Rumours and Errours
Start Spreading the News
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
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.