A limit theorem for the $1$st Betti number of layer-$1$ subgraphs in random graphs (1911.00585v1)

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.