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Rumours and Errours

Brian Hayes

My Bad

If you have already figured out where my reasoning went astray, I offer my congratulations. My own belated enlightenment came when I finally drew the matrix of all nine possible encounters of ignorants, spreaders and stiflers. As shown in Figure 4, this diagram can serve as more than just an enumeration of possible outcomes; it encodes the entire structure and operation of the model. If we make the widths of the columns and rows proportional to the sizes of the three subpopulations, then the area of each of the nine boxes gives the probability of the corresponding two-person encounter. Choosing two participants at random is equivalent to choosing a point at random within the diagram; the outcome of the interaction is then decided by which of the nine boxes the chosen point lies within. (I am again glossing over the issue of spreading a rumor to oneself; it's a minor correction.)

Figure 4. Matrix of all possible events...Click to Enlarge Image

To understand where I went wrong, it's enough to analyze the simplest case, where the three subpopulations are of equal size and all nine kinds of encounters have the same probability, namely 1/9. The boxes at the four corners of the diagram correspond to events that do not involve a spreader and that change no one's status. Two other boxes describe ignorant-spreader encounters, which therefore have a total probability of 2/9. Two more boxes correspond to spreader-stifler meetings, so those events also occur with probability 2/9. But there is only one box representing spreader-spreader interactions, which accordingly must be assigned a probability of 1/9. The key point is that ignorant-spreader and spreader-stifler events are each twice as likely as spreader-spreader meetings.

Now consider what happens in my first program for the Daley-Kendall model. By always choosing a spreader first, I confined all events to the middle row of the matrix, and the three boxes in this row were selected with equal probability. As a result, spreader-spreader interactions were twice as frequent as they should have been, and the rumor was extinguished prematurely.

From the point of view of probability theory, the error is an elementary one of failing to count cases properly. Perhaps a professional programmer would cite a different root cause: I had violated the old adage, "Don't start optimizing your program until you've finished writing it."

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