Clique complexes of multigraphs, edge inflations, and tournaplexes

TL;DR

This paper explores the topology of clique complexes of multigraphs, introducing edge inflations and tournaplexes, and provides methods for homology computation and decompositions relevant to topological data analysis.

Contribution

It generalizes clique complexes to multigraphs and introduces edge inflation techniques, enabling efficient homology computations and extending topological methods to new classes of complexes.

Findings

Homotopy wedge decompositions for edge-inflated complexes.

Parallelized homology computations for clique complexes and tournaplexes.

Functorial approaches for persistent homology calculations.

Abstract

In this paper we introduce and study the topology of clique complexes of multigraphs without loops. These clique complexes generalize tournaplexes, which were recently introduced by Govc, Levi, and Smith for the topological study of brain functional networks. We study a general construction of edge-inflated simplicial posets, which generalize clique complexes of multigraphs. The poset fiber theorem of Bj\"{o}rner, Wachs, and Welker is applied to obtain the homotopy wedge decomposition of an edge-inflated simplicial poset. The homological corollary of this result allows to parallelize the homology computations for edge inflated complexes, in particular, for clique complexes of multigraphs and tournaplexes. We provide functorial versions of some results to be used in computations of persistent homology. Finally, we introduce a general notion of simplex inflations and prove homotopy wedge decompositions for this class of spaces.