Characterize and Transfer Attention in Graph Neural Networks
Authors: Mufei Li
Hao Zhang
Xingjian Shi
Minjie Wang
Yixing Guan
Zheng Zhang
Venue:  (2020)
Keywords: Graph Neural Networks
Graph Attention Networks
Attention
Transfer Learning
Empirical Study

Abstract: Does attention matter and, if so, when and how? Our study on both inductive and transductive learning suggests that datasets have a strong influence on the effects of attention in graph neural networks. Independent of learning setting, task and attention variant, attention mostly degenerate to simple averaging for all three citation networks, whereas they behave strikingly different in the protein-protein interaction networks and molecular graphs: nodes attend to different neighbors per head and get more focused in deeper layers. Consequently, attention distributions become telltale features of the datasets themselves. We further explore the possibility of transferring attention for graph sparsification and show that, when applicable, attention-based sparsification retains enough information to obtain good performance while reducing computational and storage costs. Finally, we point out several possible directions for further study and transfer of attention.

Under review as a conference paper at ICLR 2020
CHARACTERIZE AND TRANSFER ATTENTION IN GRAPH NEURAL NETWORKS
Anonymous authors Paper under double-blind review
ABSTRACT
Does attention matter and, if so, when and how? Our study both inductive and transductive learning suggests that datasets have a strong influence on the effects of attention in graph neural networks. Independent of learning setting, task and attention variant, attention mostly degenerate to simple averaging for all three citation networks, whereas they behave strikingly different in the protein- protein-protein protein interaction networks and molecular graphs: nodes attend to different neighbors per head and get more focused in deeper layers. Consequently, attention distribu- tions become telltale features of the datasets themselves. We further explore the possibility of transferring attention for graph sparsification and show that, when applicable, attention-based sparsification retains enough information to obtain good performance while reducing computational and storage costs. Finally, we point out several possible directions for further study and transfer of attention.
INTRODUCTION
The modeling of graphs has become an active research topic in deep learning (Bronstein et al. 2017)
Dozens neural network models have been developed for exploiting the structural information of
graphs (Scarselli et al. 20099, Bruna et al., 2014 Henaff et al. 2013 Duvenaud et al. 2015 Niepert
ct al. Defferradd et al. 2016), now collectively referred to as graph neural networks (GNNs).
Built upon the success of attention in NLP (Vaswani et al. 2017) Velicković et al. (0088) proposed
the graph attention networks (GATs) to integrate multi-head self-artention into node feature update for adaptive weighting, with several extensions (Thekumparampi et al. 2018 Zhang et al. 2018
Monti et al., 2018 Svoboda et al. 2019 Trivedi et al.) 2019). While the use of attention in GNNs an attractive direction, several works also report that attention contributes little to the performance of GNNs (Zhang et al., 2018 Shchur et al. 2019). Considering the high computational cost of attention, the question is then that does attention help and, if so, when and how?
In this paper, we take a first step towards the question. We first identify the key questions for understanding attention and propose analytical paradigm. With extensive experiments, findings suggest that, although attention is motivated by inductive learning, its functionality depends highly on the characteristics of the datasets. The attention distributions across heads and layers are uniform for all citation networks (Cora, Citeseer and Pubmed) while they get more concentrated over layers on the protein-protein interaction networks (PPI) and molecular graphs, with significant diversity among heads. That the attention distribution is telltale sign of the nature of graph class is further verified with meta graph classification experiment. With attention features inputs, citation networks are indistinguishable whereas PPI and molecule graphs are.
Inspired by these findings, we hypothesize that attention carry semantic meanings when they are non-uniform and can be helpful for transfer learning. This has been the case in the NLP commu- nity (Rafford al., 2019). and is motivating many research efforts on understanding multi-head attention (Jain & Wallace 2019; Clark et al.) [2019] Voita et al., 2019). We attempt the idea of attention based sparsification sparsifying a graph by retaining edges where attention are higher, with the intuition being that the resulting graph preserves enough information. We find that not only such attention based sparsification is transferable (meaning, can work on unseen graphs), it also affords us train cheaper model without using attention to fit the downstream task. Finally, discuss several possible fruitful directions for further exploration, including theory, interpretability. and unsupervised learning.

