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).
3. defining algbraic properties according to hypergraphs.

More on the recent workshop held in person and online you can learn more from the link (see youtube video on the right).

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.

Hypergraphs algebratic representation problem

In the algebraic confernce, July 2024, we have presented some of our research on representation of hypergraphs and encoding of hypergraphs using various algebraic structures, such as universal algebras constructs [Shirshov, 1976]. These objects are general objects, which are generally used to work with algebraic structures with two operations, multiplications and summation, yet which do not neccesarily require commutativity or associativity structures.
We made the conjecture, that using such universal alebras, it is possible to use them for construction of algorithms to study hypergraph isomorphisms (Figure, right), we can encode [Tupikina, Proceed.Alg.Sbornik thesis, 2024]. We first conjecture it for k-uniform hypergraphs (as extension of WL) and then extend it for general case of hypergraphs.

Hypergraphs motifs 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

Google Sites

Report abuse