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Asymptotics for the size of the largest component scaled to “log$n$” in inhomogeneous random graphs

Tatyana S. Turova Centre for Mathematical Sciences, Lund University

Information Theory Optimization and Control Probability mathscidoc:1701.19001

Arkiv for Matematik, 51, (2), 371-403, 2012.1
We study inhomogeneous random graphs in the subcritical case. Among other results, we derive an exact formula for the size of the largest connected component scaled by log$n$, with$n$being the size of the graph. This generalizes a result for the “rank-1 case”. We also investigate branching processes associated with these graphs. In particular, we discover that the same well-known equation for the survival probability, whose positive solution determines the asymptotics of the size of the largest component in the supercritical case, also plays a crucial role in the subcritical case. However, now it is the$negative$solutions that come into play. We disclose their relationship to the distribution of the progeny of the branching process.
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@inproceedings{tatyana2012asymptotics,
  title={Asymptotics for the size of the largest component scaled to “log$n$” in inhomogeneous random graphs},
  author={Tatyana S. Turova},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203636132228049},
  booktitle={Arkiv for Matematik},
  volume={51},
  number={2},
  pages={371-403},
  year={2012},
}
Tatyana S. Turova. Asymptotics for the size of the largest component scaled to “log$n$” in inhomogeneous random graphs. 2012. Vol. 51. In Arkiv for Matematik. pp.371-403. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203636132228049.
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