Benford’s Law states that, in naturally occurring systems, the frequency of numbers’ first digits is not evenly distributed. Numbers beginning with a 1 occur roughly 30 of the time, and are six times more common than numbers beginning with a 9. We show that Benford’s Law applies to social and behavioral features of users in online social networks. Using social data from five major social networks (Facebook, Twitter, Google Plus, Pinterest, and LiveJournal), we show that the distribution of first significant digits of friend and follower counts for users in these systems follow Benford’s Law. The same is true for the number of posts users make. We extend this to egocentric networks, showing that friend counts among the people in an individual’s social network also follows the expected distribution. We discuss how this can be used to detect suspicious or fraudulent activity online and to validate datasets.
The formula for the law, @math , provides a theoretical distribution of expected first digits, shown in Table 1. On the surface, Benford's Law is quite counter-intuitive. Why would numbers beginning with 1 be any more common than those beginning with 9? However, the law holds across many variations in measurement @cite_3 . Temperatures that follow Benford's Law do so regardless of whether they are measured in Farenheit, Celsius, or Kelvin. Distances follow whether measured in miles, kilometers, or smoots.
@cite_3
2096880290 1481984145 1993141097 1503956976
The distribution of first digits in numbers series obtained from very different origins shows a marked asymmetry in favor of small digits that goes under the name of Benford's law. We analyze in detail this property for different data sets and give a general explanation for the origin of the Benford's law in terms of multiplicative processes. We show that this law can be also generalized to series of numbers generated from more complex systems like the catalogs of seismic activity. Finally, we derive a relation between the generalized Benford's law and the popular Zipf's law which characterize the rank order statistics and has been extensively applied to many problems ranging from city population to linguistics. The history, empirical evidence and classical explanations of the significant-digit (or Benford's) law are reviewed, followed by a sum- mary of recent invariant-measure characterizations. Then a new statisti- cal derivation of the law in the form of a CLT-like theorem for significant digits is presented. If distributions are selected at random (in any "unbi- ased" way) and random samples are then taken from each of these dis- tributions, the significant digits of the combined sample will converge to the logarithmic (Benford) distribution. This helps explain and predict the appearance of the significant-digit phenomenon in many different empir- ical contexts and helps justify its recent application to computer design, mathematical modelling and detection of fraud in accounting data. If asked whether certain digits in numbers collected randomly from, for example, the front pages of newspapers or from stock-market prices should occur more often than others, most people would think not. Nonetheless, in 1881, the astronomer and mathematician Simon Newcomb published a two-page article in the American Journal of Mathematics reporting an example of exactly that pheonomenon. Newcomb described his observation that books of logarithms in the library were quite dirty at the beginning and progressively cleaner throughout. From this he inferred that fellow scientists using the logarithm tables were looking up numbers starting with 1 more often than numbers starting with 2, numbers with first digit 2 more often than 3, and so on. After a short heuristic argument, Newcomb concluded that the probability that a number has a particular first significant digit (that is, first nonzero digit) d can be calculated as follows: Prob (first significant digit = d) = log10 (1 + 1 d), d = 1,2,...,9 When we examine numbers in the newspaper or magazines we might expect that their first digits are just as likely to be 8 or 9 as a 1 or a 2 and we might assume that each of the nine digits (zero is not used as a first digit of course) will occur 1 9 of the time. Unexpectedly this turns out to be untrue in many situations.