The robustness of contextuality and the contextuality cost of empirical models

In this paper, we introduce and discuss the robustness of contextuality (RoC) R_C(e) and the contextuality cost C(e) of an empirical model e. The following properties of them are proved. (i) An empirical model e is contextual if and only if R_C(e) > 0; (ii) the RoC function R_C is convex, lower semi-continuous and un-increasing under an affine mapping on the set EM of all empirical models; (iii) e is non-contextual if and only if C(e) = 0; (iv) e is contextual if and only if C(e) > 0; (v) e is strongly contextual if and only if C(e) = 1. Also, a relationship between R_C(e) and C(e) is obtained. Lastly, the RoC of three empirical models is computed and compared. Especially, the RoC of the PR boxes is obtained and the supremum 0.5 is found for the RoC of all no-signaling type (2, 2, 2) empirical models.