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A Random-Line-Graph Approach to Overlapping Line Segments

Type of publication Peer-reviewed
Publikationsform Original article (peer-reviewed)
Author Böttcher Lucas,
Project Multispecies interacting stochastic systems in biology
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Original article (peer-reviewed)

Journal Journal of Complex Networks
Volume (Issue) 8(4)
Page(s) cnaa029
Title of proceedings Journal of Complex Networks
DOI 10.1093/comnet/cnaa029

Open Access

URL https://arxiv.org/abs/1911.10679
Type of Open Access Repository (Green Open Access)

Abstract

We study graphs that are formed by independently positioned needles (i.e. line segments) in the unit square. To mathematically characterize the graph structure, we derive the probability that two line segments intersect and determine related quantities such as the distribution of intersections, given a certain number of line segments $N$. We interpret intersections between line segments as nodes and connections between them as edges in a spatial network that we refer to as random-line graph (RLG). Using methods from the study of random-geometric graphs, we show that the probability of RLGs to be connected undergoes a sharp transition if the number of lines exceeds a threshold $N^*$.
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