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Rumours and Errours

Brian Hayes

Start Spreading the News

Figure 1. The birth and death of a rumor...Click to Enlarge Image

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. By the time I found the paper, my column had already gone to press. This was a disappointment, because Belen and Pearce describe an illuminating application of the W function in a context that I found interesting in its own right. Here is how they begin:

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.

The Daley-Kendall and Maki-Thompson models simplify and formalize this social process. Both models assume a thoroughly mixed population, so that people encounter each other at random, with uniform probability. Another simplifying assumption is that people always meet two-by-two, never in larger groups. The pairwise interactions are governed by a rigid set of rules:

•Whenever a spreader meets an ignorant, the  ignorant becomes a spreader, while the original spreader continues spreading.

•When a spreader meets a stifler, the spreader becomes a stifler.

•In the case where two spreaders meet, the models differ. In the Daley-Kendall version, both spreaders become stiflers. The Maki-Thompson rules convert only one spreader into a stifler; the other continues spreading.

•All other interactions (ignorant-ignorant, ignorant-stifler, stifler-stifler) have no effect on either party.

The rules begin to explain why rumors are self-limiting in these models. Initially, spreaders are recruited from the large reservoir of ignorants, and the rumor ripples through part of the population. But as the spreaders proliferate, they start running into one another and thereby become stiflers. Because the progression from ignorant to spreader to stifler is irreversible, it's clear the rumor must eventually die out, as all spreaders wind up as stiflers in the end. What's not so obvious is why the last spreader should disappear before the supply of ignorants is exhausted, or why the permanently clueless fraction is equal to 0.203188 of the original population.

The rumor models are closely related to well-known models of epidemic disease, where the three subpopulations are usually labeled susceptibles, infectives and removed cases. But there's a difference between rumors and epidemics. In the rumor models, it's not only the disease that's contagious but also the cure, since both spreading and stifling are communicable traits.

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