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A limit theorem for the st Betti number of layer- subgraphs in random graphs
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- subgraphs for an input graph : Specifically, the
layer- subgraph at vertex , denoted by , is the induced
subgraph of over vertex set , where is shortest-path distance in . Viewing a graph as a
1-dimensional simplicial complex, we then aim to study the st Betti number
of such subgraphs. Our main result is that the st Betti number of layer-
subgraphs in Erd\H{o}s--R\'enyi random graphs satisfies a central
limit theorem.Comment: Comments are welcome