On the existence of $δ$-temporal cliques in random simple temporal graphs

Abstract

We consider random simple temporal graphs in which every edge of the complete graph $K_n$ appears once within the time interval [0,1] independently and uniformly at random. Our main result is a sharp threshold on the size of any maximum $δ$-clique (namely a clique with edges appearing at most $δ$ apart within [0,1]) in random instances of this model, for any constant~$δ$. In particular, using the probabilistic method, we prove that the size of a maximum $δ$-clique is approximately $\frac{2\log{n}}{\log{\frac{1}δ}}$ with high probability (whp). What seems surprising is that, even though the random simple temporal graph contains $Θ(n^2)$ overlapping $δ$-windows, which (when viewed separately) correspond to different random instances of the Erdos-Renyi random graphs model, the size of the maximum $δ$-clique in the former model and the maximum clique size of the latter are approximately the same. Furthermore, we show that the minimum interval containing a $δ$-clique is $δ-o(δ)$ whp. We use this result to show that any polynomial time algorithm for $δ$-TEMPORAL CLIQUE is unlikely to have very large probability of success.