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Journal articles on the topic 'Graph and hypergraph drawing'

Author: Grafiati

Published: 4 June 2021

Last updated: 4 February 2022

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1

Jia, Jun, Xiao Yuan He, and Xiao Feng Hu. "Drawing Hypergraphs in Hyperedge’s Average Degree and Multi-Rules." Applied Mechanics and Materials 713-715 (January 2015): 1682–88. http://dx.doi.org/10.4028/www.scientific.net/amm.713-715.1682.

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By analysing the past algorithms of drawing hypergraphs, this paper gives the definition of hyper graphs’ vertex degree and hyperedge’s average degree at first. Then it introduces the flow of this algorithm and particularly describes the rule set in setting the position of the vertex and the principal of minimum envelop law in drawing the hyperedge, and the complexity of this algorithm is analyzed. At last it draws a hypergraphs of scientific collaboration network successfully based on this algorithm and the result proves that the drawing algorithm of hyper graphs based on hyper edge’s average degree and multi-rules is feasible.

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2

RÖDL, V., A. RUCIŃSKI, and A. TARAZ. "Hypergraph Packing and Graph Embedding." Combinatorics, Probability and Computing 8, no. 4 (July 1999): 363–76. http://dx.doi.org/10.1017/s0963548399003879.

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We provide sufficient conditions for packing two hypergraphs. The emphasis is on the asymptotic case when one of the hypergraphs has a bounded degree and the other is dense. As an application, we give an alternative proof for the bipartite case of the recently developed Blow-up Lemma [12].

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3

Brown, Jason I., and Derek G. Corneil. "Graph properties and hypergraph colourings." Discrete Mathematics 98, no. 2 (December 1991): 81–93. http://dx.doi.org/10.1016/0012-365x(91)90034-y.

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4

Devezas, José, and Sérgio Nunes. "Hypergraph-of-entity." Open Computer Science 9, no. 1 (June 6, 2019): 103–27. http://dx.doi.org/10.1515/comp-2019-0006.

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AbstractModern search is heavily powered by knowledge bases, but users still query using keywords or natural language. As search becomes increasingly dependent on the integration of text and knowledge, novel approaches for a unified representation of combined data present the opportunity to unlock new ranking strategies. We have previously proposed the graph-of-entity as a purely graph-based representation and retrieval model, however this model would scale poorly. We tackle the scalability issue by adapting the model so that it can be represented as a hypergraph. This enables a significant reduction of the number of (hyper)edges, in regard to the number of nodes, while nearly capturing the same amount of information. Moreover, such a higher-order data structure, presents the ability to capture richer types of relations, including nary connections such as synonymy, or subsumption. We present the hypergraph-of-entity as the next step in the graph-of-entity model, where we explore a ranking approach based on biased random walks. We evaluate the approaches using a subset of the INEX 2009 Wikipedia Collection. While performance is still below the state of the art, we were, in part, able to achieve a MAP score similar to TF-IDF and greatly improve indexing efficiency over the graph-of-entity.

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Zakiyyah, Alfi Yusrotis. "Laplacian Integral of Particular Steiner System." Engineering, MAthematics and Computer Science (EMACS) Journal 3, no. 1 (January 31, 2021): 31–32. http://dx.doi.org/10.21512/emacsjournal.v3i1.6883.

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The notion of a hypergraph is motivated by a graph. In graph, every edge contains of two vertices. However, a hypergraph edges contains more than two vertices. In this article use hyperedge to mention edge of hypergraph. A finite projective plane of order n, denoted by , is a linear intersecting hypergraph. In this research finite projective plane order is Laplacian integral.

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6

Vu Dang, Nguyen Trinh, Loc Tran, and Linh Tran. "Noise-robust classification with hypergraph neural network." Indonesian Journal of Electrical Engineering and Computer Science 21, no. 3 (March 10, 2021): 1465. http://dx.doi.org/10.11591/ijeecs.v21.i3.pp1465-1473.

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<p>This paper presents a novel version of hypergraph neural network method. This method is utilized to solve the noisy label learning problem. First, we apply the PCA dimensional reduction technique to the feature matrices of the image datasets in order to reduce the “noise” and the redundant features in the feature matrices of the image datasets and to reduce the runtime constructing the hypergraph of the hypergraph neural network method. Then, the classic graph based semisupervised learning method, the classic hypergraph based semi-supervised learning method, the graph neural network, the hypergraph neural network, and our proposed hypergraph neural network are employed to solve the noisy label learning problem. The accuracies of these five methods are evaluated and compared. Experimental results show that the hypergraph neural network methods achieve the best performance when the noise level increases. Moreover, the hypergraph neural network methods are at least as good as the graph neural network.</p>

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7

Paul, Viji, and K. A. Germina. "On hypergraph coloring and 3-uniform linear hypergraph set-indexers of a graph." Discrete Mathematics, Algorithms and Applications 07, no. 02 (May 25, 2015): 1550015. http://dx.doi.org/10.1142/s1793830915500159.

