A framework for second-order eigenvector centralities and clustering coefficients

Arrigo, Francesca; Higham, Desmond J.; Tudisco, Francesco

doi:10.1098/rspa.2019.0724

Cited by 14 publications

(14 citation statements)

References 59 publications

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“…Another relatively well-established idea to represent and work with hypergraphs relies on the use of a higher-order tensors [4,7]. This approach requires a uniform hypergraph, which can be fully described by an adjacency tensor, and can be used to define the centrality scores in terms of the Perron eigenvector of the adjacency tensor.…”

Section: Motivationmentioning

confidence: 99%

Node and Edge Eigenvector Centrality for Hypergraphs

Tudisco

Higham

2021

Preprint

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Network scientists have shown that there is great value in studying pairwise interactions between components in a system. From a linear algebra point of view, this involves defining and evaluating functions of the associated adjacency matrix. Recent work indicates that there are further benefits from accounting directly for higher order interactions, notably through a hypergraph representation where an edge may involve multiple nodes. Building on these ideas, we motivate, define and analyze a class of spectral centrality measures for identifying important nodes and hyperedges in hypergraphs, generalizing existing network science concepts. By exploiting the latest developments in nonlinear Perron-Frobenius theory, we show how the resulting constrained nonlinear eigenvalue problems have unique solutions that can be computed efficiently via a nonlinear power method iteration. We illustrate the measures on realistic data sets.

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Section: Motivationmentioning

confidence: 99%

Node and Edge Eigenvector Centrality for Hypergraphs

Tudisco

Higham

2021

Preprint

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“…This approximation boils down to a mapping given by A = I − L, where L is the (possibly rescaled) normalized Laplacian matrix of the graph L = I − D −1/2 AD −1/2 , A is the adjacency matrix, and A = D −1/2 AD −1/2 is the normalized adjacency matrix. The forward model for a two-layer GCN is the Z = softmax(F ) = softmax(Aσ(AXΘ (1) )Θ (2) )…”

Section: Neural Network Approachesmentioning

confidence: 99%

“…where X ∈ R n×d is the matrix of the graph signals (the node features), Θ (1) , Θ (2) are the input-to-hidden and hidden-to-output weight matrices of the network and σ is a nonlinear activation function (typically, σ = ReLU).…”

Section: Neural Network Approachesmentioning

confidence: 99%

“…A hypergraph is a standard representation for such data, where a hyperedge can connect any number of nodes. Directly modeling these higher-order interactions has led to improvements in a number of machine learning problems [42,6,22,23,39,32,2]. Along this line, there are a number of diffusions or label spreading techniques for semi-supervised learning on hypergraphs [42,14,40,21,24,37,35], which are also built on principles of similarity or assortativity.…”

Section: Introductionmentioning

confidence: 99%

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A nonlinear diffusion method for semi-supervised learning on hypergraphs

Tudisco¹,

Prokopchik²,

Benson³

2021

Preprint

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Hypergraphs are a common model for multiway relationships in data, and hypergraph semisupervised learning is the problem of assigning labels to all nodes in a hypergraph, given labels on just a few nodes. Diffusions and label spreading are classical techniques for semi-supervised learning in the graph setting, and there are some standard ways to extend them to hypergraphs. However, these methods are linear models, and do not offer an obvious way of incorporating node features for making predictions. Here, we develop a nonlinear diffusion process on hypergraphs that spreads both features and labels following the hypergraph structure, which can be interpreted as a hypergraph equilibrium network. Even though the process is nonlinear, we show global convergence to a unique limiting point for a broad class of nonlinearities, which is the global optimum of a interpretable, regularized semi-supervised learning loss function. The limiting point serves as a node embedding from which we make predictions with a linear model. Our approach is much more accurate than several hypergraph neural networks, and also takes less time to train.

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“…On the one end, higherorder adjacency tensors have been employed to model higher-order neighborhoods made by hyperedges containing three or more nodes. This approach is typically characterized by the use of hypergraphs or simplicial complexes [3,4,11,17]. On the other hand, non Markovian stochastic processes with memory have been used to model random walks that take into account longer paths of connections [2,6,16,31].…”

Section: Introductionmentioning

confidence: 99%

Nonlocal pagerank

2021

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In this work we introduce and study a nonlocal version of the PageRank. In our approach, the random walker explores the graph using longer excursions than just moving between neighboring nodes. As a result, the corresponding ranking of the nodes, which takes into account a long-range interaction between them, does not exhibit concentration phenomena typical of spectral rankings which take into account just local interactions. We show that the predictive value of the rankings obtained using our proposals is considerably improved on different real world problems.

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A framework for second-order eigenvector centralities and clustering coefficients

Cited by 14 publications

References 59 publications

Node and Edge Eigenvector Centrality for Hypergraphs

Node and Edge Eigenvector Centrality for Hypergraphs

A nonlinear diffusion method for semi-supervised learning on hypergraphs

Nonlocal pagerank

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