A limit theorem for the 1st Betti number of layer-1 subgraphs in random graphs
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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 ∈ V, denoted by G_u; r, is the induced subgraph of G over vertex set Δ_u^r:= {v ∈ V: d_G(u,v) = r }, where d_G is shortest-path distance in G. Viewing a graph as a 1-dimensional simplicial complex, we then aim to study the 1st Betti number of such subgraphs. Our main result is that the 1st Betti number of layer-1 subgraphs in Erdős–Rényi random graphs G(n,p) satisfies a central limit theorem.
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