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Adjacency and Tensor Representation in General Hypergraphs Part 1: e-adjacency Tensor Uniformisation Using Homogeneous Polynomials
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TL;DR: This paper contributes in a uniformization process of a general hypergraph to allow the definition of an e-adjacency tensor, viewed as a hypermatrix, reflecting the general hyper graph structure.
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Abstract: Adjacency between two vertices in graphs or hypergraphs is a pairwise relationship. It is redefined in this article as 2-adjacency. In general hypergraphs, hyperedges hold for $n$-adic relationship. To keep the $n$-adic relationship the concepts of $k$-adjacency and e-adjacency are defined. In graphs 2-adjacency and e-adjacency concepts match, just as $k$-adjacency and e-adjacency do for $k$-uniform hypergraphs. For general hypergraphs these concepts are different. This paper also contributes in a uniformization process of a general hypergraph to allow the definition of an e-adjacency tensor, viewed as a hypermatrix, reflecting the general hypergraph structure. This symmetric e-adjacency hypermatrix allows to capture not only the degree of the vertices and the cardinality of the hyperedges but also makes a full separation of the different layers of a hypergraph.
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Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond
TL;DR: In this article, the authors provide a didactic treatment of the emerging topic of signal processing on higher-order networks, with a special emphasis on the concepts needed for the processing of signals supported on these structures.
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Phase transitions and stability of dynamical processes on hypergraphs
TL;DR: In this paper, the authors derive expressions for the stability of dynamical systems defined on an arbitrary hypergraph and provide a general framework for further exploration of dynamic processes on hypergraphs.
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Hypergraph Learning with Line Expansion.
TL;DR: By reducing the hypergraph to a simple graph, the proposed 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.
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Community Detection in General Hypergraph via Graph Embedding
Yaoming Zhen,Junhui Wang +1 more
TL;DR: In this paper, a null vertex is introduced to augment a non-uniform hypergraph into a uniform multi-hypergraph, and then embeds the multihop in a low-dimensional vector space such that vertices within the same community are close to each other.
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Adjacency and Tensor Representation in General Hypergraphs.Part 2: Multisets, Hb-graphs and Related e-adjacency Tensors.
TL;DR: HyperBagGraphs (hb-graphs as short) extend hypergraphs by allowing the hyperedges to be multisets by extending the definition of e-adjacency to natural hb- graphs and defining different way of building an e- adjacency tensor that is used withhypergraphs.
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