Hyperedges Of Size Research Articles

Stochastic block hypergraph model.

The stochastic block model is widely used to generate graphs with a community structure, but no simple alternative currently exists for hypergraphs, in which more than two nodes can be connected together through a hyperedge. We discuss here such a hypergraph generalization, based on the clustering connection probability P_{ij} between nodes of communities i and j, and that uses an explicit and modulable hyperedge formation process. We focus on the standard case where P_{ij}=pδ_{ij}+q(1-δ_{ij}) when 0≤q≤p (δ_{ij} is the Kronecker symbol). We propose a simple model that satisfies three criteria: it should be as simple as possible, when p=q the model should be equivalent to the standard hypergraph random model, and it should use an explicit and modulable hyperedge formation process so that the model is intuitive and can easily express different real-world formation processes. We first show that for such a model the degree distribution and hyperedge size distribution can be approximated by binomial distributions with effective parameters that depend on the number of communities and q/p. Also, the composition of hyperedges goes for q=0 from 'pure' hyperedges (comprising nodes belonging to the same community) to 'mixed' hyperedges that comprise nodes from different communities for q=p. We test various formation processes and our results suggest that when they depend on the composition of the hyperedge, they tend to favor the dominant community and lead to hyperedges with a smaller diversity. In contrast, for formation processes that are independent from the hyperedge structure, we obtain hyperedges comprising a larger diversity of communities. The advantages of the model proposed here are its simplicity and flexibility that make it a good candidate for testing community-related problems, such as their detection, impact on various dynamics, and visualization.

Almost tight bounds for online hypergraph matching

In the online hypergraph matching problem, hyperedges of size k over a common ground set arrive online in adversarial order. The goal is to obtain a maximum matching (disjoint set of hyperedges). A naïve greedy algorithm for this problem achieves a competitive ratio of 1k. We show that no (randomized) online algorithm has competitive ratio better than 2+o(1)k. If edges are allowed to be assigned fractionally, we give a deterministic online algorithm with competitive ratio 1−o(1)ln⁡(k) and show that no online algorithm can have competitive ratio strictly better than 1+o(1)ln⁡(k). Lastly, we give a 1−o(1)ln⁡(k) competitive algorithm for the fractional edge-weighted version of the problem under a free disposal assumption.

Connectivity of random hypergraphs with a given hyperedge size distribution

This article discusses random hypergraphs with varying hyperedge sizes, admitting large hyperedges with size tending to infinity, and heavy-tailed limiting hyperedge size distributions. The main result describes a threshold for the random hypergraph to be connected with high probability, and shows that the average hyperedge size suffices to characterise connectivity under mild regularity assumptions. Especially, the connectivity threshold is in most cases insensitive to the shape and higher moments of the hyperedge size distribution. Similar results are also provided for related random intersection graph models.

A generative hypergraph model for double heterogeneity

Abstract While network science has become an indispensable tool for studying complex systems, the conventional use of pairwise links often shows limitations in describing high-order interactions properly. Hypergraphs, where each edge can connect more than two nodes, have thus become a new paradigm in network science. Yet, we are still in lack of models linking network growth and hyperedge expansion, both of which are commonly observable in the real world. Here, we propose a generative hypergraph model by employing the preferential attachment mechanism in both nodes and hyperedge formation. The model can produce bi-heterogeneity, exhibiting scale-free distributions in both hyperdegree and hyperedge size. We provide a mean-field treatment that gives the expression of the two scaling exponents, which agree with the numerical simulations. Our model may help to understand the networked systems showing both types of heterogeneity and facilitate the study of complex dynamics thereon.

