  We extend previous work of max-linear models on finite directed acyclic
graphs to infinite graphs as well as random graphs, and investigate their
relations to classical percolation theory, more particularly the impact of
Bernoulli bond percolation on such models. We show that the critical
probability of percolation on the oriented square lattice graph $\mathbb{Z}^2$
describes a phase transition in the obtained model. Focus is on the dependence
introduced by this graph into the max-linear model. We discuss natural
applications in communication networks, in particular, concerning the
propagation of influences.
