Proximity Graphs: Construction and Asymptotic Characteristics

Summary

A poster on proximity-graph construction and asymptotic characteristics, including explicit area formulas for the β-lune and stepping-stone diversion kernels. For a Poisson process, it presents the empty-neighbourhood relation \(\Pr(\mathrm{empty})=e^{-\rho w\ell^2}\) and its connection to asymptotic expected degree.

Context

The poster organizes the theory around the area of the exclusion kernel as the geometric quantity controlling edge survival and large-sample graph behaviour.

Main contributions

• Describes proximity graphs through local empty-region rules on a finite point cloud.
• Presents explicit area formulas for the β-lune and stepping-stone diversion kernels.
• Connects kernel area with asymptotic degree, edge length, and graph sparsity.