Temel Kavramlar
The size of a maximum δ-temporal clique in a random simple temporal graph on n vertices is approximately 2 log n / log (1/δ) with high probability.
Özet
The paper studies random simple temporal graphs, where the underlying graph is the complete graph on n vertices and the edge labels (timestamps) are chosen uniformly at random from the interval [0, 1]. The main result is a sharp threshold on the size of any maximum δ-temporal clique, which is a clique where the time difference between any pair of edges is at most δ.
The key insights are:
Using the probabilistic method, the authors prove that the size of a maximum δ-temporal clique is approximately 2 log n / log (1/δ) with high probability.
This is surprising because the random simple temporal graph contains Θ(n^2) overlapping δ-windows, each of which corresponds to a different random instance of the Erdős-Rényi random graph model Gn,δ. Yet, the size of the maximum δ-temporal clique and the maximum clique size of Gn,δ are approximately the same.
The authors also show that the minimum interval containing a δ-temporal clique is δ - o(δ) with high probability, and use this result to argue that any polynomial-time algorithm for finding a maximum δ-temporal clique is unlikely to have a very large probability of success.