Oct. 3, 2018 EAS Presidential Alert test

Fri Oct 5 10:32:45 UTC 2018

> I wonder, if there were a real alert, what the odds are that one
> wouldn't hear about it in 1 minute, 5 minutes, etc even if they didn't
> personally get it.
> 
> Obviously edge cases are possible, you were deep in a cave with your
> soccer team, but there must be mathematical modeling of that sort of
> information dispersion.
> 
> It would have to account for other possible channels, word of mouth,
> facebook, twitter, &c posts or really any informatonal source you were
> on on the internet (e.g., news sites), TV, radio, people screaming in
> the streets, etc.

You could do this, in principle, but you’d need a whole bunch of 
assumptions. What you want is a big graph of all people with weighted
edges. The weights are the effective bitrates, or the chance per unit
time that a message is sent and received along the edge (that’s going
to mean guessing at some plausible numbers for each medium).

We don’t have such a giant graph so we’d have to construct it and claim
that for these purposes it has the right properties and represents the
real world. You do this by saying, for each person, make a “word of 
mouth” edge to another randomly chosen person with a some probability.
And so on. There’s more guessing here about those probabilities, but
this has been studied quite a bit, at least for real networks where the
graph is available (e.g. twitter and facebook are favourites among
people who research social networks).

Once you’ve got this big graph of all the people and the chance of a
message going between any two of them pairwise, you write down a big
matrix, Q = [q_ij] which tells you that a message at person i has 
such and such a chance to go to person j in one time unit. You then
pick the people who got the initial message and make a vector x_0 = [x_i]
where the entries are 0 if they didn’t get it and 1/n if they did 
(n is the number of people who got it). Now you can say,

    x(t) = exp(tQ) * x_0

and ask all the sorts of questions that you ask. That gives you the
chance at each time that each person is receiving the message. To
answer “what are the chances someone heard about it in one minute”,
sum up x*dt for all times from 0 to 1 minute, subtract out x_0 
(because they already got it) and add up the probabilities that are
left.

If Q is very big, this is expensive to compute (matrix exponentials
are expensive) but I think you could scale the whole thing down to a
representative sample population. It might be fun to do this a little
bit more seriously than a hastily written mailing list post but I think
it would always rely on a lot of guesses so would have to be taken with
a very big grain of salt. As well, this is just one way you could
model the process and there are a number of obvious criticisms
(memorylessness jumps right out).

Cheers,
-w