  We investigate the higher-order connectivity of scale-free networks using
algebraic topology. We model scale-free networks as preferential attachment
graphs, and we study the algebraic-topological properties of their clique
complexes. We focus on the Betti numbers and the homotopy-connectedness of
these complexes. We determine the asymptotic almost sure orders of magnitude of
the Betti numbers. We also establish the occurence of homotopical phase
transitions for the infinite complexes, and we determine the critical
thresholds at which the homotopy-connectivity changes. This partially verifies
Weinberger's conjecture on the homotopy type of the infinite complexes. We
conjecture that the mean-normalized Betti numbers converge to power-law
distributions, and we present numerical evidence. Our results also highlight
the subtlety of the scaling limit of topology, which arises from the tension
between topological operations and analytical limiting process. We discuss such
tension at the end of the Introduction.
