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Sat Dec 9 22:11:42 UTC 2006

specified computer, then you have that same probability of taking over
any other specified computer (before knowing whether or not you took over
the first computer).  Of course, these are not independent
probabilities, in the sense that knowing that you took over one computer
means that you are less likely to have taken over other computers because
you just used up a lot of darts.  However, the solution means that in
the range where it is probable that you have taken over a particular
computer, the correlations are negligible.  Therefore, you are likely
to have taken over a large number of computers by then.

Qualitatively, what seems to happen is that you have a negligible chance
of taking over any computers until you hit a threshold where you
rapidly take over all the computers in the network.  This threshold
is roughly the number of computers in the network.

If you take the correlations in the wash for a qualitative analysis, the
probability of taking over any single computer is roughly M * the probability
of taking over a specific computer.  However, at around N= 1/2 the "critical"
threshold, your probability scales roughly as (1/2)^P of the probability
at the critical threshold.  If P = log M, (1/2)^P = 1/M, which cancels
the factor of M you add if you only want to take over any machine.
What this means, qualitatively, is that there is a threshold, which is some
small slowly-growing factor (like 10 or 20 or so) times M.  If N is less than,
say, 1/2 of this threshold, you are unlikely to take over *any* computer.
When N hits 1/2 the threshold, you have a decent probability to take over
some random computer.  As you increase N from 1/2 the threshold to the
threshold, you rapidly take over most of the computers in the system.
(At N=10*M, you've just put an average of 10 machines in each bin, so it's
 not surprising that you've taken over the system at that point.)

Of course, that threshold is pretty high to begin with.  However, when
it starts to fail, it pretty much all fails at once.  Still, this is
somewhat ideal behavior in the sense that to take over something specific
you have to have enough firepower to pretty much take over the whole thing.

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