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 [Bau18a], 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.

Computing Persistent Homology of Directed Flag Complexes

“…See Section 2.3 for details. In particular, documentation for FLAGSER, including examples, tools and code options can be found in [17].…”

“…As such FLAGSER is an open source package, licensed under the LGPL 3.0 scheme. The code can be found on the GitHub page of Daniel Lütgehetmann who owns the copyright for the code and will maintain the page [17].…”

Topological Data Analysis of C. elegans Locomotion and Behavior

“…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 ).…”

“…Research into gene expression has used persistent homology to detect patterns or classify whether a signal is periodic ( Dequéant et al, 2008 ; Perea et al, 2015 ). 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.…”

Complexes of Tournaments, Directionality Filtrations and Persistent Homology

“…Most of our computations were carried out by purpose designed modification of the software package Flagser [7], designed for computation of persistent homology of directed flag complexes. The package which we named Tournser is designed specifically for the computation and manipulation of flag tournaplexes associated to digraphs.…”