"2020年图神经网络分类、进展与趋势"

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"2020年的图神经网络分类、进展和趋势报告对图神经网络进行了全面的分析。在这份报告中，作者分别介绍了图神经网络的分类、进展和趋势。首先，作者通过对图神经网络的分类进行了详细解释，包括基于图结构的深度学习、图卷积网络等。接着，报告详细介绍了图神经网络的最新进展，包括在社交网络、生物信息学、推荐系统等领域的应用。最后，作者对图神经网络未来的发展趋势进行了展望，强调了在图数据挖掘、图神经网络理论等方面的重要性。这份报告的发布为图神经网络的研究和应用提供了重要的参考，为相关领域的研究者提供了宝贵的指导。" "图神经网络:分类、进展和趋势"报告是由来自中国太原理工大学数据科学学院以及山西省空间信息网络工程技术研究中心的研究人员撰写的，对图神经网络进行了系统综述。该报告内容涵盖了图神经网络的分类、最新进展和未来发展趋势，在图神经网络领域具有较高的学术价值。报告中对图神经网络的分类进行了深入的解析，包括不同类型的图神经网络模型及其在实际应用中的表现。在最新进展方面，报告提到了图神经网络在社交网络分析、生物信息学、推荐系统等领域的前沿应用，说明图神经网络在多个领域取得了显著的进展。此外，报告还展望了图神经网络未来的发展趋势，强调了在图数据挖掘、图神经网络理论等方面的重要性，为该领域的研究者指明了未来的发展方向。在报告中还介绍了图神经网络的基本理念，其基础模型是如何运作的。报告的作者通过详细阐述了这些概念和模型，让读者对图神经网络有了更清晰的认识。此外，报告还指出了图神经网络的未来趋势和发展方向，对于该领域的学术研究和技术应用具有重要的指导意义。总之，"图神经网络:分类、进展和趋势"报告对图神经网络进行了全面而深入的研究，对该领域的研究者具有重要的参考价值。它不仅对图神经网络的分类和最新进展进行了系统梳理，还展望了未来的发展方向，为相关领域的学术研究和技术应用提供了宝贵的参考。在当前广泛关注的人工智能和大数据领域，图神经网络的研究和应用具有重要意义，相信这份报告能够为相关领域的研究与实践者提供重要的参考，促进图神经网络领域的不断发展和进步。"

000:8 Yu Zhou, et al.

Current Edge

Update states of edges

Aggregate states of edges

Current Node

Update states of nodes

Aggregate states of all edges

Aggregate states of all nodes

update the global state

Fig. 3. Computational framework of the spatial GCNN.

Variants of GNs——NLNNs.

NLNNs [

] give a general denition of non-local operations [

]

which is a exible building block and can be easily integrated into convolutional/recurrent layers.

Specically, the generic non-local operation is dened as

𝑦

𝑠

C(𝑥

𝑠

)



𝑡

𝑓 (𝑥

𝑠

, 𝑥

𝑡

)𝑔(𝑥

𝑡

), (10)

where

𝑓 (·, ·)

denotes the anity between

𝑥

𝑠

and

𝑥

𝑡

, and

C(𝑥

𝑠

) =

𝑡

𝑓 (𝑥

𝑠

, 𝑥

𝑡

)

is a normalization

factor. The anity function 𝑓 (·, ·) is of the following form

(1) Gaussian: 𝑓 (𝑥

𝑠

, 𝑥

𝑡

) = 𝑒

𝑥

𝑇

𝑠

𝑥

𝑡

;

(2) Embedded Gaussian: 𝑓 (𝑥

𝑠

, 𝑥

𝑡

) = 𝑒

𝜃 (𝑥

𝑠

)

𝑇

𝜂 (𝑥

𝑡

)

, where 𝜃 (𝑥

𝑠

) = 𝑊

𝜃

𝑥

𝑠

and 𝜂(𝑥

𝑡

) = 𝑊

𝜂

𝑥

𝑡

;

(3) Dot Product: 𝑓 (𝑥

𝑠

, 𝑥

𝑡

) = 𝜃 (𝑥

𝑠

)

𝑇

𝜂(𝑥

𝑡

);

(4) Concatenation: 𝑓 (𝑥

𝑠

, 𝑥

𝑡

) = ReLU(𝑤

𝑇

𝑓

[

𝜃 (𝑥

𝑠

), 𝜂 (𝑥

𝑡

)

]

The non-local building block is dened as

𝑧

𝑠

= 𝑊

𝑧

𝑦

𝑠

+𝑥

𝑠

where "

+𝑥

𝑠

" denotes a residual connection.

