Multipoint connection by long-range density interaction and short-range distance rule

Abstract

The performance of a system is influenced by the way its elements are connected. Networks of cells with high clustering and short paths communicate more efficiently than random or periodic networks of the same size. While many algorithms exist for generating networks from distributions of points in a plane, most of them are based on the oversimplification that a system’s components form connections in proportion to the inverse of their distance. The Waxman algorithm, which is based on a similar assumption, represents the gold standard for those who want to model biological networks from the spatial layout of cells. This assumption, however, does not allow to reproduce accurately the complexity of physical or biological systems, where elements establish both short and long-range connections, the combination of the two resulting in non-trivial topological features, including small-world characteristics. Here, we present a wiring algorithm that connects elements of a system using the logical connective between two disjoint probabilities, one correlated to the inverse of their distance, as in Waxman, and one associated to the density of points in the neighborhood of the system’s element. The first probability regulates the development of links or edges among adjacent nodes, while the latter governs interactions between cluster centers, where the density of points is often higher. We demonstrate that, by varying the parameters of the model, one can obtain networks with wanted values of small-world-ness, ranging from ∼1 (random graphs) to ∼14 (small world networks).