Homology and Homotopy Properties of Scale-Free Networks

Many real-world networks are believed to be scale-free. We study the random model of preferential attachment for such networks. Our results show that preferential attachment favors higher-order connectivity, in the sense that it drives the growth of Betti numbers in the finite-graph setting, and it annihilates homotopy groups in the infinite-graph setting. More precisely, we determined the high-probability growth rates of the Betti numbers of the clique complexes of finite preferential attachment graphs, as well as the sharp threshold at which the infinite clique complex becomes homotopy-connected almost surely. This is joint work with Gennady Samorodnitsky, Christina Lee Yu, Rongyi He and Avhan Misra. The talk is based on the preprint [https://arxiv.org/abs/2305.11259] and a work in preparation.