Percolation on a directed Network. Most of the time, it refers to percolation on a directed lattice where the direction of the edges is constrained to certain directions in the space. This is applied for instance to water percolating down a porous medium, where in this case gravity imposes a preferred direction.
It is in a different universality class than undirected bond percolation. It can be described by an stochastic process similar to models describing Epidemics on networks.
https://en.wikipedia.org/wiki/Directed_percolation
http://guava.physics.uiuc.edu/~nigel/courses/563/Essays_2013/PDF/wolin.pdf
Percolation in directed scale-free networks
New universality for spatially disordered cellular automata and directed percolation
Exhaustive percolation on random networks
See also Dynamical systems on networks. It also has relations to sandpile models in the study Self-organized criticality
Directed percolation and Sandpile models - 1 by Deepak Dhar
New universality for spatially disordered cellular automata and directed percolation
Directed percolation and Reggeon field theory
Equivalence of Cellular Automata to Ising Models and Directed Percolation See Cellular automata
Are Damage Spreading Transitions Generically in the Universality Class of Directed Percolation?
Understand the different kinds of universality classes better (see Critical phenomena).