  We have characterized the scaling behavior of the first-passage percolation
(FPP) model on two types of discrete networks, the regular square lattice and
the disordered Delaunay lattice, thereby addressing the effect of the
underlying topology. Several distribution functions for the link-times were
considered. The asymptotic behavior of the fluctuations for both the minimal
arrival time and the lateral deviation of the geodesic path are in perfect
agreement with the Kardar-Parisi-Zhang (KPZ) universality class regardless of
the type of the link-time distribution and of the lattice topology.
Pre-asymptotic behavior, on the other hand, is found to depend on the
uniqueness of geodesics in absence of disorder in the local crossing times, a
topological property of lattice directions that we term geodesic degeneracy.
This property has important consequences on the model, as for example the
well-known anisotropic growth in regular lattices. In this work we provide a
framework to understand its effect as well as to characterize its extent.
