C3

Persistent homology (PH) is an operation which, loosely speaking, describes the different holes in a point cloud via a collection of intervals called a barcode. The two most frequently used variants of persistent homology for point clouds are called Čech PH and Vietoris-Rips PH. How much information is lost when we apply these kinds of PH to a point cloud? We investigate this question by studying the subspace of point clouds with the same barcodes under these operations. We establish upper and lower bounds on the dimension of this space, and find that the question of when the persistence map is identifiable has close ties to rigidity theory. For example, we show that a generic point cloud being locally identifiable under Vietoris-Rips persistence is equivalent to a certain graph being rigid on the same point cloud.

Partition diversity refers to the concept that for some networks there may be multiple, similarly plausible ways to group the nodes, rather than one single best partition. In this talk, I will present two projects that address this idea from different but complementary angles. The first introduces the benchmark stochastic cross-block model (SCBM), a generative model designed to create synthetic networks with two distinct 'ground-truth' partitions. This allows us to study the extent to which existing methods for partition detection are able to reveal the coexistence of multiple underlying structures. The second project builds on this benchmark and paves the way for a Bayesian inference framework to directly detect coexisting partitions in empirical networks. By formulating this model as a microcanonical variant of the SCBM, we can evaluate how well it fits a given network compared to existing models. We find that our method more reliably detects partition diversity in synthetic networks with planted coexisting partitions, compared to methods designed to detect a single optimal partition. Together, the two projects contribute to a broader understanding of partition diversity by offering tools to explore the ambiguity of network structure.

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