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For a graph G = (V, E) and a nonempty set X, a linear hypergraph set-indexer (LHSI) is a function f : V(G) → 2X satisfying the following conditions: (i) f is injective (ii) the ordered pair Hf(G) = (X, f(V)), where f(V) = {f(v) : v ∈ V(G)}, is a linear hypergraph (iii) the induced function f⊕ : E → 2X defined by f⊕(uv) = f(u) ⊕ f(v), for all uv ∈ E is injective and (iv) Hf⊕(G) = (X, f⊕(E)), where f⊕(E) = {f⊕(e) : e ∈ E}, is a linear hypergraph. It is shown that a 3-uniform LHSI of a graph, corresponding to upper LHSI number, is unique up to isomorphism and possible relations between the coloring numbers of a given graph and the two linear set-indexing hypergraphs are established. Also, we show that the two hypergraphs associated with an extremal 3-uniform LHSI of a graph are 2-colorable and the corresponding induced edge hypergraphs are self-dual.

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8

Cowling, Peter. "The total graph of a hypergraph." Discrete Mathematics 167-168 (April 1997): 215–36. http://dx.doi.org/10.1016/s0012-365x(96)00230-0.

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9

D'Atri, Alessandro, and Marina Moscarini. "On hypergraph acyclicity and graph chordality." Information Processing Letters 29, no. 5 (November 1988): 271–74. http://dx.doi.org/10.1016/0020-0190(88)90121-4.

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10

Narayanamoorthy, S., and A. Tamilselvi. "Bipolar Fuzzy Line Graph of a Bipolar Fuzzy Hypergraph." Cybernetics and Information Technologies 13, no. 1 (March 1, 2013): 13–17. http://dx.doi.org/10.2478/cait-2013-0002.

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Abstract This paper introduces the concept of a bipolar fuzzy line graph of a bipolar fuzzy hypergraph and some of the properties of the bipolar fuzzy line graph of a bipolar fuzzy hypergraph are also examined.

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11

Eschbach, Thomas, Wolfgang Guenther, and Bernd Becker. "Orthogonal Hypergraph Drawing for Improved Visibility." Journal of Graph Algorithms and Applications 10, no. 2 (2006): 141–57. http://dx.doi.org/10.7155/jgaa.00122.

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12

Quaddoura, Ruzayn. "On 2-Colorability Problem for Hypergraphs with P_8-free Incidence Graphs." International Arab Journal of Information Technology 17, no. 2 (February 28, 2019): 257–63. http://dx.doi.org/10.34028/iajit/17/2/14.

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A 2-coloring of a hypergraph is a mapping from its vertex set to a set of two colors such that no edge is monochromatic. The hypergraph 2- Coloring Problem is the question whether a given hypergraph is 2-colorable. It is known that deciding the 2-colorability of hypergraphs is NP-complete even for hypergraphs whose hyperedges have size at most 3. In this paper, we present a polynomial time algorithm for deciding if a hypergraph, whose incidence graph is P_8-free and has a dominating set isomorphic to C_8, is 2-colorable or not. This algorithm is semi generalization of the 2-colorability algorithm for hypergraph, whose incidence graph is P_7-free presented by Camby and Schaudt.

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13

Fu, Sichao, Weifeng Liu, Yicong Zhou, and Liqiang Nie. "HpLapGCN: Hypergraph p-Laplacian graph convolutional networks." Neurocomputing 362 (October 2019): 166–74. http://dx.doi.org/10.1016/j.neucom.2019.06.068.

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14

Bellaachia, Abdelghani, and Mohammed Al-Dhelaan. "Random Walks in Hypergraph." International Journal of Education and Information Technologies 15 (March 10, 2021): 13–20. http://dx.doi.org/10.46300/9109.2021.15.2.

