Abstract After hypergraph structure generation for the data, next step is how to conduct data analysis on hypergraph. In this chapter, we introduce four typical computation tasks, including label propagation, clustering, imbalance learning, and link prediction. The first task which predict labels vertices, i.e., assigning a each unlabeled vertex in hypergraph, based labeled information. general...

Let (X,B) be a set system in which B is a set of 3-subsets of X. Then (X,B) is a friendship 3hypergraph if it satisfies the following property: for all distinct elements u, v, w ∈ X, there exists a unique fourth element x ∈ X such that {u, v, x}, {u,w, x}, {v, w, x} ∈ B. The element x is called the completion of u, v, w and we say u, v, w is completed by x. If a friendship 3-hypergraph contains...

Graph-based learning methods explicitly consider the relations between two entities (i.e., vertices) for learning the prediction function. They have been widely used in semi-supervised learning, manifold ranking, and clustering, among other tasks. Enhancing the expressiveness of simple graphs, hypergraphs formulate an edge as a link to multiple vertices, so as to model the higher-order relation...

We provide an exposition of hypergraph models for parallelizing sparse matrix-vector multiplies. Our aim is to emphasize the expressive power of hypergraph models. First, we set forth an elementary hypergraph model for parallel matrix-vector multiply based on one-dimensional (1D) matrix partitioning. In the elementary model, the vertices represent the data of a matrix-vector multiply, and the n...

In this paper, we present a new hypergraphpartitioning algorithm that is based on the multilevel paradigm. In the multilevel paradigm, a sequence of successively coarser hypergraphs is constructed. A bisection of the smallest hypergraph is computed and it is used to obtain a bisection of the original hypergraph by successively projecting and refining the bisection to the next level finer hyperg...

We consider two natural generalizations of the notion of transversal to a finite hypergraph, arising in data-mining and machine learning, the so called multiple and partial transversals. We show that for an arbitrary hypergraph the families of multiple and partial transversals are both dual-bounded in the sense that the size of the corresponding dual hypergraph is bounded by a polynomial in the...

We prove a hypergraph analogue of the Erdős-Gallai theorem: for any δ > 0 there exists n0 such that for any 0 ≤ α ≤ 1, any k-uniform hypergraph H on n ≥ n0 vertices with at least (α+ δ) ( n k ) edges contains a tight cycle of length at least αn. Moreover, H contains tight cycles of all lengths up to αn which are divisible by k. A key tool in our proof is a ‘regular slice lemma’, which is derive...

The Füredi–Hajnal conjecture, originally posed in [6] and solved by Marcus and Tardos in [10], can be cast as an extremal problem on ordered graphs. It states that any n-vertex bipartite graph that avoids a fixed permutation graph has O(n) edges. Balogh, Bollobás and Morris [1] recently generalized this result to show that any n-vertex, ordered hypergraph that avoids a fixed permutation graph h...

The expansion of a hypergraph, a natural extension of the notion of expansion in graphs, is defined as the minimum over all cuts in the hypergraph of the ratio of the number of the hyperedges cut to the size of the smaller side of the cut. We study the Hypergraph Small Set Expansion problem, which, for a parameter δ ∈ (0, 1/2], asks to compute the cut having the least expansion while having at ...