The asymptotic topology of the multineighbor complex of a random graph

TL;DR

This paper introduces the multineighbor complex for graphs, analyzes its homological properties in random graphs, and applies it to topological data analysis with promising results.

Contribution

It extends the concept of neighbor complexes to multiple neighbors and studies their asymptotic topological properties in Erdős-Rényi graphs.

Findings

Identifies conditions for homology vanishing and persistence in the complex.

Demonstrates the complex's effectiveness in classifying noisy data.

Provides theoretical insights into the topology of multineighbor complexes.

Abstract

We introduce the multineighbor complex of a graph, which is a simplicial complex in which a simplex is a subset of the graph with a sufficient number of mutual neighbors. We investigate the asymptotic homological properties of such complexes for the Erdos-Renyi random graphs and obtain a number of vanishing and nonvanishing results. We use this construction to perform a topological data analysis classification of noisy synthetic point clouds obtaining favorable accuracy as obtained by the standard methods. The case when there is a single neighbor has been studied earlier by Mathew Kahle.