A hypergraph or “packed forest” is a compact data structure that uses structure-sharing to represent exponentially many trees in polynomial space. A probabilistic/weighted hypergraph also defines a probability (or other weight) for each tree, and can be used to represent the hypothesis space considered (for a given input) by a monolingual parser or a tree-based translation system (e.g., tree to...

We present the first streaming algorithm for counting an arbitrary hypergraph H of constant size in a massive hypergraph G. Our algorithm can handle both edge-insertions and edge-deletions, and is applicable for the distributed setting. Moreover, our approach provides the first family of graph polynomials for the hypergraph counting problem. Because of the close relationship between hypergraphs...

Hypergraph multiway cut problem is a problem of finding a minimum capacity set of hyperedges whose removal divides a given hypergraph into a specified number of connected components. We present an algorithm for this problem which runs in strongly polynomial-time if both the specified number of connected components and the maximum size of hyperedges in the hypergraph are constants. Our algorithm...

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...