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Random walks on graphs have been extensively used for a variety of graph-based problems such as ranking vertices, predicting links, recommendations, and clustering. However, many complex problems mandate a high-order graph representation to accurately capture the relationship structure inherent in them. Hypergraphs are particularly useful for such models due to the density of information stored in their structure. In this paper, we propose a novel extension to defining random walks on hypergraphs. Our proposed approach combines the weights of destination vertices and hyperedges in a probabilistic manner to accurately capture transition probabilities. We study and analyze our generalized form of random walks suitable for the structure of hypergraphs. We show the effectiveness of our model by conducting a text ranking experiment on a real world data set with a 9% to 33% improvement in precision and a range of 7% to 50% improvement in Bpref over other random walk approaches.

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15

Guedes, André Luiz Pires, and Lilian Markenzon. "Directed hypergraph planarity." Pesquisa Operacional 25, no. 3 (December 2005): 383–90. http://dx.doi.org/10.1590/s0101-74382005000300005.

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Directed hypergraphs are generalizations of digraphs and can be used to model binary relations among subsets of a given set. Planarity of hypergraphs was studied by Johnson and Pollak; in this paper we extend the planarity concept to directed hypergraphs. It is known that the planarity of a digraph relies on the planarity of its underlying graph. However, for directed hypergraphs, this property do not apply and we propose a new approach which generalizes the usual concept. We also show that the complexity of the recognition of a directed hypergraph as planar is linear on the size of the hypergraph.

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16

Li, Yalan, and Bo Deng. "A New Method to Find the Wiener Index of Hypergraphs." Discrete Dynamics in Nature and Society 2020 (June 11, 2020): 1–6. http://dx.doi.org/10.1155/2020/8138942.

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The Wiener index is defined as the summation of distances between all pairs of vertices in a graph or in a hypergraph. Both models—graph-theoretical and hypergraph-theoretical—are used in mathematical chemistry for quantitatively studying physical and chemical properties of classical and nonclassical organic compounds. In this paper, we consider relationships between hypertrees and trees and hypercycles and cycles with respect to their Wiener indices.

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17

Emtander, E., R. Fröberg, F. Mohammadi, and S. Moradi. "Poincaré Series of some Hypergraph Algebras." MATHEMATICA SCANDINAVICA 112, no. 1 (March 1, 2013): 5. http://dx.doi.org/10.7146/math.scand.a-15229.

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A hypergraph $H=(V,E)$, where $V=\{x_1,\ldots,x_n\}$ and $E\subseteq 2^V$ defines a hypergraph algebra $R_H=k[x_1,\ldots, x_n]/(x_{i_1}\cdots x_{i_k}; \{i_1,\ldots,i_k\}\in E)$. All our hypergraphs are $d$-uniform, i.e., $|e_i|=d$ for all $e_i\in E$. We determine the Poincaré series $P_{R_H}(t)=\sum_{i=1}^\infty\dim_k\mathrm{Tor}_i^{R_H}(k,k)t^i$ for some hypergraphs generalizing lines, cycles, and stars. We finish by calculating the graded Betti numbers and the Poincaré series of the graph algebra of the wheel graph.

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18

Ma, Jichao, Chunyu Du, Weifeng Liu, and Yanjiang Wang. "Numerical Simulation of Higher-Order Nonlinearity of Human Brain Functional Connectivity Using Hypergraph p-Laplacian." Mathematics 9, no. 18 (September 21, 2021): 2345. http://dx.doi.org/10.3390/math9182345.

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Unravelling how the human brain structure gives rise to function is a central question in neuroscience and remains partially answered. Recent studies show that the graph Laplacian of the human brain’s structural connectivity (SC) plays a dominant role in shaping the pattern of resting-state functional connectivity (FC). The modeling of FC using the graph Laplacian of the brain’s SC is limited, owing to the sparseness of the Laplacian matrix. It is unable to model the negative functional correlations. We extended the graph Laplacian to the hypergraph p-Laplacian in order to describe better the nonlinear and high-order relations between SC and FC. First we estimated those possible links showing negative correlations between the brain areas shared across subjects by statistical analysis. Then we presented a hypergraph p-Laplacian model by embedding the two matrices referring to the sign of the correlations between the brain areas relying on the brain structural connectome. We tested the model on two experimental connectome datasets and evaluated the predicted FC by estimating its Pearson correlation with the empirical FC matrices. The results showed that the proposed diffusion model based on hypergraph p-Laplacian can predict functional correlations more accurately than the models using graph Laplacian as well as hypergraph Laplacian.

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19

Ibrahim, Rania, and David F. Gleich. "Local hypergraph clustering using capacity releasing diffusion." PLOS ONE 15, no. 12 (December 23, 2020): e0243485. http://dx.doi.org/10.1371/journal.pone.0243485.

