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Note that with this derivation, if accept the assumptions (which obviously do not always hold), then all there is to 'viral' growth are three numbers, the current number of customers y(0), the eventual number of customers b, and the constant k. This situation holds also in the case of some customers leaving and never coming back (just by some adjustments in b and k). For k, might fit to past data. For given y(0) and b, all k does is adjust how fast the curve rises to the asymptote. So basically all we are doing is interpolating between y(0) and b. Otherwise, all viral curves are the same. So, an advantage of my derivation is a simple, explicit equation for a fairly general solution. The article has a comment claiming that biology addresses a similar problem and gets a 'logistic' curve. The comment didn't say just what was meant by a logistic curve, but I suspect that my solution here is an example. If so, then here we have an 'axiomatic' derivation of the logistic curve. It is true that the growth of some products, e.g., TV sets, look to the eye very much like one of the curves from my solution for selected values of y(0), b, and k. Could also make a Markov assumption: So, assume that get new customers (and, if wish, lose old customers) at some 'rates' and, thus, get a continuous time, discrete state space Markov process. Then as is well known the solution is a matrix exponential. Could evaluate the matrix exponential or just use Monte Carlo to generate a few thousand sample paths. Then could put some confidence limits on the deterministic solution. Since no one guessed the war story, the startup was FedEx, the SVP was Mike Basch, the CEO, of course, was Fred Smith, the person who called on the phone was Roger Frock, and the investor was General Dynamics. The arithmetic was courtesy of an HP-35. So, HP might run an ad saying how they saved FedEx! |