Bounding the Number of Hyperedges in Friendship rr-Hypergraphs

For $r \ge 2$, an $r$-uniform hypergraph is called a friendship $r$-hypergraph if every set $R$ of $r$ vertices has a unique 'friend' - that is, there exists a unique vertex $x \notin R$ with the property that for each subset $A \subseteq R$ of size $r-1$, the set $A \cup \{x\}$ is a hyperedge. We show that for $r \geq 3$, the number of hyperedges in a friendship $r$-hypergraph is at least $\frac{r+1}{r} \binom{n-1}{r-1}$, and we characterise those hypergraphs which achieve this bound. This generalises a result given by Li and van Rees in the case when $r = 3$. We also obtain a new upper bound on the number of hyperedges in a friendship $r$-hypergraph, which improves on a known bound given by Li, van Rees, Seo and Singhi when $r=3$.Comment: 14 page

Hypergraph Learning with Line Expansion

Previous hypergraph expansions are solely carried out on either vertex level or hyperedge level, thereby missing the symmetric nature of data co-occurrence, and resulting in information loss. To address the problem, this paper treats vertices and hyperedges equally and proposes a new hypergraph formulation named the \emph{line expansion (LE)} for hypergraphs learning. The new expansion bijectively induces a homogeneous structure from the hypergraph by treating vertex-hyperedge pairs as "line nodes". By reducing the hypergraph to a simple graph, the proposed \emph{line expansion} makes existing graph learning algorithms compatible with the higher-order structure and has been proven as a unifying framework for various hypergraph expansions. We evaluate the proposed line expansion on five hypergraph datasets, the results show that our method beats SOTA baselines by a significant margin

Higher-dimensional models of networks

Networks are often studied as graphs, where the vertices stand for entities in the world and the edges stand for connections between them. While relatively easy to study, graphs are often inadequate for modeling real-world situations, especially those that include contexts of more than two entities. For these situations, one typically uses hypergraphs or simplicial complexes. In this paper, we provide a precise framework in which graphs, hypergraphs, simplicial complexes, and many other categories, all of which model higher graphs, can be studied side-by-side. We show how to transform a hypergraph into its nearest simplicial analogue, for example. Our framework includes many new categories as well, such as one that models broadcasting networks. We give several examples and applications of these ideas

Hypergraph model of social tagging networks

The past few years have witnessed the great success of a new family of paradigms, so-called folksonomy, which allows users to freely associate tags to resources and efficiently manage them. In order to uncover the underlying structures and user behaviors in folksonomy, in this paper, we propose an evolutionary hypergrah model to explain the emerging statistical properties. The present model introduces a novel mechanism that one can not only assign tags to resources, but also retrieve resources via collaborative tags. We then compare the model with a real-world dataset: \emph{Del.icio.us}. Indeed, the present model shows considerable agreement with the empirical data in following aspects: power-law hyperdegree distributions, negtive correlation between clustering coefficients and hyperdegrees, and small average distances. Furthermore, the model indicates that most tagging behaviors are motivated by labeling tags to resources, and tags play a significant role in effectively retrieving interesting resources and making acquaintance with congenial friends. The proposed model may shed some light on the in-depth understanding of the structure and function of folksonomy.Comment: 7 pages,7 figures, 32 reference