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Local graph clustering is an important machine learning task that aims to find a well-connected cluster near a set of seed nodes. Recent results have revealed that incorporating higher order information significantly enhances the results of graph clustering techniques. The majority of existing research in this area focuses on spectral graph theory-based techniques. However, an alternative perspective on local graph clustering arises from using max-flow and min-cut on the objectives, which offer distinctly different guarantees. For instance, a new method called capacity releasing diffusion (CRD) was recently proposed and shown to preserve local structure around the seeds better than spectral methods. The method was also the first local clustering technique that is not subject to the quadratic Cheeger inequality by assuming a good cluster near the seed nodes. In this paper, we propose a local hypergraph clustering technique called hypergraph CRD (HG-CRD) by extending the CRD process to cluster based on higher order patterns, encoded as hyperedges of a hypergraph. Moreover, we theoretically show that HG-CRD gives results about a quantity called motif conductance, rather than a biased version used in previous experiments. Experimental results on synthetic datasets and real world graphs show that HG-CRD enhances the clustering quality.

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20

Jouili, Salim, and Salvatore Tabbone. "Hypergraph-based image retrieval for graph-based representation." Pattern Recognition 45, no. 11 (November 2012): 4054–68. http://dx.doi.org/10.1016/j.patcog.2012.04.016.

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21

Ould-Khaoua, M., L. M. Mackenzie, R. J. Sutherland, and R. Sotudeh. "Constraint-based evaluation of hypergraph and graph networks." Simulation Practice and Theory 4, no. 2-3 (May 1996): 119–40. http://dx.doi.org/10.1016/0928-4869(95)00043-7.

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22

Bertolazzi, P., G. Di Battista, and G. Liotta. "Parametric graph drawing." IEEE Transactions on Software Engineering 21, no. 8 (1995): 662–73. http://dx.doi.org/10.1109/32.403790.

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23

Gong, Zengtai, and Junhu Wang. "Hesitant fuzzy graphs, hesitant fuzzy hypergraphs and fuzzy graph decisions1." Journal of Intelligent & Fuzzy Systems 40, no. 1 (January 4, 2021): 865–75. http://dx.doi.org/10.3233/jifs-201016.

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Up to now, there have been a lot of research results about multi-attribute decision making problems by fuzzy graph theory. However, there are few investigations about multi-attribute decision making problems under the background of indecisiveness. The main reason is that the difference of cognition and the complexity of thinking by decision makers, for the same question have different opinions. In this paper, we proposed a hesitant fuzzy hypergraph model based on hesitant fuzzy sets and fuzzy hypergraphs. At the same time, some basic graph operations of hesitant fuzzy hypergraphs are investigated and several equivalence relationship between hesitant fuzzy hypergraphs, hesitant fuzzy formal concept analysis and hesitant fuzzy information systems are discussed. Since granular computing can deal with multi-attribute decision-making problems well, we considered the hesitant fuzzy hypergraph model of granular computing, and established an algorithm of multi-attribute decision-making problem based on hesitant fuzzy hypergraph model. Finally an example is given to illustrate the effectiveness of the algorithm.

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24

Putri, Faiq Fauziya, Triyani Triyani, and Ari Wardayani. "KONSEP DASAR HIPERGRAF DAN SIFAT-SIFATNYA." Jurnal Ilmiah Matematika dan Pendidikan Matematika 12, no. 2 (February 11, 2021): 49. http://dx.doi.org/10.20884/1.jmp.2020.12.2.3619.

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This article discusses fundamental properties of hypergraphs. Hypergraphs are generalization of graph which hyperedges, edges in hypergraph, can join more than two vertices. The fundamental properties in this article are the vertices degrees, connection in hypergraphs, and dual hypergraph. connectivity in hypergraphs in this article are walks, trails, strict trails, path, and cycles. In the end of this article, we present a few examples of problems that can be represented by hypergraph.

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Feng, Yifan, Haoxuan You, Zizhao Zhang, Rongrong Ji, and Yue Gao. "Hypergraph Neural Networks." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 3558–65. http://dx.doi.org/10.1609/aaai.v33i01.33013558.

