When studying flocking/swarming behaviors in animals one is interested in quantifying and comparing the dynamics of the clustering induced by the coalescence and disbanding of animals in different groups.  Motivated by this, we study the problem of obtaining persistent homology based summaries of time-dependent metric data. Given a finite dynamic metric space (DMS ), we construct the zigzag simplicial filtration arising from applying the Rips simplicial complex construction (with a fixed scale parameter) to this finite DMS. Upon passing to 0-th homology with field coefficients, we obtain a zigzag persistence module and, based on standard results, we in turn obtain a persistence diagram or barcode from this zigzag persistence module. We prove that these barcodes are stable under perturbations in the input DMS. In order to formalize the notion of perturbation we introduce a suitable distance between DMSs and we then prove that the value of this distance between any two DMSs admits as a lower bound the bottleneck distance between the Rips barcodes associated to each of two input DMSs. This lower bound can be computed in polynomial time from the DMS inputs. Along the way, we propose a summarization of dynamic metric spaces that captures their time-dependent clustering features which we call formigrams. These set-valued functions generalize the notion of dendrogram, a prevalent tool for hierarchical clustering. In order to elucidate the relationship between our distance between two dynamic metric spaces and the bottleneck distance between their Rips zigzag barcodes, we exploit recent advances in the stability of zigzag persistence ( due to Botnan and Lesnick ). By providing explicit constructions, we prove that for each integer k\geq 1 there exist pairs of DMSs at finite interleaving distance whose k-th persistent homology barcodes are at infinite barcode distance .
When studying flocking/swarming behaviors in animals one is interested in quantifying and comparing the dynamics of the clustering induced by the coalescence and disbanding of animals in different groups. In a similar vein, studying the dynamics of social networks leads to the problem of characterizing groups/communities as they form and disperse throughout time. Motivated by this, we study the problem of obtaining persistent homology based summaries of time-dependent data. Given a finite dynamic graph (DG ), we first construct a zigzag persistence module arising from linearizing the dynamic transitive graph naturally induced from the input DG. Based on standard results, we then obtain a persistence diagram or barcode from this zigzag persistence module. We prove that these barcodes are stable under perturbations in the input DG under a suitable distance between DGs that we identify. More precisely, our stability theorem can be interpreted as providing a lower bound for the distance between DGs. Since it relies on barcodes, and their bottleneck distance, this lower bound can be computed in polynomial time from the DG inputs. Since DGs can be given rise by applying the Rips functor (with a fixed threshold) to dynamic metric spaces, we are also able to derive related stable invariants for these richer class of dynamic objects. Along the way, we propose a summarization of dynamic graphs that captures their time-dependent clustering features which we call formigrams. These set-valued functions generalize the notion of dendrogram, a prevalent tool for hierarchical clustering. In order to elucidate the relationship between our distance between two DGs and the bottleneck distance between their associated barcodes, we exploit recent advances in the stability of zigzag persistence due to Botnan and Lesnick , and to Bjerkevik .
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