2021/05/17 by Alexander P. Kartun-Giles, Kartun-Giles, Alexander P., Kostas Koufos +3
Computer Science · Mathematics · #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #Data Management and Algorithms #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Physics and Society (physics.soc-ph) #Probability (math.PR) #Social and Information Networks (cs.SI) #Statistical Mechanics (cond-mat.stat-mech) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2105.07731
openalex publication_date 2021/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A 1d random geometric graph (1d RGG) is built by joining a random sample of n points from an interval of the real line with probability p. We count the number of k-hop paths between two vertices of the graph in the case where the space is the 1d interval [0,1]. We show how the k-hop path count between two vertices at Euclidean distance |x-y| is in bijection with the volume enclosed by a uniformly random d-dimensional lattice path joining the corners of a (k-1)-dimensional hyperrectangular lattice. We are able to provide the probability generating function and distribution of this k-hop path count as a sum over lattice paths, incorporating the idea of restricted integer partitions with limited number of parts. We therefore demonstrate and describe an important link between spatial random graphs, and lattice path combinatorics, where the d-dimensional lattice paths correspond to spatial permutations of the geometric points on the line.