A Graph Representation Learning Framework Predicting Potential Multivariate Interactions

Link prediction is a widely adopted method for extracting valuable data insights from graphs, primarily aimed at predicting interactions between two nodes. However, there are not only pairwise interactions but also multivariate interactions in real life. For example, reactions between multiple proteins, multiple compounds, and multiple metabolites cannot be mined effectively using link prediction. A hypergraph is a higher-order network composed of nodes and hyperedges, where hyperedges can be composed of multiple nodes, and can be used to depict multivariate interactions. The interactions between multiple nodes can be predicted by hyperlink prediction methods. Since hyperlink prediction requires predicting the interactions between multiple nodes, it makes the study of hyperlink prediction much more complicated than that of other complex networks, thus resulting in relatively limited attention being devoted to this field. The existing hyperlink prediction can only predict potential hyperlinks in uniform hypergraphs, or need to predict hyperlinks based on the candidate hyperlink sets, or only study hyperlink prediction for undirected hypergraphs. Therefore, a hyperlink prediction framework for predicting multivariate interactions based on graph representation learning is proposed to solve the above problems, and then the framework is extended to directed hyperlink prediction (e.g., directed metabolic reaction networks). Furthermore, any size of hyperedges can be predicted by the proposed hyperlink prediction algorithm framework, whose performance is not affected by the number of nodes or the number of hyperedges. Finally, the proposed framework is applied to both the biological metabolic reaction network and the organic chemical reaction network, and experimental analysis has demonstrated that the hyperlinks can be predicted efficiently by the proposed hyperlink prediction framework with relatively low time complexity, and the prediction performance has been improved by up to 40% compared with the baselines.

Generating real-world hypergraphs via deep generative models

Hypergraph generation has found wide applications in data analysis, social networks, and communication networks. Although various hypergraph generation methods have been proposed, many challenges remain. On the one hand, traditional hypergraph generation methods are time-consuming since they rely on computing various statistics about hypergraph distributions prior to generation. On the other hand, they store many intermediate variables in the process of hypergraph generation, thus depleting a large number of memory resources, which could lead to memory explosion. To address these issues, we present a novel deep generative model for hypergraph generation (DGMH), which consists of a node selection module and a hyperedge size decision module. In DGMH, each hyperedge is treated as an embedding in a high-dimensional space, where values along specific dimensions represent the probabilities of node inclusion in the hyperedge. The node selection module captures the distribution of hyperedge embeddings and identifies other latent hyperedge embeddings within the space. In addition, the hyperedge size decision module determines the hyperedge size based on the hyperedge embedding. We sample a certain number of nodes from the categorical distribution parameterized by the embedding and assign these nodes to the hyperedge, an essential step in hyperedge generation. Once a sufficient number of hyperedges are generated, they are connected through common nodes to form a complete hypergraph. Experimental results demonstrate the effectiveness of DGMH in capturing the structural distribution of hypergraphs in a short time while maintaining a modest memory footprint during the generation process.

Impartial games with decreasing Sprague–Grundy function and their hypergraph compound

AbstractThe Sprague–Grundy (SG) theory reduces the disjunctive compound of impartial games to the classical game of NIM. We generalize this concept by introducing hypergraph compounds of impartial games. An impartial game is called SG-decreasing if its SG value is decreased by every move. Extending the SG theory, we reduce hypergraph compounds of SG-decreasing games to hypergraph compounds of single-pile NIM games. We show that this reduction works only if all games involved in the compound are SG-decreasing. A hypergraph is called SG-decreasing if the corresponding hypergraph compound of single-pile NIM games is an SG-decreasing game. We provide some necessary and some sufficient conditions for a hypergraph to be SG-decreasing. In particular, for hypergraphs with hyperedges of size at most 3 we obtain a necessary and sufficient condition verifiable in polynomial time.

Model-based clustering for random hypergraphs

A probabilistic model for random hypergraphs is introduced to represent unary, binary and higher order interactions among objects in real-world problems. This model is an extension of the latent class analysis model that introduces two clustering structures for hyperedges and captures variation in the size of hyperedges. An expectation maximization algorithm with minorization maximization steps is developed to perform parameter estimation. Model selection using Bayesian Information Criterion is proposed. The model is applied to simulated data and two real-world data sets where interesting results are obtained.

Betweenness centrality of teams in social networks.

