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Rumours and Errours

Brian Hayes

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Not all of my confusion was cleared up by the discovery of this error. In particular, the algorithm that I now knew to be incorrect for the Daley-Kendall model still seemed to give the right answer for the Maki-Thompson rules. To make sense of this situation, I finally went back to the library to find out what the original authors had said.

Daley and Kendall are Daryl J. Daley, now of the Australian National University, and David G. Kendall, a distinguished statistician and probabilist at the University of Cambridge. Their paper, published in 1965, is a model of lucid exposition, which would have spared me all my stumbling in the dark—and for that reason I'm glad I didn't see it sooner. The correct calculation of probabilities is stated very clearly (there's a factor of 1/2 in the expression for the spreader-spreader interaction). Furthermore, the origin of the mysteriously precise number 0.203188 is made plain. Those six digits come not from a discrete-event simulation like the ones I had designed but from a continuous, differential-equation version of the model. The number θ is a solution of the equation:

θ e 2(1- θ) = 1.

(This equation brings us back almost to the Lambert W function, WeW .)

Maki and Thompson are Daniel P. Maki and Maynard Thompson of Indiana University, who discussed rumors in a 1973 textbook, Mathematical Models and Applications. They described the rumor-passing process in terms of telephone calls, and they limited their attention to calls placed by spreaders; because of this asymmetry, only the middle row of the matrix in Figure 4 enters into the calculation, and my first program was in fact a correct implementation of their model. (At least I got something right.) It is almost a coincidence that Maki and Thompson arrive at the same value of θ as Daley and Kendall: Their spreader-spreader interactions are twice as likely but have only half the effect.

The paper by Belen and Pearce that launched me on this adventure also deserves a further comment. The phrase "general initial conditions" in their title refers to rumors initiated not by a single spreader but by many. One might guess that with enough spreaders, the rumor must surely permeate the entire society, but Belen and Pearce show otherwise. Measuring the fraction of those originally ignorant who remain ignorant when the rumor has finished, they find that this fraction actually increases along with the number of initial spreaders, tending to a maximum of 1/e, or about 0.368. In other words, as more people spread the news, more people fail to hear it. The reason is simply that the multitude of spreaders quickly stifle one another.

By now the mathematics of rumors has acquired a vast literature. Variant models track competing rumors and counter-rumors or allow people to meet more than two at a time. Still more models study the progress of rumors through networks or lattices rather than structureless mixed populations. I have not yet had a chance to make any errors in exploring these systems.

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