Rank of contextuality

We propose a new measure of statistical Kochen-Specker contextuality, called rank of contextuality. The rank of contextuality is the minimal number of noncontextual boxes (input-output devices admitting a non-contextual hidden variable model) that are needed to switch between in order to simulate a contextual box. We show that the logarithm of the rank of contextuality is additive, faithful measure of contextuality, monotonous under simple wirings. We also provide a construction of contextual boxes with arbitrary high rank of contextuality, exhibiting thereby extremely high logical contradiction.