Limits of Dense Simplicial Complexes
By T. Mitchell Roddenberry and Santiago Segarra; 24(225):1−42, 2023.
Abstract
In this paper, we present a theory on the limits of sequences of dense abstract simplicial complexes. Convergence of a sequence is determined by the convergence of its homomorphism densities. The limit objects are represented by stacks of measurable $[0,1]$-valued functions on unit cubes of increasing dimension, with each function corresponding to a dimension of the abstract simplicial complex. We demonstrate that convergence in homomorphism density is equivalent to convergence in a cut-metric, and vice versa. Additionally, we show that sampled simplicial complexes from the limit objects closely resemble their structure. Furthermore, we apply this framework to partially characterize the convergence of nonuniform hypergraphs.
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