Hypergraphs analysis
Main topics of hypergraphs research
We are working on several directions of hypergraph theory: 1. hypergraph coloring using some elements of hypergraph rewriting systems (see below), 2. hypergraph geometrical properties (see on the separate page here).
Hypergraphs coloring problem
We consider several problems here from hypergraph coloring. In particular, how coloring of nodes can support the encoding of higher order structures. We also aim to use higher-order structures as the language for dissecting the embeddings and dimensionality reduction algorithms.
Our work get its motivation from the previously emphasized lacking understanding higher-order structures, still mostly rooted in binary structures (vectors, binary multiplication, incidence matrices etc.), and incorporates recent developments in higher-arity structures [Zapata-Carratala et al. 2022] and coloring of hypergraphs [Raigorodsky et al. 2000], using and developing on the previous ideas of the many-valued logic ideas, hypergraphs coloring and non-associative algebraic structures [Kerner et al. 2002]. See some references and extended bibliography here and on the arity science project page, as well explained in the article here.
Coloring hypergraphs and rewriting systems
We are interested to describe connections between hypergraphs coloring and non-associative algebraic structures. We incorporate developments in higher n-arity algebraic structures such as theory of Zn graded forms, hyper-sets coloring properties [Raigorodskii et al. 2011] and translate some of the challenges outlined in the recent higher-arity papers [Grochow et al.2019, Zapata-Carratala et al. 2022]. We are also extensively using the graph and hypergraph rewriting systems, as we are developing the algebraic structures associated with the geometric constructions of ternary colored hypergraphs [2].
In order to describe the algebraic properties of hypergraph coloring we are using the hypergraph rewriting techniques developed and elaborated in work by N.Behr e.g. in the work overview [Behr et al. 2016].
Transformations of the hypergraph structures and possible variations of such motif structures starting from specific configurations.
Diagram algebra is an algebraic structure in which operations are performed using diagrams rather than traditional techniques. We use diagrammatic equations to describe coloring of hypergraphs and groupoids. Main work done on the diagram algberas in the work of N.Behr. Illustration from Braid algebras
