On the subgraphs of percolated random geometric graphs and the associated random complexes

TL;DR

This paper studies the properties of subgraphs and topological features of percolated random geometric graphs, providing asymptotic and approximation results for subgraph counts and Betti numbers.

Contribution

It introduces new asymptotic formulas and Poisson approximation techniques for subgraph counts and Betti numbers in percolated random geometric graphs and complexes.

Findings

Asymptotic expected number of subgraphs derived

Poisson approximation for subgraph counts established

Expected Betti numbers of complexes analyzed

Abstract

In this paper, we investigate the induced subgraphs of percolated random geometric graphs, and get some asymptotic results for the expected number of the subgraphs. Moreover, we get the Poisson approximation for the counting by Stein's method. We also present some similar results for the expectation of Betti number of the associated percolated random geometric complexes.