Finding patterns in the degree distribution of real-world complex networks: going beyond power law

  • Swarup ChattopadhyayEmail author
  • Asit K. Das
  • Kuntal Ghosh
Industrial and Commercial Application


The most important structural characteristics in the study of large-scale real-world complex networks in pattern analysis are degree distribution. Empirical observations on the pattern of the real-world networks have led to the claim that their degree distributions follow, in general, a single power law. However, a closer observation, while fitting, shows that the single power-law distribution is often inadequate to meet the data characteristics properly. Since the degree distribution in the log–log scale actually displays, under inspection, two different slopes unlike what happens while fitting with the single power law. These two slopes with a transition in between closely resemble the pattern of the sigmoid function. This motivates us to derive a novel double power-law distribution for accurately modeling the real-world networks based on the sigmoid function. The proposed modeling approach further leads to the identification of a transition degree which, it has been demonstrated, may have a significant implication in analyzing the complex networks. The applicability, as well as effectiveness of the proposed methodology, is shown using rigorous experiments and also validated using statistical tests.


Degree distribution Power-law distribution Sigmoid function Hyperbolic tangent function KL-divergence Goodness-of-fit 



The authors gratefully acknowledge the financial assistance received from Indian Statistical Institute (I. S. I.) and Visvesvaraya PhD Scheme awarded by the Government of India.

Supplementary material

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Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  • Swarup Chattopadhyay
    • 1
    • 2
    Email author
  • Asit K. Das
    • 2
  • Kuntal Ghosh
    • 1
  1. 1.Machine Intelligence UnitIndian Statistical InstituteKolkataIndia
  2. 2.Department of Computer Science and TechnologyIndian Institute of Engineering Science and TechnologyShibpur, HowrahIndia

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