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In this paper, we present a hypergraph neural networks (HGNN) framework for data representation learning, which can encode high-order data correlation in a hypergraph structure. Confronting the challenges of learning representation for complex data in real practice, we propose to incorporate such data structure in a hypergraph, which is more flexible on data modeling, especially when dealing with complex data. In this method, a hyperedge convolution operation is designed to handle the data correlation during representation learning. In this way, traditional hypergraph learning procedure can be conducted using hyperedge convolution operations efficiently. HGNN is able to learn the hidden layer representation considering the high-order data structure, which is a general framework considering the complex data correlations. We have conducted experiments on citation network classification and visual object recognition tasks and compared HGNN with graph convolutional networks and other traditional methods. Experimental results demonstrate that the proposed HGNN method outperforms recent state-of-theart methods. We can also reveal from the results that the proposed HGNN is superior when dealing with multi-modal data compared with existing methods.

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26

Goncharenko, I. V. "DRSA: a non-hierarchical clustering algorithm using k-NN graph and its application in vegetation classification." Vegetation of Russia, no. 27 (2015): 125–38. http://dx.doi.org/10.31111/vegrus/2015.27.125.

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In this article we proposed a new method of non-hierarchical cluster analysis using k-nearest-neighbor graph and discussed it with respect to vegetation classification. The method of k-nearest neighbor (k-NN) classiﬁcation was originally developed in 1951 (Fix, Hodges, 1951). Later a term “k-NN graph” and a few algorithms of k-NN clustering appeared (Cover, Hart, 1967; Brito et al., 1997). In biology k-NN is used in analysis of protein structures and genome sequences. Most of k-NN clustering algorithms build «excessive» graph firstly, so called hypergraph, and then truncate it to subgraphs, just partitioning and coarsening hypergraph. We developed other strategy, the “upward” clustering in forming (assembling consequentially) one cluster after the other. Until today graph-based cluster analysis has not been considered concerning classification of vegetation datasets.

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27

LENZ, JOHN, and DHRUV MUBAYI. "Perfect Packings in Quasirandom Hypergraphs II." Combinatorics, Probability and Computing 25, no. 4 (October 27, 2015): 595–611. http://dx.doi.org/10.1017/s0963548315000267.

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For each of the notions of hypergraph quasirandomness that have been studied, we identify a large class of hypergraphs F so that every quasirandom hypergraph H admits a perfect F-packing. An informal statement of a special case of our general result for 3-uniform hypergraphs is as follows. Fix an integer r ⩾ 4 and 0 < p < 1. Suppose that H is an n-vertex triple system with r|n and the following two properties: •for every graph G with V(G) = V(H), at least p proportion of the triangles in G are also edges of H,•for every vertex x of H, the link graph of x is a quasirandom graph with density at least p. Then H has a perfect Kr(3)-packing. Moreover, we show that neither of the hypotheses above can be weakened, so in this sense our result is tight. A similar conclusion for this special case can be proved by Keevash's Hypergraph Blow-up Lemma, with a slightly stronger hypothesis on H.

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28

Liotta, Giuseppe, and Henk Meijer. "Advances in graph drawing: The 11th International Symposium on Graph Drawing." Discrete Applied Mathematics 155, no. 9 (May 2007): 1077. http://dx.doi.org/10.1016/j.dam.2006.10.002.

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29

Hao, Yong-Jing, Ying-Lian Gao, Mi-Xiao Hou, Ling-Yun Dai, and Jin-Xing Liu. "Hypergraph Regularized Discriminative Nonnegative Matrix Factorization on Sample Classification and Co-Differentially Expressed Gene Selection." Complexity 2019 (August 19, 2019): 1–12. http://dx.doi.org/10.1155/2019/7081674.

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Nonnegative Matrix Factorization (NMF) is a significant big data analysis technique. However, standard NMF regularized by simple graph does not have discriminative function, and traditional graph models cannot accurately reflect the problem of multigeometry information between data. To solve the above problem, this paper proposed a new method called Hypergraph Regularized Discriminative Nonnegative Matrix Factorization (HDNMF), which captures intrinsic geometry by constructing hypergraphs rather than simple graphs. The introduction of the hypergraph method allows high-order relationships between samples to be considered, and the introduction of label information enables the method to have discriminative effect. Both the hypergraph Laplace and the discriminative label information are utilized together to learn the projection matrix in the standard method. In addition, we offered a corresponding multiplication update solution for the optimization. Experiments indicate that the method proposed is more effective by comparing with the earlier methods.

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30

Golumbic, Martin Charles, and Amir Wassermann. "Complexity and Algorithms for Graph and Hypergraph Sandwich Problems." Graphs and Combinatorics 14, no. 3 (August 31, 1998): 223–39. http://dx.doi.org/10.1007/s003730050028.

