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The preferential attachment model is a natural and popular random graph modelfor a growing network that contains very well-connected ``hubs''. We study thehigher-order connectivity of such a network by investigating the topologicalproperties of its clique complex. We concentrate on the expected Betti numbers,a sequence of topological invariants of the complex related to the numbers ofholes of different dimensions. We determine the asymptotic growth rates of theexpected Betti numbers, and prove that the expected Betti number at dimension 1grows linearly fast, while those at higher dimensions grow sublinearly fast.Our theoretical results are illustrated by simulations.

The Asymptotics of the Expected Betti Numbers of Preferential Attachment Clique Complexes