Simplicial complexes can be viewed as high dimensional generalizations of graphs that explicitly encode multi-way ordered relations between vertices at different resolutions, all once. This concept is central towards detection higher topological features data, to which graphs, encoding only pairwise relationships, remain oblivious. While attempts have been made extend Graph Neural Networks (GNN...

We show how to prove combinatorially the Splitting Necklace Theorem by Alon for any number of thieves. Such a proof requires developing a combinatorial theory for abstract simplotopal complexes and simplotopal maps, which generalizes the theory of abstract simplicial complexes and abstract simplicial maps. Notions like orientation, subdivision, and chain maps are defined combinatorially, withou...

Abstract Motivated by applications in Topological Data Analysis, we consider decompositions of a simplicial complex induced cover its vertices. We study how the homotopy type such approximates itself. The difference between and an approximation is quantitatively measured means so called obstruction complexes. Our general machinery then specialized to clique complexes, Vietoris-Rips complexes me...

There is a canonical way to associate two simplicial complexes K, L to any relation R ⊂ X×Y . Moreover, the geometric realizations of K and L are homotopy equivalent. This was studied in the fifties by C.H. Dowker [8]. In this article we prove a Galois-type correspondence for relations R ⊂ X × Y when X is fixed and use these constructions to investigate finite posets (or equivalently, finite T0...

In this paper, we study the problem of verifying dynamic coverage in mobile sensor networks using certain switched linear systems. These switched systems describe the flow of discrete differential forms on time-evolving simplicial complexes. The simplicial complexes model the connectivity of agents in the network, and the homology groups of the simplicial complexes provides information about th...

We study a general model of random dynamical simplicial complexes and derive formula for the asymptotic degree distribution. This generalises results number existing models, including Apollonian networks weighted recursive tree. It also confirms on scale-free nature complex quantum network manifolds in dimensions d>2, special types geometry with Flavour models studied physics literature by Bian...

Towards a Spectral Theory for Simplicial Complexes