Power Law Applications: Benford’s Law
Benford’s Law, also known as the first-digit law or leading digit phenomenon, predicts the distribution frequency of the leading digits for certain data sets. For data sets that lie outside the bounds of human influence, such as city populations or land mass by state, the leading digit for each data set is predicted to follow a certain frequency following Benford’s law, given by the formula P(D) = log10(1+1/D) where P(D) is the frequency of digit D for 1<D<9. Thus, for a randomized data set, the number 1 should appear as the leading digit with a frequency of 30.11%, given by P(1) = log10(1+1/1) = log10(2) = 30.10. The law therefore predicts that lower digits like 1-3 appear much more often than numbers 7-9. Conversely, for a data set governed by human intervention, such as zip codes or street addresses, Benford’s Law does not apply, since the leading digits are arbitrarily chosen. To clarify, for a data set consisting of 100 entries, the number 1 should appear as the leading digit approximately 30 times. Benford’s Law can be algebraically rearranged to produce the following power law: 10^P(D) – 1 = 1/D. Here, following the class notes for a generic power law f(k) = a/k^c, the constant a and exponent c are both equal to 1. This phenomena has some very unique and powerful applications, such as identifying tax fraud.
The IRS used Benford’s law to predict tax fraudulent tax returns. When unlawfully filing tax returns, people often choose an even distribution for their leading digits, meaning that the frequency of each leading digit is 1/9 = 0.11. However, Benford’s Law says that the frequency of each digit should decrease exponentially starting with 1. Thus, illicit tax returns will deviate heavily from Benford’s Law, a substantial indicator for further investigation into the account. Furthermore, political scientists employ Bedford’s Law to identify voter fraud, and engineers use the law to find altered images. As a result, a simple power law that merely predicts the frequency of leading digits contains very important applications, especially in identifying illegal activity.
A Basic Theory of Benford’s Law: https://projecteuclid.org/download/pdfview_1/euclid.ps/1311860830