It is noted that

𝑓 (·, ·)

and

𝑔(·)

play the role of

𝜙

𝑒

(𝑒

𝑘

, 𝑣

𝑟

𝑘

, 𝑣

𝑠

𝑘

, 𝑢)

, and the summation in Eq. (10) plays

the role of 𝜓

𝑒→𝑣

(𝐸

′

𝑖

Variants of GNs——GraphSAGE.

GraphSAGE (SAmple and aggreGatE) [

204

] is a general

inductive framework capitalizing on node feature information to eciently generate node embed-

ding vectors for previously unseen nodes. Specically, GraphSAGE is composed of an aggregation

function Aggregate

(𝑙)

(·) and an update function Update

(𝑙)

, i.e.

𝑥

(𝑙)

𝑁

𝐺

(𝑣)

= Aggregate

(𝑙)

n

𝑥

(𝑙−1)

𝑢

: 𝑢 ∈ 𝑁

𝐺

(𝑣)

o

𝑥

(𝑙)

𝑣

= Update

(𝑙)

n

𝑥

(𝑙−1)

𝑣

, 𝑥

(𝑙)

𝑁

𝐺

(𝑣)

o

where

𝑁

𝐺

(𝑣)

denotes a xed-size set of neighbors of

𝑣

uniformly sampling from its whole neighbors.

The aggregation function is of the following form

(1) Mean Aggregator: 𝑥

(𝑙)

𝑣

= 𝜎



𝑊 · Mean

n

𝑥

(𝑙−1)

𝑣

∪

𝑥

(𝑙−1)

𝑢

: 𝑢 ∈ 𝑁

𝐺

(𝑣)

o

;

ACM Trans. Intell. Syst. Technol., Vol. 00, No. 0, Article 000. Publication date: 2020.

Graph Neural Networks: Taxonomy, Advances and Trends 000:9

(2) LSTM Aggregator: applying the LSTM [171] to aggregate the neighbors of 𝑣;

(3) Pooling Aggregator: Aggregate

(𝑙)

= max

n

𝜎



𝑊 𝑥

(𝑙)

𝑢

+𝑏



: 𝑢 ∈ 𝑁

𝐺

(𝑣)

o

Note that the aggregation function and update function play the role of

𝜓

𝑒→𝑣

(𝐸

′

𝑖

)

and

𝜙

𝑣

(𝑣

𝑖

, 𝑒

′

𝑖

, 𝑢)

in formula (9) respectively.

Variants of GNs——Hyperbolic GCNNs.

The Euclidean GCNNs aim to embed nodes in a graph

into a Euclidean space. This will incur a large distortion especially when embedding real-world

graphs with scale-free and hierarchical structure. Hyperbolic GCNNs pave an alternative way of

embedding with little distortion. The

𝑛-dimensional

hyperbolic space [

155

], denoted as

𝑛

𝐾

, is a

unique, complete, simply connected

𝑑-dimensional

Riemannian manifold with constant negative

sectional curvature −

𝐾

, i.e.