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31

Wu, Longcan, Daling Wang, Kaisong Song, Shi Feng, Yifei Zhang, and Ge Yu. "Dual-view hypergraph neural networks for attributed graph learning." Knowledge-Based Systems 227 (September 2021): 107185. http://dx.doi.org/10.1016/j.knosys.2021.107185.

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32

Chodrow, Philip S., Nate Veldt, and Austin R. Benson. "Generative hypergraph clustering: From blockmodels to modularity." Science Advances 7, no. 28 (July 2021): eabh1303. http://dx.doi.org/10.1126/sciadv.abh1303.

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Hypergraphs are a natural modeling paradigm for networked systems with multiway interactions. A standard task in network analysis is the identification of closely related or densely interconnected nodes. We propose a probabilistic generative model of clustered hypergraphs with heterogeneous node degrees and edge sizes. Approximate maximum likelihood inference in this model leads to a clustering objective that generalizes the popular modularity objective for graphs. From this, we derive an inference algorithm that generalizes the Louvain graph community detection method, and a faster, specialized variant in which edges are expected to lie fully within clusters. Using synthetic and empirical data, we demonstrate that the specialized method is highly scalable and can detect clusters where graph-based methods fail. We also use our model to find interpretable higher-order structure in school contact networks, U.S. congressional bill cosponsorship and committees, product categories in copurchasing behavior, and hotel locations from web browsing sessions.

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33

McCreary, C. L., R. O. Chapman, and F. S. Shieh. "Using graph parsing for automatic graph drawing." IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans 28, no. 5 (1998): 545–61. http://dx.doi.org/10.1109/3468.709599.

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34

Jing-wei, Huang, and Wei Wen-fang. "Evolutionary graph drawing algorithms." Wuhan University Journal of Natural Sciences 8, no. 1 (March 2003): 212–16. http://dx.doi.org/10.1007/bf02899481.

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35

Friedrich, Carsten, and Peter Eades. "Graph Drawing in Motion." Journal of Graph Algorithms and Applications 6, no. 3 (2002): 353–70. http://dx.doi.org/10.7155/jgaa.00057.

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36

Tantau, Till. "Graph Drawing in TikZ." Journal of Graph Algorithms and Applications 17, no. 4 (2013): 495–513. http://dx.doi.org/10.7155/jgaa.00301.

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37

Frishman, Y., and A. Tal. "Online Dynamic Graph Drawing." IEEE Transactions on Visualization and Computer Graphics 14, no. 4 (July 2008): 727–40. http://dx.doi.org/10.1109/tvcg.2008.11.

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38

Cohen, Robert F. "Dynamic graph drawing (abstract)." ACM SIGACT News 24, no. 1 (January 15, 1993): 60. http://dx.doi.org/10.1145/152992.153005.

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39

Papakostas, A., and I. G. Tollis. "Interactive orthogonal graph drawing." IEEE Transactions on Computers 47, no. 11 (1998): 1297–309. http://dx.doi.org/10.1109/12.736444.

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40

Cohen, R. F., P. Eades, Tao Lin, and F. Ruskey. "Three-dimensional graph drawing." Algorithmica 17, no. 2 (February 1997): 199–208. http://dx.doi.org/10.1007/bf02522826.

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41

Johnson, D. S., and H. O. Pollak. "Hypergraph planarity and the complexity of drawing venn diagrams." Journal of Graph Theory 11, no. 3 (1987): 309–25. http://dx.doi.org/10.1002/jgt.3190110306.

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42

Li, Xiaodong, Farhan Khan, Gohar Ali, Lubna Gul, and Muhammad Sarwar. "Hypergraphical Metric Spaces and Fixed Point Theorems." Mathematical Problems in Engineering 2021 (June 25, 2021): 1–7. http://dx.doi.org/10.1155/2021/9961734.

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Hypergraph is a generalization of graph in which an edge can join any number of vertices. Hypergraph is used for combinatorial structures which generalize graphs. In this research work, the notion of hypergraphical metric spaces is introduced, which generalizes many existing spaces. Some fixed point theorems are studied in the corresponding spaces. To show the authenticity of the established work, nontrivial examples and applications are also provided.

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43

Zhang, Yubo, Nan Wang, Yufeng Chen, Changqing Zou, Hai Wan, Xinbin Zhao, and Yue Gao. "Hypergraph Label Propagation Network." Proceedings of the AAAI Conference on Artificial Intelligence 34, no. 04 (April 3, 2020): 6885–92. http://dx.doi.org/10.1609/aaai.v34i04.6170.

