As a follow-up to [What are the Applications of Hypergraphs](https://mathoverflow.net/questions/13750/what-are-the-applications-of-hypergraphs), the linked article:


[Learning with Hypergraphs: Clustering, Classification, and Embedding](https://www.microsoft.com/en-us/research/publication/learning-hypergraphs-clustering-classification-embedding/?from=http%3A%2F%2Fresearch.microsoft.com%2Fen-us%2Fum%2Fpeople%2Fdenzho%2Fpapers%2Fhyper.pdf)


Mentions this:



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> Let us consider a problem of grouping a collection of articles into different topics. Given an article, assume the only information that we have is who wrote this article. One may construct an undirected graph in which **two vertices are joined together by an edge if there is at least one common author of their corresponding articles**...
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> A natural way of remedying the information loss issue occurring in the above methodology is to represent the data as a hypergraph instead. A hypergraph is a graph in which an edge can connect more than two vertices [2]. In other words, **an edge is a subset of vertices**. In what follows, we shall unifiedly refer to the usual undirected or directed graphs as simple graphs. Moreover, without special mentioning, the referred simple graphs are undirected. It is obvious that a simple graph is a special kind of hypergraph with each edge containing two vertices only. In the problem of clustering articles stated before, it is quite straightforward to construct a hypergraph with the vertices representing the articles, and the edges the authors (Figure 1). **Each edge contains all articles by its corresponding author**.
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What I'm wondering is, instead of using *1 edge with many vertices*, why not use *many edges*. That way it's just like a normal graph, not a hypergraph.


I don't yet understand the value in making 1 edge become essentially like a new type of node, if you could do the same thing by creating many edges. In hypergraphs they even call these new things "tentacles" or "attachments", connecting the edge to many things. Then it's like an edge is just another node. Hoping for some clarification on why this is necessary or what advantage this provides that multiple edges can't accomplish.



Why not have many edges instead of one edge connecting to many nodes (in hypergraphs)
The hypergraph gives strictly more information than the graph: it can distinguish between the case where Larry, Curly, and Moe wrote a joint paper from the case where Larry and Curly wrote a paper and Curly and Moe wrote a different paper.


