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.
@cite_18: Social media generates a prodigious wealth of real-time content at an incessant rate. From all the content that people create and share, only a few topics manage to attract enough attention to rise to the top and become temporal trends which are displayed to users. The question of what factors cause the formation and persistence of trends is an important one that has not been answered yet. In this paper, we conduct an intensive study of trending topics on Twitter and provide a theoretical basis for the formation, persistence and decay of trends. We also demonstrate empirically how factors such as user activity and number of followers do not contribute strongly to trend creation and its propagation. In fact, we find that the resonance of the content with the users of the social network plays a major role in causing trends.
@cite_14: 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
@cite_10: 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.
@cite_1: Systems as diverse as genetic networks or the World Wide Web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature was found to be a consequence of two generic mechanisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach preferentially to sites that are already well connected. A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.
@cite_5: 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.
@cite_13: 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.
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 .