A study of viral social-media posts records reshare counts above a recording…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Probability, statistics and stochastic methods · Probability, statistics and stochastic methods

Many Parts, One Behaviour, Ch Heavy tails: power laws and how to measure them

Problem

A study of viral social-media posts records n=2000n=2000 reshare counts above a recording threshold xmin⁡=5x_{\min}=5 reshares, and finds ∑i=1nln⁡(xi/xmin⁡)=1000\sum_{i=1}^{n}\ln(x_i/x_{\min}) = 1000, consistent with a power-law density p(x)∝x−αp(x)\propto x^{-\alpha}. ((a)) Compute the maximum-likelihood estimate α^\hat\alpha and its standard error. ((b)) Decide, taking the standard error into account, whether these data establish that the mean and the variance of the reshare-count distribution are finite, giving the condition used in each case. ((c)) A follow-up study records ten times as many posts (n=2×104n=2\times10^{4}) above the same threshold. By what factor is the expected largest reshare count larger, using the scaling of the sample maximum with sample size?

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