well, it's my first blog post attempt in this new blog, and i had wanted to write on something like the inaccuracies involved in "proof" in nonrigorous arguments via predisposition (ie, the use of the desired conclusion to flavor "reasonable" steps in an argument). but that might take a bit more thought before i want to compose it so here's something else for this little soap box that might interest people.
in the concept of word-of-mouth, one hopes that one's idea or advertisement will be propagated through the networks of friendship that connect people; i tell some friends about something, and they tell their friends, etc. well, i'm afraid i shall have to rain on that parade somewhat.
starting over the summer, i did some work for a nonprofit called givology; web programming stuff. not to stray too far offtopic, but our hope was that if one person visited the site or if one person heard about givology, then he would tell his friends, and they would tell theirs, etc. unfortunately, it didn't really work out that way. our biggest growth times have been in response to things happening in media; as an example, we had a couple weeks of good traffic after being mentioned in the blog of a new york times something-or-other, whose viewers then had some probability of visiting the site and registering and donating.
anyway, the whole word of mouth thing is a bit unlikely to occur, and i can explain with some very back-of-the-envelope math.
if one performs a breadth-first search over the network of friendship, considering the effect of friends connected to friends, the growth of each "layer" on the search is very roughly exponential. it's not straightforward: for example, there's the issue that of the friends of two friends, there will generally be a large intersection, so it's not exactly the same thing as simply growing by a multiple of the number of friends each person has, but it's still a roughly exponential growth.
consider the number of friends that someone might talk to that haven't already been told about something. there is some probability that the topic will come up in conversation. at layer k of the breadth-first search, the number of people just now hearing about the bit of viral knowledge is then the product of those two values, raised to the kth power. to get the total number of people who have heard about this, we have essentially a geometric series. now, if the product of the number of friends and the probability of telling is greater than one, well hey, we've got an infinite growth here and everything is wonderful. unfortunately, while we, as social humans, know quite a few people, we have a pretty low chance of talking about any one particular subject; thus, the probability of propagation is quite low. geometric serieses will converge swiftly (when the converge at all) and not too large a crowd will hear about the information.
there's a second problem here; the probability of successfully converting someone decreases as we move out from the initial persons. people further from them are less likely to have the same enthusiasm, and so will be less likely to convince their friends. thus, the growth will be decreasing over time; even if the growth rate starts out well, it might dissipate over distance in the network. so even if the base of the exponent would be greater than one initially, it will likely drop lower further out.
so, this very very rough analysis suggests that word-of-mouth won't reach out as well as one might hope.
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