Latticedirected

Latticedirected is a term used in mathematics and network science to describe directed structures embedded on a regular lattice, where edge directions are determined by rules tied to lattice coordinates. Nodes are lattice points and edges carry a direction according to a global orientation or a local height-based rule. When the orientation is strictly height-increasing, the resulting graph is acyclic and defines a partial order on the lattice.

Construction methods vary. A common approach on a d-dimensional cubic lattice Z^d assigns to each adjacent pair of sites a direction from x to y if a height function h: Z^d → R satisfies h(y) > h(x). This yields a latticedirected acyclic graph. Alternatively, a fixed set of directions (for example, east and north in 2D) models directed percolation or anisotropic transport.

Key properties include reachability under the directed edges and the induced partial order. In acyclic cases, a topological ordering exists and directed path lengths can be analyzed analogously to monotone growth processes.

Applications include models of anisotropic diffusion, grid-based routing in networks, and theoretical studies of growth and percolation on lattices. Latticedirected structures relate to directed acyclic graphs, oriented lattices, and percolation theory, while remaining distinct from undirected lattice models.