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In recent years, with the explosion of information on the Internet, there has been a large amount of data produced, and analyzing these data is useful and has been widely employed in real world applications. Since data labeling is costly, lots of research has focused on how to efficiently label data through semi-supervised learning. Among the methods, graph and hypergraph based label propagation algorithms have been a widely used method. However, traditional hypergraph learning methods may suffer from their high computational cost. In this paper, we propose a Hypergraph Label Propagation Network (HLPN) which combines hypergraph-based label propagation and deep neural networks in order to optimize the feature embedding for optimal hypergraph learning through an end-to-end architecture. The proposed method is more effective and also efficient for data labeling compared with traditional hypergraph learning methods. We verify the effectiveness of our proposed HLPN method on a real-world microblog dataset gathered from Sina Weibo. Experiments demonstrate that the proposed method can significantly outperform the state-of-the-art methods and alternative approaches.

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Malvestuto, Francesco M., Mauro Mezzini, and Marina Moscarini. "Equivalence between Hypergraph Convexities." ISRN Discrete Mathematics 2011 (January 15, 2011): 1–22. http://dx.doi.org/10.5402/2011/806193.

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Let G be a connected graph on V. A subset X of V is all-paths convex (or ap -convex) if X contains each vertex on every path joining two vertices in X and is monophonically convex (or m-convex) if X contains each vertex on every chordless path joining two vertices in X. First of all, we prove that ap -convexity and m-convexity coincide in G if and only if G is a tree. Next, in order to generalize this result to a connected hypergraph H, in addition to the hypergraph versions of ap -convexity and m-convexity, we consider canonical convexity (or c-convexity) and simple-path convexity (or sp -convexity) for which it is well known that m-convexity is finer than both c-convexity and sp -convexity and sp -convexity is finer than ap -convexity. After proving sp -convexity is coarser than c-convexity, we characterize the hypergraphs in which each pair of the four convexities above is equivalent. As a result, we obtain a convexity-theoretic characterization of Berge-acyclic hypergraphs and of γ-acyclic hypergraphs.

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Devezas, José. "Graph-based entity-oriented search." ACM SIGIR Forum 55, no. 1 (June 2021): 1–2. http://dx.doi.org/10.1145/3476415.3476430.

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Entity-oriented search has revolutionized search engines. In the era of Google Knowledge Graph and Microsoft Satori, users demand an effortless process of search. Whether they express an information need through a keyword query, expecting documents and entities, or through a clicked entity, expecting related entities, there is an inherent need for the combination of corpora and knowledge bases to obtain an answer. Such integration frequently relies on independent signals extracted from inverted indexes, and from quad indexes indirectly accessed through queries to a triplestore. However, relying on two separate representation models inhibits the effective cross-referencing of information, discarding otherwise available relations that could lead to a better ranking. Moreover, different retrieval tasks often demand separate implementations, although the problem is, at its core, the same. With the goal of harnessing all available information to optimize retrieval, we explore joint representation models of documents and entities, while taking a step towards the definition of a more general retrieval approach. Specifically, we propose that graphs should be used to incorporate explicit and implicit information derived from the relations between text found in corpora and entities found in knowledge bases. We also take advantage of this framework to elaborate a general model for entity-oriented search, proposing a universal ranking function for the tasks of ad hoc document retrieval (leveraging entities), ad hoc entity retrieval, and entity list completion. At a conceptual stage, we begin by proposing the graph-of-entity, based on the relations between combinations of term and entity nodes. We introduce the entity weight as the corresponding ranking function, relying on the idea of seed nodes for representing the query, either directly through term nodes, or based on the expansion to adjacent entity nodes. The score is computed based on a series of geodesic distances to the remaining nodes, providing a ranking for the documents (or entities) in the graph. In order to improve on the low scalability of the graph-of-entity, we then redesigned this model in a way that reduced the number of edges in relation to the number of nodes, by relying on the hypergraph data structure. The resulting model, which we called hypergraph-of-entity, is the main contribution of this thesis. The obtained reduction was achieved by replacing binary edges with n -ary relations based on sets of nodes and entities (undirected document hyperedges), sets of entities (undirected hyperedges, either based on cooccurrence or a grouping by semantic subject), and pairs of a set of terms and a set of one entity (directed hyperedges, mapping text to an object). We introduce the random walk score as the corresponding ranking function, relying on the same idea of seed nodes, similar to the entity weight in the graph-of-entity. Scoring based on this function is highly reliant on the structure of the hypergraph, which we call representation-driven retrieval. As such, we explore several extensions of the hypergraph-of-entity, including relations of synonymy, or contextual similarity, as well as different weighting functions per node and hyperedge type. We also propose TF-bins as a discretization for representing term frequency in the hypergraph-of-entity. For the random walk score, we propose and explore several parameters, including length and repeats, with or without seed node expansion, direction, or weights, and with or without a certain degree of node and/or hyperedge fatigue, a concept that we also propose. For evaluation, we took advantage of TREC 2017 OpenSearch track, which relied on an online evaluation process based on the Living Labs API, and we also participated in TREC 2018 Common Core track, which was based on the newly introduced TREC Washington Post Corpus. Our main experiments were supported on the INEX 2009 Wikipedia collection, which proved to be a fundamental test collection for assessing retrieval effectiveness across multiple tasks. At first, our experiments solely focused on ad hoc document retrieval, ensuring that the model performed adequately for a classical task. We then expanded the work to cover all three entity-oriented search tasks. Results supported the viability of a general retrieval model, opening novel challenges in information retrieval, and proposing a new path towards generality in this area.

