Complex networks and Benford’s disctribution

Morzy Mikołaj, Kajdanowicz Tomasz, Boleslaw Szymanski

1. Institute of Informatics, Poznan University of Technology, Poznan, Poland
2. Department of Computational Intelligence, Wroclaw University of Science and Technology, Wrocław,
3. Network Science and Technology Center, Rensselaer Polytechnic Institute, Troy, USA

keywords: Benford’s distribution, Benford’s Law, complex networks

Many collections of numbers do not have a uniform distribution of the leading digit, but conform to a very particular pattern known as Benford’s distribution. This distribution has been found in numerous areas such as accounting data, voting registers, census data, and even in natural phenomena. Recently it has been reported that Benford’s law applies to online social networks. Here we introduce a set of rigorous tests for adherence to Benford’s law, see Fig. 1, and apply it to verification of this claim, extending the scope of the experiment to various complex networks and to artificial networks created by several popular generative models. Our findings are that neither for real nor for artificial networks there is sufficient evidence for common conformity of network structural properties with Benford’s distribution. We find very weak evidence suggesting that three measures, degree centrality, betweenness centrality and local clustering coefficient, could adhere to Benford’s law for scalefree networks but only for very narrow range of their parameters [1].

Figure 1: Benford’s distribution purity by means of tests: (t1) Chi-Square Test for Benford Distribution (t2) Euclidean Distance Test for Benford Distribution Joint Digits Test (t3) JP-Square Correlation Statistic Test for Benford Distribution (t4) K-S Test for Benford Distribution (t5) Chebyshev Distance Test for Benford Distribution (t6) Freedman-Watson U-squared Test for Benford Distribution (t7) Judge-Schechter Mean Deviation Test for Benford Distribution (t8) Mantissa Arc Test (t9) Distortion Factor

[1] M. Morzy, T. Kajdanowicz, and B. K. Szymanski. Benford’s Distribution in Complex Networks. Scientific reports, 6:34917, 2016.

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