Random topology: The topology of preferential attachment graphs

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. Such generalizations serve as the probabilistic foundation of topological data analysis.

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 articles "The asymptotics of the expected Betti numbers of preferential attachment clique complexes" (Adv Appl Probab 2025) and "The topological behavior of preferential attachment graphs" (arXiv 2025).