Nonlinear Higher-Order Label Spreading

Tudisco, Francesco; Benson, Austin R.; Prokopchik, Konstantin

doi:10.1145/3442381.3450035

Cited by 16 publications

(10 citation statements)

References 46 publications

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“…As a consequence, since U is entrywise positive, there exists a M > 0 such that Φ(F ) ≤ M U for all F such that ϕ(F ) = 1. The thesis thus follows from Theorem 3.1 in [35].…”

Section: The Modelmentioning

confidence: 86%

“…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. However, these methods are designed for cases where only labels are available, and do not take advantage of rich features or metadata associated with hypergraphs that are potentially useful for making accurate predictions.…”

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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“…As a consequence, since U is entrywise positive, there exists a M > 0 such that Φ(F ) ≤ M U for all F such that ϕ(F ) = 1. The thesis thus follows from Theorem 3.1 in [35].…”

Section: The Modelmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

A nonlinear diffusion method for semi-supervised learning on hypergraphs

Tudisco¹,

Prokopchik²,

Benson³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…We note that this is somewhat automatically obtained by our choice of smooth function f . In fact, while in the graph setting the non-smooth limit case q → ∞ is to be preferred [48] as each edge contains exactly either one, two or no core nodes, we argue that large but finite values of q are better suited to hypergraphs. In fact, when 1 q < ∞, the cost function f (x) naturally handles possible ambiguity due to the presence of hyperedges with more than two nodes in the core: While the infinity norm x| e ∞ is large if there is at least one core node in e but ignores the presence of a larger number of core nodes, x| e q (for large but finite q), is large when there is at least one core node in e but grows when the hyperedge contains a larger number of such nodes.…”

Section: P(e E)mentioning

confidence: 89%

“…In particular, in [48] this type of hypergraph mapping is used to define generalized eigenvector centrality scores for hypergraphs which include as special cases hypergraph centralities based on tensor eigenvectors [10]. Thus, the proposed hypergraph core-periphery score can be interpreted as a particular hypergraph centrality designed specifically for core-periphery problems and gives mathematical support to the intuition that centrality measures for hypergraphs may be an indication of core and periphery [3].…”

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confidence: 95%

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Core-periphery detection in hypergraphs

Tudisco¹,

Higham²

2022

Preprint

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Core-periphery detection is a key task in exploratory network analysis where one aims to find a core, a set of nodes well-connected internally and with the periphery, and a periphery, a set of nodes connected only (or mostly) with the core. In this work we propose a model of core-periphery for higher-order networks modeled as hypergraphs and we propose a method for computing a core-score vector that quantifies how close each node is to the core. In particular, we show that this method solves the corresponding non-convex core-periphery optimization problem globally to an arbitrary precision. This method turns out to coincide with the computation of the Perron eigenvector of a nonlinear hypergraph operator, suitably defined in term of the incidence matrix of the hypergraph, generalizing recently proposed centrality models for hypergraphs. We perform several experiments on synthetic and real-world hypergraphs showing that the proposed method outperforms alternative core-periphery detection algorithms, in particular those obtained by transferring established graph methods to the hypergraph setting via clique expansion.

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Quantifying the Structural Stability of Simplicial Homology

2023

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Simplicial complexes are generalizations of classical graphs. Their homology groups are widely used to characterize the structure and the topology of data in e.g. chemistry, neuroscience, and transportation networks. In this work we assume we are given a simplicial complex and that we can act on its underlying graph, formed by the set of 1-simplices, and we investigate the stability of its homology with respect to perturbations of the edges in such graph. Precisely, exploiting the isomorphism between the homology groups and the higher-order Laplacian operators, we propose a numerical method to compute the smallest graph perturbation sufficient to change the dimension of the simplex’s Hodge homology. Our approach is based on a matrix nearness problem formulated as a matrix differential equation, which requires an appropriate weighting and normalizing procedure for the boundary operators acting on the Hodge algebra’s homology groups. We develop a bilevel optimization procedure suitable for the formulated matrix nearness problem and illustrate the method’s performance on a variety of synthetic quasi-triangulation datasets and real-world transportation networks.

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Nonlinear Higher-Order Label Spreading

Cited by 16 publications

References 46 publications

A nonlinear diffusion method for semi-supervised learning on hypergraphs

A nonlinear diffusion method for semi-supervised learning on hypergraphs

Core-periphery detection in hypergraphs

Quantifying the Structural Stability of Simplicial Homology

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