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.
Most persuasively, Hill provided a proof for the Benford's Law in 1995 @cite_10 . He suggests that skeptics jot down all the numbers that appear on the front pages of several news papers or randomly select data from the Farmer's Almanac as a simple experiment and personal demonstration that the law holds @cite_14 . Benford's Law has been shown to describe many naturally occurring sets of numbers. In addition to those already mentioned, there is some research specifically relevant to this paper. It is known that Benford will often apply to systems that follow a power law distribution @cite_13 ; power laws are commonly found in social network structure @cite_1 and social media @cite_18 . While no work has investigated how well Benford's Law describes social networks (online or offline) or social media, research has shown it describes human behavior online through price distributions in eBay auctions @cite_5 .
@cite_18 @cite_14 @cite_10 @cite_1 @cite_5 @cite_13
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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. Even though people in our contemporary technological society are depending on communication, our understanding of the underlying laws of human communicational behavior continues to be poorly understood. Here we investigate the communication patterns in 2 social Internet communities in search of statistical laws in human interaction activity. This research reveals that human communication networks dynamically follow scaling laws that may also explain the observed trends in economic growth. Specifically, we identify a generalized version of Gibrat's law of social activity expressed as a scaling law between the fluctuations in the number of messages sent by members and their level of activity. Gibrat's law has been essential in understanding economic growth patterns, yet without an underlying general principle for its origin. We attribute this scaling law to long-term correlation patterns in human activity, which surprisingly span from days to the entire period of the available data of more than 1 year. Further, we provide a mathematical framework that relates the generalized version of Gibrat's law to the long-term correlated dynamics, which suggests that the same underlying mechanism could be the source of Gibrat's law in economics, ranging from large firms, research and development expenditures, gross domestic product of countries, to city population growth. These findings are also of importance for designing communication networks and for the understanding of the dynamics of social systems in which communication plays a role, such as economic markets and political systems. 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. We show that certain winning bids for certain ebaY auctions obey Benford's Law. One implication of this is that it is unlikely that these bids are subjected to collusion among bidders, or 'shilling' on the part of sellers. Parenthetically, we also show that numbers from the naturally occurring Fibonacci and Lucas sequences also obey Benford's Law.