Higher order correlations using hypergraphs or simplicial complexes

[muellerbjoern]

Regular graphs have the inherent limitation to only work for pairwise measures because single edges can only connect two vertices each. Hypergraph structures remove this limitation by allowing edges to connect multiple vertices. For correlation strucutures a subset of hypergraphs is relevant: the simplicial complex, that requires every subset of vertices connected by a hyperedge (simplex) to be connected by a lower order simplex. This holds true for correlation structures: If A, B and C are highly correlated, it's impossible that A and B are less correlated. However it is possible that A and B have some additional pairwise correlation that C is not part of. A weighted simplicial complex requires exactly that. The weights of sub-simplexes have to be larger than or equal to those of the super-simplex.
References:
www.sci.kyoto-u.ac.jp/ja/_upimg/kce/dULKCg/files/Kyoto-July-17-Prelim.pdf
https://link.springer.com/article/10.1007/s10827-016-0608-6 or https://arxiv.org/abs/1601.01704

Ideas for calculations based on this:

• Algorithm: Calculate center of every simplex and make this a gravitational center. For sub-simplexes do the same (maybe reduced by the effects already accounted for by the super-simplexes), pairwise correlation stays the same
• Maybe: Clustering based on single order of correlation at a time (i.e. only pairwise, or only triplewise, or only n-tuplewise), enables comparison