On the existence of $δ$-temporal cliques in random simple temporal graphs
George B. Mertzios, Sotiris Nikoletseas, Christoforos Raptopoulos, Paul G. Spirakis
TL;DR
Addresses the existence and size of $\delta$-temporal cliques in random simple temporal graphs on $n$ vertices with edge times uniform in $[0,1]$. The authors derive the probability that a fixed subgraph $H$ with $h$ edges forms a $\delta$-window clique via a lemma giving $\Pr(|\lambda(e)-\lambda(e')|\le\delta,\forall e,e'\in E(H)) = h\delta^{h-1}(1-\delta)+\delta^h$, and apply the first-moment method to obtain a sharp threshold for the maximum $\delta$-temporal clique size at $k_0=\frac{2\log n}{\log(1/\delta)}$, whp. They also show that the minimal interval containing a $\delta$-clique is $\delta-o(\delta)$ whp and discuss implications for the average-case hardness of $\delta$-Temporal Clique. Surprisingly, the threshold behavior mirrors that of the static Erdős–Rényi analogue $\mathcal{G}_{n,\delta}$ despite the presence of $\Theta(n^2)$ overlapping $\delta$-windows, suggesting further avenues for studying the computational complexity of temporal clique problems.
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.