Cascading Ice: How to Model the Spread of the ALS Ice Bucket Challenge
https://www.wired.com/2014/08/how-long-would-it-take-the-whole-world-to-do-the-ice-bucket-challenge/
In this article, the author, Physicist Rhett Allain, created two different mathematical models to predict how long it would take the whole world to do the ALS Ice Bucket Challenge, a challenge that spreads because every person who does the challenge must nominate three others to do the challenge after them. For both Allain’s models, he assumed that each nominee would do the challenge and nominate 3 people within two days, and that one person started the challenge.
In the first model, Allain assumed any person who was nominated had not already completed the challenge. He then did a simple recursion to calculate how many rounds of nominations it would take to surpass the world population. The number who had completed the challenge at each round grew exponentially and he estimated it would take just under 35 days for 7 billion people to do the challenge.
For his second model, Allain made some changes to attempt to correct some of the issues with the first model—namely the fact that there’s a chance someone could nominate a person who had already done the challenge. He assumed that the probability of picking a person who hadn’t yet done the challenge depended on the fraction of the world population that had completed the challenge already. Since that fraction did not become significantly large until the last round or two of nominations (due to the exponential growth), it didn’t change the model that much and he still ended up predicting about 35 days.
In truth, neither of Allain’s models were realistic. The probability of picking someone who hadn’t done the challenge yet was important, but it did not only depend on the fraction of the world population that had done the challenge. Even if we assumed everybody did what they were supposed to—complete the challenge within two days and nominate three other people—there is no way that a person could nominate any other random person in the world. In reality, people would nominate people they know, and because of the structure of friendship networks, the people that a nominator knows probably overlap significantly with the people each of their nominees knows. The fraction of a person’s friends who had completed the challenge already—a better determinant of the probability of someone nominating the same person again—would be significant at each step of the nomination process then, lengthening Allain’s estimate greatly. In reality, the ALS Ice Bucket Challenge is an example of a cascade where the structure of the network of friendship underlying it is critical in modeling its spread.