Mathematical Modelling

Friday 4 Oct. 2019

Towards a New Multi-Scale And Multi-Relational Network Theory

Venue: Nanyang Technological University
Thursday 23 January 2020
Hierachical Hypergraphs - A Fundamental Scientific Data Structure.

We introduce the mathematical concept of a hierachical hypergraph, as an ordinary hypergraph plus a discrete linear or partial order based on the natural numbers \(\mathbb{N}\). This structural concept is immensely important when dealing with the mathematical abstraction of 'composition' in general. Ordinary graphs, as used in network theory, have two major defects: (i) they are based on binary relationships, and (ii) have no multi-scale structure, i.e. all the vertices (nodes) are assumed to be positioned on the same level of system description. But the idea of composition, or equivalently, group membership, is central to most real-world systems. Mathematically, this problem can be solved by hypergraphs, as they are able to describe arbitrary \(n\)-relations. However, the multi-scale composition problem needs a further ingredient, and this is an order relation between vertices of the hypergraph. We introduce and investigate the mathematical structure of this idea, the hierachical hypergraph, in this talk. Applications in all areas of science are briefly discussed at the end, to give the audience an impression of the descriptive power of this novel relational model.