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The asymptotics of the expected Betti numbers of preferential attachment clique complexes

Chunyin Siu et al.

Advances in Applied Probability2025https://doi.org/10.1017/apr.2024.66article
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0.46

Abstract

The preferential attachment model is a natural and popular random graph model for a growing network that contains very well-connected ‘hubs’. We study the higher-order connectivity of such a network by investigating the topological properties of its clique complex. We concentrate on the Betti numbers, a sequence of topological invariants of the complex related to the numbers of holes (equivalently, repeated connections) of different dimensions. We prove that the expected Betti numbers grow sublinearly fast, with the trivial exceptions of those at dimensions 0 and 1. Our result also shows that preferential attachment graphs undergo infinitely many phase transitions within the parameter regime where the limiting degree distribution has an infinite variance. Regarding higher-order connectivity, our result shows that preferential attachment favors higher-order connectivity. We illustrate our theoretical results with simulations.

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https://doi.org/https://doi.org/10.1017/apr.2024.66

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@article{chunyin2025,
  title        = {{The asymptotics of the expected Betti numbers of preferential attachment clique complexes}},
  author       = {Chunyin Siu et al.},
  journal      = {Advances in Applied Probability},
  year         = {2025},
  doi          = {https://doi.org/https://doi.org/10.1017/apr.2024.66},
}

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The asymptotics of the expected Betti numbers of preferential attachment clique complexes

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0.46

Balanced mode · F 0.40 / M 0.15 / V 0.05 / R 0.40

F · citation impact0.37 × 0.4 = 0.15
M · momentum0.60 × 0.15 = 0.09
V · venue signal0.50 × 0.05 = 0.03
R · text relevance †0.50 × 0.4 = 0.20

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