## Hypergraphs

The mathematics subject classification is 05C And more importantly, take a look at What's new and search for "hypergraphs" to see a lot of other results that involve hypergraph methods in their proofs. One more thing, for real world applications, hypergraph methods appear in various places including declustering problems which are quite important to scale up the performance of parallel databases.

### 1st Edition

Since 3-Sat is one of the most important algorithmic problems in computational complexity theory, hypergraphs play an important role there. For just one of many possible examples, take a look at the following paper of Feige, Kim, and Ofek:. Matroids and more generally, greedoids , are special classes of hypergraphs. For these classes greedy algorithms give optimal solutions for optimization problems, and have low polynomial time complexity. Special cases are. Directed hypergraphs are used to model chemical reaction networks.

This is closely related to the biological application Peter Arndt mentions in his answer.

The reaction network and the underlying hypergraph are related via the stoichiometry matrix , which is a matrix consisting of one's, zeros and minus ones which generalizes the adjacency matrix of a graph. One obvious question you might ask about such a network is "are there any feedback loops"? This translates into the mathematical problem of finding directed hypercircuits in a directed hypergraph. Hypergraphs can arise as Bruhat-Tits buildings of groups, see e. Some real world applications: In this article the authors list some applications to biology. Any minimal solution is a tree all of whose leaves are terminals a so-called Steiner tree. Hypergraphs are useful because there is a "full component decomposition" of any Steiner tree into subtrees; the problem of reconstructing a min-cost Steiner tree from the set of all possible full components is the same as the min-cost spanning connected hypergraph problem a.

That's the approach used by many modern algorithms for the Steiner tree problem whether they are integer-program based exact algorithms that are actually implemented, or non-implemented approximation algorithms with good provable approximation guarantees. I like this application since one must view the hypergraph as "like a graph" want it to be connected and acyclic and not like a set system.

It makes me feel that hypergraphs aren't a strict subset. This theorem is generalized and used in a sieving method to find large intervals having only composite numbers. Every finite geometry projective planes, generalized polygons, polar spaces, near polygons, etc. So, all the applications of those objects can be considered as applications of hypergraphs.

Inside mathematics, there are many other applications of these hypergraphs in group theory, extremal combinatorics and graph theory. I believe a hypergraph can implement, or at least represent the transition states of, a nondeterministic Turing machine. Can't yet find any literature demonstrating that though, which makes me wonder. Indeed, on the first page of this paper we find the quote: "It is to be mentioned, that I can not describe this construction for graphs using only graphs and no set systems.

Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. What are the Applications of Hypergraphs Ask Question. Asked 9 years, 7 months ago. Active 2 years ago. Viewed 19k times. For example, topologies and measurable spaces are both technically special cases of hypergraphs. So any theorems or applications necessarily need to focus on special cases. Some polynomial-time algorithms for graphs turn into NP-complete problems when you try to generalize them to hypergraphs e.

We often use graphs to model symmetric binary relations, and symmetric binary relations appear much more frequently than symmetric ternary relations and beyond.

## Link Prediction in Hypergraphs using Graph Convolutional Networks | OpenReview

Subset of the powerset? Arrgh, complex stuff way over my head.

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Usual graphs are only good for modelling of the pairwise interaction. But oftentimes for example in statistical physics and effective theories one works with general interactions that depend on more than two particles. In this situation there is usually some restriction on the number of vertices a hyperedge can contain and this important class of hypergraphs is no longer absurdly general.

Would you mind adding some simple diagrams of modelling the data as data and graph rewriting? A short, well-illustrated white paper on the topic would indeed make this point of view easier to explain. Gjergji Zaimi. Wondering why not just have one edge per article instead. Ryan O'Donnell. Each hyperedge represents the single set of literals that is forbidden by some clause. These structures have also been studied in constraint satisfaction, under the name microstructure complements.