Explore the higher-order connectivity of preferential attachment networks through an in-depth analysis of Betti numbers in clique complexes. Delve into a 46-minute lecture that examines the asymptotic growth rates of expected Betti numbers as network size increases, revealing sublinear growth patterns with a notable exception of linear growth for the first Betti number. Gain insights from theoretical findings supported by simulation results, based on collaborative research presented in the associated preprint. Enhance understanding of complex network structures and their topological properties in this Applied Algebraic Topology Network presentation.