Random clique complex process inside the critical window

Abstract: We consider the random clique complex process - the process of clique complexes induced by the complete graph with i.i.d. Uniform edge weights. We investigate the evolution of the Betti numbers of the clique complex process in the critical window and in particular, show a process-level convergence of the Betti numbers to a Poisson process. Our proof technique gives easily an hitting time result i.e, with high probability, the $k$th cohomology becomes trivial when there are no more isolated $k$-faces. Our results imply that the thresholds for vanishing of cohomology of the clique complex process coincides with that of the threshold for vanishing of `instantaneous' homology determined by \citet{SVT}. We also give a lower bound for the probability of clique complex process to have Kazhdan's property $(T)$. These results show a different behaviour for the clique complex process compared to the \v{C}ech complex process investigated in the geometric setting by \citet{B19}.