𝑛

𝐾



𝑥 ∈ R

𝑛+1

: ⟨𝑥, 𝑥⟩

= −𝐾, 𝑥

> 0



where the Minkowski inner produce

⟨𝑥, 𝑦⟩

= −𝑥

𝑦

𝑑

𝑗=1

𝑥

𝑗

𝑦

𝑗

, ∀𝑥, 𝑦 ∈ R

𝑛+1

. Its tangent space

centered at point

𝑥

is denoted as

𝑥

𝑛

𝐾



𝑣 ∈ R

𝑛+1

: ⟨𝑥, 𝑣⟩

= 0



. Given

𝑥 ∈ H

𝑛

𝐾

, let

𝑢 ∈ T

𝑥

𝑛

𝐾

be unit-speed. The unique unit-speed geodesic 𝛾

𝑥→𝑢

(·) such that 𝛾

𝑥→𝑢

(0) = 𝑥 and ¤𝛾

𝑥→𝑢

(0) = 𝑢 is

denoted as

𝛾

𝑥→𝑢

(𝑡) = cosh



𝑡

√

𝐾



𝑥 +

√

𝐾 sinh



𝑡

√

𝐾



, 𝑢, 𝑡 >

The intrinsic distance between two

points 𝑥, 𝑦 ∈ H

𝑛

𝐾

is then equal to

𝑑

𝐾

(

𝑥,𝑦

)

√

𝐾arcosh



−

⟨𝑥, 𝑦⟩

𝐾



Therefore, the above

𝑛-dimensional

hyperbolic space with constant negative sectional curvature

−

𝐾

is usually denoted as



𝑛

𝐾

, 𝑑

𝐾

(·, ·)



. In particular,

𝑛

, i.e.

𝐾 =

1, is called the hyperboloid

model of the hyperbolic space. Hyperbolic Graph Convolutional Networks (HGCN) [

] benet

from the expressiveness of both GCNNs and hyperbolic embedding. It employs the exponential

and logarithmic maps of the hyperboloid model, respectively denoted as

exp

𝐾

𝑥

(·)

and

log

𝐾

𝑥

(·)

, to

realize the mutual transformation between Euclidean features and hyperbolic ones. Let

∥𝑣 ∥

⟨𝑣, 𝑣⟩

, 𝑣 ∈ T

𝑥

𝑛

𝐾

. The exp

𝐾

𝑥

(·) and log

𝐾

𝑥

(·) are respectively dened to be

exp

𝐾

𝑥

(𝑣) = cosh



∥𝑣 ∥

√

𝐾



𝑥 +

√

𝐾 sinh



∥𝑣 ∥

√

𝐾



𝑣

∥𝑣 ∥

log

𝐾

𝑥

(𝑦) = 𝑑

𝐾

(𝑥, 𝑦)

𝑦 +

𝐾

⟨𝑥, 𝑦⟩

𝑥



𝑦 +

𝐾

⟨𝑥, 𝑦⟩

𝑥



where

𝑥 ∈ H

𝑛

𝐾

𝑣 ∈ T

𝑥

𝑛

𝐾

and

𝑦 ∈ H

𝑛

𝐾

such that

𝑦 ≠

0 and

𝑦 ≠ 𝑥

. The HGCN architecture

is composed of three components: a Hyperbolic Feature Transform (HFT), an Attention-Based

Aggregation (ABA) and a Non-Linear Activation with Dierent Curvatures (NLADC). They are

respectively dened as

ℎ

𝐻,𝑙

𝑗



𝑊

(𝑙)

⊗

𝐾

𝑙−1

𝑥

𝐻,𝑙 −1

𝑗



⊕

𝐾

𝑙−1

𝑏

(𝑙)

(HFT),

𝑦

𝐻,𝑙

𝑗

= Aggregate

𝐾

𝑙−1

(ℎ

𝐻,𝑙

)

𝑗

(ABA),

𝑥

𝐻,𝑙

𝑗

= exp

𝐾

𝑙

𝑜



𝜌



log

𝐾

𝑙−1

𝑜



𝑦

𝐻,𝑙

𝑗



(NLADC),

where

𝑜 =



√

𝐾, 0, ··· , 0



∈ H

𝑛

𝐾

, the subscript

𝑗

denotes the indices of nodes, the superscript

𝑙

denotes the layer of the HGCN. The linear transform in hyperboloid manifold is dened to be

𝑊 ⊗

𝐾

ACM Trans. Intell. Syst. Technol., Vol. 00, No. 0, Article 000. Publication date: 2020.

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