Relations between Ollivier and Bakry Emery curvature on graphs

Ollivier and Bakry Emery curvature are very similar in spirit, however their defnitions
look completely different. Both curvature notions share the applicability of the maximum
principle yielding semigroup characterizations strong enough to prove spectral gap and
diameter bounds under positive curvature. But only Bakry-Emery curvature allows a
Ledoux approach which guarantees Harnack inequality, Liouville property and Buser in-
equality. We will also discuss exponential modifcations of both curvature notions yielding
Li-Yau inequality in the Bakry-Emery case.