Persistent Homology on a lattice of multigraphs
Joaquin Diaz Boils
TL;DR
This work develops an algebraic and topological framework for studying persistent homology on a lattice of coloured multigraphs, built via two composition operators ⊗ and ⊙. It extends the clique construction to multigraphs, producing multicomplexes and a monoidal, lattice-ordered partial monoid structure that underpins a new interaction filtration based on ⊙-merging. The paper then defines multiboundaries and edge colorings within this setting, and establishes a generalized incremental method to compute Betti numbers across the interaction filtration, enabling analysis of holes as networks merge. The approach aims to provide a rigorous, algebraic/topological toolkit for applications in neuroscience Embodiment, with potential to reveal how topological features persist or vanish as complex networks evolve.
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.