Maximal persistence in random clique complexes

We study the persistent homology of an Erdős–Rényi random clique complex filtration on n vertices. Here, each edge e appears independently at a uniform time $$p_e \in [0,1]$$ , and persistence cycle $$\sigma $$ is defined as $$p_2(\sigma ) / p_1(\sigma )$$ where $$p_1(\sigma are birth death times . show that if $$k \ge 1$$ fixed, then with high probability maximal k-cycle order $$n^{1/k(k+1)}$$