Betweenness centrality (BC) was proposed as an indicator of the extent of an individual's influence in a social network. It is measured by counting how many times a vertex (i.e., an individual) appears on all the shortest paths between pairs of vertices. A question naturally arises as to how the influence of a team or group in a social network can be measured. Here, we propose a method of measuring this influence on a bipartite graph comprising vertices (individuals) and hyperedges (teams). When the hyperedge size varies, the number of shortest paths between two vertices in a hypergraph can be larger than that in a binary graph. Thus, the power-law behavior of the team BC distribution breaks down in scale-free hypergraphs. However, when the weight of each hyperedge, for example, the performance per team member, is counted, the team BC distribution is found to exhibit power-law behavior. We find that a team with a widely connected member is highly influential.

FORMAN–RICCI CURVATURE FOR HYPERGRAPHS

Hypergraphs serve as models of complex networks that capture more general structures than binary relations. For graphs, a wide array of statistics has been devised to gauge different aspects of their structures. Hypergraphs lack behind in this respect. The Forman–Ricci curvature is a statistics for graphs based on Riemannian geometry, which stresses the relational character of vertices in a network by focusing on the edges rather than on the vertices. Despite many successful applications of this measure to graphs, Forman–Ricci curvature has not been introduced for hypergraphs. Here, we define the Forman–Ricci curvature for directed and undirected hypergraphs such that the curvature for graphs is recovered as a special case. It quantifies the trade-off between hyperedge (arc) size and the degree of participation of hyperedge (arc) vertices in other hyperedges (arcs). Here, we determine upper and lower bounds for Forman–Ricci curvature both for hypergraphs in general and for graphs in particular. The measure is then applied to two large networks: the Wikipedia vote network and the metabolic network of the bacterium Escherichia coli. In the first case, the curvature is governed by the size of the hyperedges, while in the second example, it is dominated by the hyperedge degree. We found that the number of users involved in Wikipedia elections goes hand-in-hand with the participation of experienced users. The curvature values of the metabolic network allowed detecting redundant and bottle neck reactions. It is found that ADP phosphorylation is the metabolic bottle neck reaction but that the reverse reaction is not similarly central for the metabolism. Furthermore, we show the utility of the Forman–Ricci curvature for quantification of assortativity in hypergraphs and illustrate the idea by investigating three metabolic networks.

Coloring Delaunay-edges and their generalizations

We consider geometric hypergraphs whose vertex set is a finite set of points (e.g., in the plane), and whose hyperedges are the intersections of this set with a family of geometric regions (e.g., axis-parallel rectangles). A typical coloring problem for such geometric hypergraphs asks, given an integer k, for the existence of an integer m=m(k), such that every set of points can be k-colored such that every hyperedge of size at least m contains points of different (or all k) colors.We generalize this notion by introducing coloring of t-subsets of points such that every hyperedge that contains enough points contains t-subsets of different (or all) colors. In particular, we consider all t-subsets and t-subsets that are themselves hyperedges. The latter, with t=2, is equivalent to coloring the edges of the so-called Delaunay-graph.In this paper we study colorings of Delaunay-edges with respect to halfplanes, pseudo-disks, axis-parallel and bottomless rectangles, and also discuss colorings of t-subsets of geometric and abstract hypergraphs, and connections between the standard coloring of vertices and coloring of t-subsets of vertices.

Ear-Slicing for Matchings in Hypergraphs

We study when a given edge of a factor-critical graph is contained in a matching avoiding exactly one, pregiven vertex of the graph. We then apply the results to always partition the vertex-set of a 3-regular, 3-uniform hypergraph into at most one triangle (hyperedge of size 3) and edges (subsets of size 2 of hyperedges), corresponding to the intuition, and providing new insight to triangle and edge packings of Cornuejols’ and Pulleyblank’s. The existence of such a packing can be considered to be a hypergraph variant of Petersen’s theorem on perfect matchings, and leads to a simple proof for a sharpening of Lu’s theorem on antifactors of graphs.

