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This lecture explores the homology and homotopy properties of scale-free networks through the random model of preferential attachment. Discover how preferential attachment mechanisms favor higher-order connectivity, driving the growth of Betti numbers in finite-graph settings while annihilating homotopy groups in infinite-graph contexts. Learn about the high-probability growth rates of Betti numbers in clique complexes of finite preferential attachment graphs and the sharp threshold at which infinite clique complexes become homotopy-connected almost surely. The presentation covers joint research with Gennady Samorodnitsky, Christina Lee Yu, and Rongyi He, based on their preprints available on arXiv.
