Random Topology: The Topology of Preferential Attachment Graphs
Abstract: The probability community has obtained fruitful results about the connectivity of random graphs in the last 50 years. Random topology is an emerging field that studies higher-order connectivity of random simplicial complexes, which are higher-order generalizations of graphs. Many classical results have higher-dimensional generalizations that shed further insight into the complicated behavior of random combinatorial objects.
In this talk, we focus on the preferential attachment model, 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 algebraic-topological properties of its clique complex. By determining the asymptotic growth rates of the Betti numbers, we discover that the graph undergoes higher-order phase transitions within the infinite-variance regime. This is a joint work with Gennady Samorodnitsky, Christina Lee Yu and Rongyi He. This talk is based on the preprints https://arxiv.org/abs/2305.11259 (to appear in Advances in Applied Probability in Sep 2025) and https://arxiv.org/abs/2406.17619.