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We study the algorithmic complexity of computing persistent homology of arandomly chosen filtration. Specifically, we prove upper bounds for the averagefill-up (number of non-zero entries) of the boundary matrix on Erd\"os-R\'enyiand Vietoris-Rips filtrations after matrix reduction. Our bounds show that, inboth cases, the reduced matrix is expected to be significantly sparser thanwhat the general worst-case predicts. Our method is based on previous resultson the expected first Betti numbers of corresponding complexes. We establish alink between these results to the fill-up of the boundary matrix. Our bound forVietoris-Rips complexes is asymptotically tight up to logarithmic factors. Wealso provide an Erd\"os-R\'enyi filtration realising the worst-case.

Average complexity of matrix reduction for clique filtrations