Diagonal and orthogonal neighbours in grid-based simulations: Buffon's stick after 200 years

Abstract

Rectangular grids are frequently used in spatial simulations, often with nearest neighbour interactions, but each cell has diagonal as well as orthogonal nearest neighbours. Here, a simple, abstract model of weed spread demonstrates that the relative strength of diagonal and orthogonal interactions affects simulation outcomes, by determining the threshold conditions required for spread from isolated and aggregated colonized cells. The relative strength of diagonal and orthogonal interactions implicitly represents the range of interaction processes. The von Neumann neighbourhood, which has no diagonal interactions, represents processes with zero or negligible range, such as contact processes; increasing the relative strength of diagonal interactions represents processes with increasing range, such as seed dispersal. Diagonal interactions are only likely to equal orthogonal interactions for processes with ranges that allow significant interactions beyond the nearest neighbours of a cell. Thus, the Moore neighbourhood of diagonal and orthogonal nearest neighbours with equal weight may be considered an inaccurate approximation to a larger neighbourhood. Accurate diagonal and orthogonal nearest neighbour interactions can be calculated by a method proposed by Buffon in the 18th century. This method is also useful for calculating the impacts of rescaling a grid on intercellular interactions. If the area represented by each cell in the grid is increased, diagonal interactions should be reduced more than orthogonal interactions. In a rectangular grid, setting diagonal interactions to half the strength of orthogonal interactions can achieve a good match to an equivalent simulation in a hexagonal grid in some cases, but not always.