A limit theorem for the $1$st Betti number of layer-$1$ subgraphs in random graphs

TL;DR

This paper proves that the first Betti number of layer-1 subgraphs in Erdős–Rényi random graphs follows a central limit theorem, advancing understanding of local topological features in random graph models.

Contribution

It introduces the study of local topology in random graphs and establishes a central limit theorem for the first Betti number of layer-1 subgraphs in Erdős–Rényi graphs.

Findings

First Betti number of layer-1 subgraphs satisfies a CLT

Provides a new perspective on local graph topology

Advances understanding of topological motifs in random graphs

Abstract

We initiate the study of local topology of random graphs. The high level goal is to characterize local "motifs" in graphs. In this paper, we consider what we call the layer-$r$ subgraphs for an input graph $G = (V,E)$: Specifically, the layer-$r$ subgraph at vertex $u \in V$, denoted by $G_{u; r}$, is the induced subgraph of $G$ over vertex set $\Delta_{u}^{r}:= \left\{v \in V: d_G(u,v) = r \right\}$, where $d_G$ is shortest-path distance in $G$. Viewing a graph as a 1-dimensional simplicial complex, we then aim to study the $1$st Betti number of such subgraphs. Our main result is that the $1$st Betti number of layer-$1$ subgraphs in Erd\H{o}s--R\'enyi random graphs $G(n,p)$ satisfies a central limit theorem.