Persistent Homology on a lattice of multigraphs

TL;DR

This paper introduces a multicomplex structure based on a lattice of multigraphs to analyze persistent homology, providing an extended incremental algorithm and exploring its implications in neuroscience.

Contribution

It develops a novel multicomplex framework from multigraph lattices and extends algorithms for Betti number computation, linking topology with neuroscience applications.

Findings

Extended incremental algorithm for Betti numbers

Multicomplex structure from multigraph lattices

Application to neuroscience embodiment models

Abstract

A multicomplex structure is defined from an ordered lattice of multigraphs. This structure will help us to observe the features of Persistent Homology in this context, its interaction with the ordering and the repercussions of the process of merging multigraphs in the calculation of the Betti numbers. For the latter, an extended version of the incremental algorithm is provided. The ideas here developed are mainly oriented to the original example described in [10] and used more extensively in [11] in the context of the formalization of the notion of embodiment in Neuroscience.