2Dnät

2Dnät, or two-dimensional networks, are networks that can be embedded in a plane. In practice they are often modeled as lattice graphs, where vertices occupy points on a two-dimensional grid and edges connect neighboring points. The square lattice is the most common example, but 2Dnät also include triangular and hexagonal lattices. Networks can be finite with boundaries or infinite, and they may use open or periodic boundary conditions. They can be undirected or directed, weighted or unweighted, and may be categorized further as bond or site models in percolation theory.

Construction and variants: In a typical square 2Dnät, vertices lie at integer coordinates and edges connect adjacent sites, such as horizontal or vertical neighbors. Variants include irregular or randomized 2D networks, anisotropic edges, or weighted connections to reflect differing strengths or distances. Percolation models study the emergence of a connected cluster when edges or sites are randomly retained with a given probability.

Applications: 2Dnät are used to model a wide range of spatial processes, including diffusion and transport, electrical resistor networks, urban and transportation grids, wireless sensor deployments, image processing, and simulations in statistical mechanics. They also serve as foundational structures in mesh generation for finite-element methods and in various computer graphics algorithms.

Analysis and properties: 2Dnät are typically planar graphs with characteristic degree patterns (for example, interior nodes in a square lattice have four neighbors). Percolation thresholds depend on lattice type (e.g., bond percolation threshold on the square lattice is 0.5; site percolation is about 0.5927). Computational methods include Monte Carlo simulations, transfer-matrix techniques, and network reliability calculations.