On the shortest separating cycle

According to a result of Arkin et al. (2016), given n point pairs in the plane, there exists a simple polygonal cycle that separates the two points in each pair to different sides; moreover, a O(n)-factor approximation with respect to the minimum length can be computed in polynomial time. Here the following results are obtained:(I) We extend the problem to geometric hypergraphs and obtain the following characterization of feasibility. Given a geometric hypergraph on points in the plane with hyperedges of size at least 2, there exists a simple polygonal cycle that separates each hyperedge if and only if the hypergraph is 2-colorable.(II) We extend the O(n)-factor approximation in the length measure as follows: Given a geometric graph G=(V,E), a separating cycle (if it exists) can be computed in O(m+nlog⁡n) time, where |V|=n, |E|=m. Moreover, a O(n)-approximation of the shortest separating cycle can be found in polynomial time. Given a geometric graph G=(V,E) in R3, a separating polyhedron (if it exists) can be found in O(m+nlog⁡n) time, where |V|=n, |E|=m. Moreover, a O(n2/3)-approximation of a separating polyhedron of minimum perimeter can be found in polynomial time.(III) Given a set of n point pairs in convex position in the plane, we show that a (1+ε)-approximation of a shortest separating cycle can be computed in time nO(ε−1/2). In this regard, we prove a lemma on convex polygon approximation that is of independent interest.

A Note on Saturation for Berge-G Hypergraphs

For a graph $$G=(V,E)$$ , a hypergraph H is called Berge-G if there is a hypergraph $$H'$$ , isomorphic to H, so that $$V(G)\subseteq V(H')$$ and there is a bijection $$\phi : E(G) \rightarrow E(H')$$ such that for each $$e\in E(G)$$ , $$e \subseteq \phi (e)$$ . The set of all Berge-G hypergraphs is denoted $$\mathcal {B}(G)$$ . A hypergraph H is called Berge-Gsaturated if it does not contain any subhypergraph from $$\mathcal {B}(G)$$ , but adding any new hyperedge of size at least 2 to H creates such a subhypergraph. Since each Berge-G hypergraph contains |E(G)| hypergedges, it follows that each Berge-G saturated hypergraph must have at least $$|E(G)|-1$$ edges. We show that for each graph G that is not a certain star and for any $$n\ge |V(G)|$$ , there are Berge-G saturated hypergraphs on n vertices and exactly $$|E(G)|-1$$ hyperedges. This solves a problem of finding a saturated hypergraph with the smallest number of edges exactly.

Counting hypergraph matchings up to uniqueness threshold

We study the problem of approximately counting matchings in hypergraphs of bounded maximum degree and maximum size of hyperedges. With an activity parameter λ, each matching M is assigned a weight λ|M|. The counting problem is formulated as computing a partition function that gives the sum of the weights of all matchings in a hypergraph. This problem unifies two extensively studied statistical physics models in approximate counting: the hardcore model (graph independent sets) and the monomer–dimer model (graph matchings).For this problem, the critical activity λc=ddk(d−1)d+1 is the threshold for the uniqueness of Gibbs measures on the infinite (d+1)-uniform (k+1)-regular hypertree. Consider hypergraphs of maximum degree at most k+1 and maximum size of hyperedges at most d+1. We show that when λ<λc, there is an FPTAS for computing the partition function; and when λ=λc, there is a PTAS for computing the log-partition function. These algorithms are based on the decay of correlation (strong spatial mixing) property of Gibbs distributions. When λ>2λc, there is no PRAS for the partition function or the log-partition function unless NP = RP.Towards obtaining a sharp transition of computational complexity of approximate counting, we study the local convergence from a sequence of finite hypergraphs to the infinite lattice with specified symmetry. We show a surprising connection between the local convergence and the reversibility of a natural random walk. This leads us to a barrier for the hardness result: The non-uniqueness of infinite Gibbs measure is not realizable by any finite gadgets.

LP-based pivoting algorithm for higher-order correlation clustering

Correlation clustering is an approach for clustering a set of objects from given pairwise information. In this approach, the given pairwise information is usually represented by an undirected graph with nodes corresponding to the objects, where each edge in the graph is assigned a nonnegative weight, and either the positive or negative label. Then, a clustering is obtained by solving an optimization problem of finding a partition of the node set that minimizes the disagreement or maximizes the agreement with the pairwise information. In this paper, we extend correlation clustering with disagreement minimization to deal with higher-order relationships represented by hypergraphs. We give two pivoting algorithms based on a linear programming relaxation of the problem. One achieves an O(k log n)-approximation, where n is the number of nodes and k is the maximum size of hyperedges with the negative labels. This algorithm can be applied to any hyperedges with arbitrary weights. The other is an O(r)-approximation for complete r-partite hypergraphs with uniform weights. This type of hypergraphs arise from the coclustering setting of correlation clustering.

