Brian Hayes in the American Scientist:
The story begins with a loose end from my column on the Lambert W function in the March-April issue of American Scientist. I had been looking for a paper with the curious title “Rumours with general initial conditions,” by Selma Belen and C. E. M. Pearce of the University of Adelaide, published in The ANZIAM Journal, which is also known as The Australia and New Zealand Industrial and Applied Mathematics Journal.
“The stochastic theory of rumours, with interacting subpopulations of ignorants, spreaders and stiflers, began with the seminal paper of Daley and Kendall. The most striking result in the area—that if there is one spreader initially, then the proportion of the population never to hear the rumour converges almost surely to a proportion 0.203188 of the population size as the latter tends to infinity—was first signalled in that article. This result occurs also in the variant stochastic model of Maki and Thompson, although a typographic error has resulted in the value 0.238 being cited in a number of consequent papers”.
I was intrigued and a little puzzled to learn that a rumor would die out while “almost surely” leaving a fifth of the people untouched. Why wouldn’t it reach everyone eventually? And that number 0.203188, with its formidable six decimal places of precision—where did that come from? I read on far enough to get the details of the models. The premise, I discovered, is that rumor-mongering is fun only if you know the rumor and your audience doesn’t; there’s no thrill in passing on stale news. In terms of the three subpopulations, people remain spreaders of a rumor as long as they continue to meet ignorants who are eager to receive it; after that, the spreaders become stiflers, who still know the rumor but have lost interest in propagating it.
Read more here.