Revealing the microstructure of the giant component in random graph ensembles

Physical Review. E
Ido TishbyReimer Kühn

Abstract

The microstructure of the giant component of the Erdős-Rényi network and other configuration model networks is analyzed using generating function methods. While configuration model networks are uncorrelated, the giant component exhibits a degree distribution which is different from the overall degree distribution of the network and includes degree-degree correlations of all orders. We present exact analytical results for the degree distributions as well as higher-order degree-degree correlations on the giant components of configuration model networks. We show that the degree-degree correlations are essential for the integrity of the giant component, in the sense that the degree distribution alone cannot guarantee that it will consist of a single connected component. To demonstrate the importance and broad applicability of these results, we apply them to the study of the distribution of shortest path lengths on the giant component, percolation on the giant component, and spectra of sparse matrices defined on the giant component. We show that by using the degree distribution on the giant component one obtains high quality results for these properties, which can be further improved by taking the degree-degree correlations into acc...Continue Reading

References

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Jan 20, 2018·Physical Review. E·Haggai BonneauEytan Katzav

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Citations

Aug 17, 2018·Physical Review. E·Eytan KatzavAlexander K Hartmann
Aug 17, 2020·Physical Review. E·Haggai BonneauEytan Katzav
Apr 3, 2019·Physical Review. E·Francesco CoghiHugo Touchette
Jul 22, 2020·Physical Review. E·Takehisa Hasegawa, Shogo Mizutaka
Oct 21, 2020·Physical Review. E·Reimer Kühn, Jort van Mourik
Jan 10, 2021·Scientific Reports·Hao Hua, Ludger Hovestadt
May 20, 2021·Physical Review. E·Haggai BonneauReimer Kühn

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