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.
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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.