Computing Persistent Homology of Directed Flag Complexes

Abstract

We present a new computing package Flagser, designed to construct the directed flag complex of a finite directed graph, and compute persistent homology for flexibly defined filtrations on the graph and the resulting complex. The persistent homology computation part of Flagser is based on the program Ripser by U. Bauer, but is optimised specifically for large computations. The construction of the directed flag complex is done in a way that allows easy parallelisation by arbitrarily many cores. Flagser also has the option of working with undirected graphs. For homology computations Flagser has an approximate option, which shortens compute time with remarkable accuracy. We demonstrate the power of Flagser by applying it to the construction of the directed flag complex of digital reconstructions of brain microcircuitry by the Blue Brain Project and several other examples. In some instances we perform computation of homology. For a more complete performance analysis, we also apply Flagser to some other data collections. In all cases the hardware used in the computation, the use of memory and the compute time are recorded.

Promises and pitfalls of Topological Data Analysis for brain connectivity analysis

“…For the calculations of persistent homology groups of undirected networks we have used the python library Ripser Tralie et al (2018). For the computation of persistent homology of the directed graphs, we have used the library Flagser by Lütgehetmann et al (2020) (via the python API reference pyflagser of the Giotto-tda Tauzin et al (2020)), with the default settings. For computing the PIs, we have used the software that accompanies the paper of Adams et al (2017); we set the variance of the Gaussians to 0.0001 and keep all the other parameters as by default.…”

“…3.4), an extension of the PH pipeline to asymmetric networks might provide a more refined family of invariants. Some novel approaches have already been proposed in this direction by Masulli and Villa (2016); Reimann et al (2017); Chowd-hury and Mémoli (2018); Turner (2019); Aktas et al (2019); Lütgehetmann et al (2020), although not yet widely adopted; notably the early applications include the analysis of brain structural connectivity networks (Reimann et al, 2017).…”

Topological Data Analysis of C. elegans Locomotion and Behavior

“…Research into gene expression has used persistent homology to detect patterns or classify whether a signal is periodic (Dequéant et al, 2008;. Frequently, persistence has been used to study neural data (Petri et al, 2013;Backholm et al, 2015;Stolz et al, 2017;Sizemore et al, 2019;Helm et al, 2020;Lütgehetmann et al, 2020), and in many cases neural data from C. elegans, but the analysis tends to rely on clique complexes as the topological space of interest instead of sliding window embeddings.…”

“…Persistence has been successfully used to study high-dimensional time series, especially those that exhibit some quasi-periodic behavior like the undulation of C. elegans (Tralie, 2016;Tralie and Perea, 2018). But to the authors' knowledge, persistent homology has not been previously used to analyze C. elegans behavior, though it and similar techniques have been used to study C. elegans neural data (Petri et al, 2013;Backholm et al, 2015;Sizemore et al, 2019;Helm et al, 2020;Lütgehetmann et al, 2020).…”

Ripserer.jl: flexible and efficient persistent homology computation in Julia

“…The introduction of Ripser has spawned a whole cottage industry of extensions and wrappers. Some examples include Ripser++ (Zhang, Xiao, & Wang, 2020), Lock-free Ripser (Morozov & Nigmetov, 2020), Ripser.py (Tralie, Saul, & Bar-On, 2018), Cubical Ripser (Kaji, Sudo, & Ahara, 2020), and Flagser (Lütgehetmann, Govc, Smith, & Levi, 2020).…”