A note on Markov normalized magnetic eigenmaps

Cloninger, Alexander

doi:10.1016/j.acha.2016.11.002

Cited by 8 publications

(4 citation statements)

References 3 publications

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“…[5,32,35,40] But thanks to its Hermitian properties, the magnetic Laplacian has been utilized in directed graph studies. [14,34] Some application studies such as graph clustering [4,6] node [54] and graph representation learning [12] are delivered. MagNet [54] proved the positive semidefinite property of the magnetic Laplacian and proposed a spectral graph convolution.…”

Section: Magnetic Laplacianmentioning

confidence: 99%

Signed Directed Graph Contrastive Learning with Laplacian Augmentation

Ko¹,

Choi²,

Kim³

2023

Preprint

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Graph contrastive learning has become a powerful technique for several graph mining tasks. It learns discriminative representation from different perspectives of augmented graphs. Ubiquitous in our daily life, singed-directed graphs are the most complex and tricky to analyze among various graph types. That is why singed-directed graph contrastive learning has not been studied much yet, while there are many contrastive studies for unsigned and undirected. Thus, this paper proposes a novel signed-directed graph contrastive learning, SDGCL. It makes two different structurally perturbed graph views and gets node representations via magnetic Laplacian perturbation. We use a node-level contrastive loss to maximize the mutual information between the two graph views. The model is jointly learned with contrastive and supervised objectives. The graph encoder of SDGCL does not depend on social theories or predefined assumptions. Therefore it does not require finding triads or selecting neighbors to aggregate. It leverages only the edge signs and directions via magnetic Laplacian. To the best of our knowledge, it is the first to introduce magnetic Laplacian perturbation and signed spectral graph contrastive learning. The superiority of the proposed model is demonstrated through exhaustive experiments on four real-world datasets. SDGCL shows better performance than other state-of-theart on four evaluation metrics.

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Section: Magnetic Laplacianmentioning

confidence: 99%

Signed Directed Graph Contrastive Learning with Laplacian Augmentation

Ko¹,

Choi²,

Kim³

2023

Preprint

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“…This matrix represents the undirected geometry of the graph in the magnitude of its entries and incorporates directional information in the phases. It has been studied by the graph signal processing community [30] and also applied to numerous data science applications such as clustering and community detection [27,16,25,26]. Recently, [76] showed that the Magnetic Laplacian could be effectively incorporated into a graph neural network.…”

Section: Signed Graphs Directed Graphs and Hypergraphsmentioning

confidence: 99%

Geometric Scattering on Measure Spaces

Chew¹,

Hirn²,

Krishnaswamy³

et al. 2022

Preprint

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The scattering transform is a multilayered, wavelet-based transform initially introduced as a mathematical model of convolutional neural networks (CNNs) that has played a foundational role in our understanding of these networks' stability and invariance properties. In subsequent years, there has been widespread interest in extending the success of CNNs to data sets with non-Euclidean structure, such as graphs and manifolds, leading to the emerging field of geometric deep learning. In order to improve our understanding of the architectures used in this new field, several papers have proposed generalizations of the scattering transform for non-Euclidean data structures such as undirected graphs and compact Riemannian manifolds without boundary. Analogous to the original scattering transform, these works prove that these variants of the scattering transform have desirable stability and invariance properties and aim to improve our understanding of the neural networks used in geometric deep learning.In this paper, we introduce a general, unified model for geometric scattering on measure spaces. Our proposed framework includes previous work on compact Riemannian manifolds without boundary and undirected graphs as special cases but also applies to more general settings such as directed graphs, signed graphs, and manifolds with boundary. We propose a new criterion that identifies to which groups a useful representation should be invariant and show that this criterion is sufficient to guarantee that the scattering transform has desirable stability and invariance properties. Additionally, we consider finite measure spaces that are obtained from randomly sampling an unknown manifold. We propose two methods for constructing a data-driven graph on which the associated graph scattering transform approximates the scattering transform on the underlying manifold. Moreover, we use a diffusion-maps based approach to prove quantitative estimates on the rate of convergence of one of these approximations as the number of sample points tends to infinity. Lastly, we showcase the utility of our method on spherical images, a directed graph stochastic block model, and on high-dimensional single-cell data.

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“…However, a user may choose to also tune q through a standard cross-validation procedure as in [13]. Moreover, one can readily address the latter issue by normalizing the adjacency matrix via a preprocessing step (e.g., [30]). In contrast to our magnetic signed Laplacian, in the case where the graph is not signed but is weighted and directed, the matrix proposed in [29] does not reduce to the magnetic Laplacian considered in [13].…”

Section: Related Workmentioning

confidence: 99%

MSGNN: A Spectral Graph Neural Network Based on a Novel Magnetic Signed Laplacian

He¹,

Permultter²,

Reinert³

et al. 2022

Preprint

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Signed and directed networks are ubiquitous in real-world applications. However, there has been relatively little work proposing spectral graph neural networks (GNNs) for such networks. Here we introduce a signed directed Laplacian matrix, which we call the magnetic signed Laplacian, as a natural generalization of both the signed Laplacian on signed graphs and the magnetic Laplacian on directed graphs. We then use this matrix to construct a novel efficient spectral GNN architecture and conduct extensive experiments on both node clustering and link prediction tasks. In these experiments, we consider tasks related to signed information, tasks related to directional information, and tasks related to both signed and directional information. We demonstrate that our proposed spectral GNN is effective for incorporating both signed and directional information, and attains leading performance on a wide range of data sets. Additionally, we provide a novel synthetic network model, which we refer to as the signed directed stochastic block model, and a number of novel real-world data sets based on lead-lag relationships in financial time series.Preprint. Preliminary work.

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A note on Markov normalized magnetic eigenmaps

Cited by 8 publications

References 3 publications

Signed Directed Graph Contrastive Learning with Laplacian Augmentation

Signed Directed Graph Contrastive Learning with Laplacian Augmentation

Geometric Scattering on Measure Spaces

MSGNN: A Spectral Graph Neural Network Based on a Novel Magnetic Signed Laplacian

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