On the Average Number of Edges in Theta Graphs

TL;DR

This paper derives exact formulas for the average degree of theta graphs generated by random points and demonstrates their concentration around the mean, with implications for various applications in computational geometry.

Contribution

Provides closed-form expressions for the average degree of theta graphs in Poisson and uniform point distributions, with concentration results for finite sets.

Findings

Exact formulas for average degree in Poisson models

Bounds applicable to finite uniform point sets

Edge count concentration around the expected value

Abstract

Theta graphs are important geometric graphs that have many applications, including wireless networking, motion planning, real-time animation, and minimum-spanning tree construction. We give closed form expressions for the average degree of theta graphs of a homogeneous Poisson point process over the plane. We then show that essentially the same bounds---with vanishing error terms---hold for theta graphs of finite sets of points that are uniformly distributed in a square. Finally, we show that the number of edges in a theta graph of points uniformly distributed in a square is concentrated around its expected value.