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Torres-Jimenez, Jose, Jose Carlos Perez-Torres, and Gildardo Maldonado-Martinez. "hClique: An exact algorithm for maximum clique problem in uniform hypergraphs." Discrete Mathematics, Algorithms and Applications 09, no. 06 (December 2017): 1750078. http://dx.doi.org/10.1142/s1793830917500781.

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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.

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Ballantine, Cristina M. "A Hypergraph with Commuting Partial Laplacians." Canadian Mathematical Bulletin 44, no. 4 (December 1, 2001): 385–97. http://dx.doi.org/10.4153/cmb-2001-039-x.

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AbstractLetFbe a totally real number field and let GLnbe the general linear group of rank n overF. Let р be a prime ideal ofFand Fрthe completion ofFwith respect to the valuation induced by р. We will consider a finite quotient of the affine building of the group GLnover the field Fр. We will view this object as a hypergraph and find a set of commuting operators whose sum will be the usual adjacency operator of the graph underlying the hypergraph.

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Wei, WeiYi, and Hui Chen. "Salient Object Detection Based on Weighted Hypergraph and Random Walk." Mathematical Problems in Engineering 2020 (July 23, 2020): 1–14. http://dx.doi.org/10.1155/2020/2073140.

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Recently, salient object detection based on the graph model has attracted extensive research interest in computer vision because the graph model can represent the relationship between two regions better. However, it is difficult to capture the high-level relationship between multiple regions. In this algorithm, the input image is segmented into superpixels first. Then, a weighted hypergraph model is established using fuzzy C-means clustering algorithm and a new weighting strategy. Finally, the random walk algorithm is used to sort all superpixels on the weighted hypergraph model to obtain the salient object. The experimental results on three benchmark datasets demonstrate that the proposed method performs better than some other state-of-the-art methods.

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CONLON, DAVID. "Hypergraph Packing and Sparse Bipartite Ramsey Numbers." Combinatorics, Probability and Computing 18, no. 6 (June 22, 2009): 913–23. http://dx.doi.org/10.1017/s0963548309990174.

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We prove that there exists a constant c such that, for any integer Δ, the Ramsey number of a bipartite graph on n vertices with maximum degree Δ is less than 2cΔn. A probabilistic argument due to Graham, Rödl and Ruciński implies that this result is essentially sharp, up to the constant c in the exponent. Our proof hinges upon a quantitative form of a hypergraph packing result of Rödl, Ruciński and Taraz.

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Emtander, Eric. "A class of hypergraphs that generalizes chordal graphs." MATHEMATICA SCANDINAVICA 106, no. 1 (March 1, 2010): 50. http://dx.doi.org/10.7146/math.scand.a-15124.

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In this paper we introduce a class of hypergraphs that we call chordal. We also extend the definition of triangulated hypergraphs, given by H. T. Hà and A. Van Tuyl, so that a triangulated hypergraph, according to our definition, is a natural generalization of a chordal (rigid circuit) graph. R. Fröberg has showed that the chordal graphs corresponds to graph algebras, $R/I(\mathcal{G})$, with linear resolutions. We extend Fröberg's method and show that the hypergraph algebras of generalized chordal hypergraphs, a class of hypergraphs that includes the chordal hypergraphs, have linear resolutions. The definitions we give, yield a natural higher dimensional version of the well known flag property of simplicial complexes. We obtain what we call $d$-flag complexes.

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