HClique: An exact algorithm for maximum clique problem in uniform hypergraphs

A hypergraph [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text] differs from a graph in that an edge can connect more than two vertices. An r-uniform hypergraph [Formula: see text] is a hypergraph with hyperedges of size [Formula: see text]. For an r-uniform hypergraph [Formula: see text], an r-uniform clique is a subset [Formula: see text] of [Formula: see text] such as every subset of [Formula: see text] elements of [Formula: see text] belongs to [Formula: see text]. We present hClique, an exact algorithm to find a maximum r-uniform clique for [Formula: see text]-uniform graphs. In order to evidence the performance of hClique, 32 random [Formula: see text]-graphs were solved.

Clustering with Hypergraphs: The Case for Large Hyperedges.

The extension of conventional clustering to hypergraph clustering, which involves higher order similarities instead of pairwise similarities, is increasingly gaining attention in computer vision. This is due to the fact that many clustering problems require an affinity measure that must involve a subset of data of size more than two. In the context of hypergraph clustering, the calculation of such higher order similarities on data subsets gives rise to hyperedges. Almost all previous work on hypergraph clustering in computer vision, however, has considered the smallest possible hyperedge size, due to a lack of study into the potential benefits of large hyperedges and effective algorithms to generate them. In this paper, we show that large hyperedges are better from both a theoretical and an empirical standpoint. We then propose a novel guided sampling strategy for large hyperedges, based on the concept of random cluster models. Our method can generate large pure hyperedges that significantly improve grouping accuracy without exponential increases in sampling costs. We demonstrate the efficacy of our technique on various higher-order grouping problems. In particular, we show that our approach improves the accuracy and efficiency of motion segmentation from dense, long-term, trajectories.

Positive independence densities of finite rank countable hypergraphs are achieved by finite hypergraphs

The independence density of a finite hypergraph is the probability that a subset of vertices, chosen uniformly at random contains no hyperedges. Independence densities can be generalized to countable hypergraphs using limits. We show that, in fact, every positive independence density of a countably infinite hypergraph with hyperedges of bounded size is equal to the independence density of some finite hypergraph whose hyperedges are no larger than those in the infinite hypergraph. This answers a question of Bonato, Brown, Kemkes, and Prałat about independence densities of graphs. Furthermore, we show that for any k, the set of independence densities of hypergraphs with hyperedges of size at most k is closed and contains no infinite increasing sequences.

Density of range capturing hypergraphs

$\newcommand{\cH}{\mathcal{H}}$For a finite set $X$ of points in the plane, a set $S$ in the plane, and a positive integer $k$, we say that a $k$-element subset $Y$ of $X$ is captured by $S$ if there is a homothetic copy $S'$ of $S$ such that $X\cap S' = Y$, i.e., $S'$ contains exactly $k$ elements from $X$. A $k$-uniform $S$-capturing hypergraph $\cH = \cH(X,S,k)$ has a vertex set $X$ and a hyperedge set consisting of all $k$-element subsets of $X$ captured by $S$. In case when $k=2$ and $S$ is convex these graphs are planar graphs, known as \emph{convex distance function Delaunay graphs}. In this paper we prove that for any $k\geq 2$, any $X$, and any convex compact set $S$, the number of hyperedges in $\cH(X,S,k)$ is at most $(2k-1)|X| - k^2 + 1 - \sum_{i=1}^{k-1}a_i$, where $a_i$ is the number of $i$-element subsets of $X$ that can be separated from the rest of $X$ with a straight line. In particular, this bound is independent of $S$ and indeed the bound is tight for all ``round'' sets $S$ and point sets $X$ in general position with respect to $S$. This refines a general result of Buzaglo, Pinchasi and Rote (2013) stating that every pseudodisc topological hypergraph with vertex set $X$ has $O(k^2|X|)$ hyperedges of size $k$ or less.