A Random-Line-Graph Approach to Overlapping Line Segments

TL;DR

This paper analyzes the structure of random-line graphs formed by overlapping line segments in a unit square, deriving intersection probabilities and identifying a sharp connectivity transition as the number of lines increases.

Contribution

It introduces a mathematical framework for random-line graphs, including intersection probabilities and a phase transition in connectivity based on line count.

Findings

Derived the probability of line segment intersections.

Characterized the distribution of intersections in the graph.

Identified a sharp connectivity transition at a threshold line number.

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 $N^*$.