A MAGNETIC FRAMELET-BASED CONVOLUTIONAL NEURAL NETWORK FOR DIRECTED GRAPHS Lequan Lin and Junbin Gao

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A MAGNETIC FRAMELET-BASED CONVOLUTIONAL NEURAL NETWORK FOR

DIRECTED GRAPHS

Lequan Lin and Junbin Gao

Discipline of Business Analytics, The University of Sydney Business School

The University of Sydney, Camperdown, NSW 2006, Australia

llin0615@uni.sydney.edu.au, junbin.gao@sydney.edu.au

ABSTRACT

Spectral Graph Convolutional Networks (spectral GCNNs),

a powerful tool for analyzing and processing graph data,

typically apply frequency ﬁltering via Fourier transform to

obtain representations with selective information. Although

research shows that spectral GCNNs can be enhanced by

framelet-based ﬁltering, the massive majority of such re-

search only considers undirected graphs. In this paper, we

introduce Framelet-MagNet, a magnetic framelet-based spec-

tral GCNN for directed graphs (digraphs). The model applies

the framelet transform to digraph signals to form a more

sophisticated representation for ﬁltering. Digraph framelets

are constructed with the complex-valued magnetic Laplacian,

simultaneously leading to signal processing in both real and

complex domains. We empirically validate the predictive

power of Framelet-MagNet over a range of state-of-the-art

models in node classiﬁcation, link prediction, and denoising.

Index Terms—Directed Graph, Graph Convolutional

Neural Network, Magnetic Laplacian, Graph Framelets,

Graph Framelet Transform

1. INTRODUCTION

Recent years have witnessed the surging popularity of re-

search on graph convolutional neural networks (GCNNs)

[1]. Through the integration of graph signals and topolog-

ical structures in graph convolution, GCNNs usually pro-

duce more valuable insights than models that analyze data

in isolation. Especially, spectral GCNNs deﬁne their graph

convolution in the frequency domain, enabling the ﬁltering of

different frequency components in graph signals. However,

the majority of studies on signal processing in spectral GC-

NNs only focus on undirected graphs [2, 3, 4]. In this paper,

we aim to extend framelet-based signal processing to spectral

GCNNs for directed graphs (digraphs).

Many directional relationships are naturally modelled

as digraphs, such as citation relationships [5], website hy-

perlinks [6], and road directions [7]. Using graph edges

to represent directional information backbones the explo-

ration of more aspects of the underlying data, which usually

provides more useful ﬁndings. Nonetheless, while spec-

tral GCNNs assume the eigendecomposition of symmetric

graph Laplacian to provide real-valued eigenvalues and or-

thonormal eigenvectors, the digraph Laplacian is asymmetric.

Converting digraphs to undirected graphs facilitates the ex-

tension of spectral methods to digraphs, but destroys the

digraph structure. Therefore, many recent studies engage

in designing symmetric digraph Laplacian that can preserve

the directional information [8, 9, 10, 11]. Magnetic Lapla-

cian [11, 12] is one of the most successful instances. It is

a complex-valued Hermitian matrix, whose real part shows

edge existence, and imaginary part indicates edge directions.

Magnetic Laplacian-based digraph networks exploit magnetic

Laplacian in classic spectral GCNN architectures and have

demonstrated their power in various graph tasks [11].

Classic spectral GCNNs adopt Fourier transform in their

convolutional layer. Converting graphs signals to the Fourier

frequency domain allows the processing of signal frequen-

cies, but only from a global perspective. More speciﬁcally,

although we can detect signal frequencies, we cannot iden-

tify their position in the graph. To investigate both global

and local information, we can ensemble spectral GCNNs with

framelet transform instead. The framelet frequency domain is

composed of graph framelets, which are constructed through

dilation and translation of a set of localized scaling functions.

Nevertheless, most existing wavelet/framelet-based networks

are solely applicable to undirected graphs [2, 3, 4]. Although

SVD-GCN [13] accomplishes digraph framelet transform via

singular value decomposition (SVD) of the asymmetric di-

graph Laplacian, its theoretical rationale is very vague, for

example, how the Laplacian frequency can be linked to the

signal frequency in the SVD domain.

In this paper, we propose Framelet-MagNet, a magnetic

Laplacian-assisted framelet-based spectral GCNN for di-

graphs. Multiresolution Analysis enables us to construct

digraph framelets with the magnetic Laplacian and a ﬁlter

bank [14, 15]. In addition, we also construct quasi-framelet

directly in the frequency domain to impose double regulation

on digraph signals [4]. We exploit Chebyshev polynomial

approximation for fast framelet transform and reconstruction.

arXiv:2210.10993v2 [cs.LG] 2 May 2023

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摘要：

AMAGNETICFRAMELET-BASEDCONVOLUTIONALNEURALNETWORKFORDIRECTEDGRAPHSLequanLinandJunbinGaoDisciplineofBusinessAnalytics,TheUniversityofSydneyBusinessSchoolTheUniversityofSydney,Camperdown,NSW2006,Australiallin0615@uni.sydney.edu.au,junbin.gao@sydney.edu.auABSTRACTSpectralGraphConvolutionalNetworks(spec...

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A MAGNETIC FRAMELET-BASED CONVOLUTIONAL NEURAL NETWORK FOR DIRECTED GRAPHS Lequan Lin and Junbin Gao

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