  We investigate fast diffusions on finite directed graphs. We prove results in
a way dual to presented in Bobrowski, A. Ann. Henri Poincar\'e (2012) 13(6):
1501-1510 and Bobrowski, A., Morawska, K. DCDS-B (2012), 17(7): 2313-2327, and
obtain asymptotic behaviour of a diffusion semigroup on a graph in $ L^1 $ and
$ L^2 $ as the diffusions' speed increases and the probability of a particle
passing through a vertex decreases.
