  We study the Ollivier-Ricci curvature of graphs as a function of the chosen
idleness. We show that this idleness function is concave and piecewise linear
with at most $3$ linear parts, with at most $2$ linear parts in the case of a
regular graph. We then apply our result to show that the idleness function of
the Cartesian product of two regular graphs is completely determined by the
idleness functions of the factors.
