatment.
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