Facundo Memoli

• 
  A review of Persistence barcodes from simplicial filtrations.
  Facundo Memoli

  Abstract not provided

• 
  Stability of Persistence. The case of the Rips filtration.
  Facundo Memoli
  Abstract not provided
• 
  Facundo Memoli - A review of Persistence barcodes from simplicial filtrations.
  Facundo Memoli
  Facundo Memoli - A review of Persistence barcodes from simplicial filtrations.
• 
  Stable Signatures for Dynamic Networks via Zigzag Persistence
  Facundo Memoli

  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.

  
  

  
  

  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.

  
  

  
  

  This is joint work with Woojin Kim.

  
  

  https://research.math.osu.edu/networks/formigrams/

  
  
• 
  Persistent Homology of Directed Networks via Dowker Filtrations
  Facundo Memoli
  We study methods for computing two network features with topological underpinnings: the Rips and Dowker Persistent Homology Diagrams. Our formulations work for general networks, which may be asymmetric and may have any real number as an edge weight. We study the sensitivity of Dowker persistence diagrams to intrinsic asymmetry in the data, and investigate the theoretical stability properties of both the Dowker and Rips persistence diagrams. We show experimental results on a variety of simulated and real world datasets using our methods. In particular, we apply both methods to a classification task on a database of networks.
• 
  A brief introduction to persistent homology via an application to neuroscience
  Facundo Memoli

  I will describe some ideas related to Persistent Homology by following an example related to the structure of the hippocampal neural code.