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2 RELATED WORK
Visualize and understand attention Several works attempted to visualize the learned attention by coloring edges or nodes based on the attention magnitudes (Veličković al. 2018; Qiu et al. 2018); Thekumparampil et al. (2018) studied the averaged attention values between nodes with different or the same class labels. Shanthamallu et al. (2018) studied the attention GAT learned on two citation networks Cora and Citeseer with interquartile range metric and showed that they are near uniform. Knyazev et al. (2019) investigated the effectiveness of attentional pooling and find that attention only effective when it is close to optimal. Our work differs in that we propose general paradigm for analyzing graph attention, including how to characterize the overall attention statistics and how to measure the layer-wise and head-wiss wise differences of the learned attention.
Transfer attention In the computer vision community, transferring the attention maps using a sacher-student network improve the downstream tasks is well-studied technique (Zagoruyko & Komodakis 2017 Li et al. 2019). Our approach for transferring attention is different from these works in that signal. Yang et al. (2018) proposed to transfer the relational structure within the data, which is represented set of attention weights, boost the performance of other tasks. Our transferring strategy is different from their work because we reduce the sparsity of the affinity matrix by removing the entries with smaller attention weights. Thus, we can train cheap graph sparsifier accelerate the training and testing speed.
the trained graph attention network as a graph sparsifier instead of teaching
3 BACKGROUND
3.1 GRAPH NEURAL NETWORKS
Let G be an undirected graph with node set V and edge set E, where each node E V has a feature he R" In a wide class of GNNs (Kipf & Welling 2017 Hamilton al. 2017 Veličković et al. 2018), the basic feature update function for node E at the 1-th GNN layer takes the form
1+1 t h1+1 = Σ
JEN(
I+1 IR is the where is activation function, Nit) is set containing and its neighbors, o attention weight edge (j. i) in updating the feature of node i, W E R xx is the projection matrix, and M1 4+1 are corresponding node features after the 1-th and the I + 1-th layer. With sparse implementation, it has time complexity of O(V|n4417 + [8]m)11)-
Graph Convolutional Network (GCN) (Kipf & Welling 2017) and the mean variant of Graph-
SAGE (Hamilton et al. 2017) static attention
and [. which separately
VW((i VINTz)
refer to as GCN and uniform attention.
GAT Veličković et al. 2018) uses parameterized subnetwork to output the attention weights os,s8. Rather than using single attention head as Eqn. equation 3.1 GAT aggregates the outputs of multiple heads:
exp (score k))
111. Oi
h
111, =a
1+1 & witt1,t ais
EvENTO exp score (h4 (1, h. = 1
(EN())
(Aggregate'*+ ++1 (1111, . (41KK4
where k is the index of the attention head and K++1 is the number of attention heads in the 1 + 1-th layer. Aggregate" aggregates all head results the I + 1-th layer and we follow the approach of Veličković et al. (2018) to use concatenation for intermediate layers and average for the final layer.
Attention variants As in Luong et al. 2015). there are multiple ways to calculate the attention scores. In this paper, focus the following three types of attention, namely concat, dot product,
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Table 1: Dataset task and learning setting
Cora
Citeseer
Pubmed
PPI
CEP
HIV
Node Classification
Task
Graph Prediction
/
/
Transductive Learning
/
Setting
Inductive Learning
and general:
LReLU(a [Wh;]] Wh;)) (concat), (Wh:)* Why (dot product), (Wh,)* BWh; (general)
GATs uses the concat attention. Zhang et al. (2018) and Ryu et al. (0018) separately explores dor product and general attention in GNNs. Also, (.Rel.(.))
the leaky ReL.U activation.
Graph-level prediction Based on the node representations generated by GNNs, we can also compute graph representation (Li et al., 2018) for graph-level prediction problems like graph classification and regression:
ha
Σ VEV
Sigmoid (g (K?)) ReLU (S (k!)),
where is the number of GNN layers, h the representation of node output by the last layer of GNN, g()) calculates the impact of node on the graph representation, and f(() is a linear projection.
3.2 TASKS AND DATASETS
We consider the tasks of node classification and graph-level prediction. For modeling, we treat all graphs as undirected with untyped nodes and edges. Self loops are added preserve information from previous node features. As in Veličković et al. (2018), we consider four datasets citation networks Cora, Citeseer (Sen al. 2008), Pubmed (Namata al. 2012) and PP/ (Zitnik & Leskoved 2017).
Additionally, we include two more datasets of molecular graphs for graph-level prediction.
The Harvard Clean Energy Project (CEP) (Hachmann et al. 201T) estimates the photovoltaic efficiency of organic molecules. We use subset of it pre-processed by Duvenaud al. (2015): Ryu et al. (2018). The HIV dataset was initially introduced by the Drug Therapeutics Program (DTP) AIDS Antiviral Screen HIV]for testing the ability of compounds to inhibit HIV replication. It was later included in the MoleculeNet benchmark (Wu et al. (0188) as a binary classification task. For both datasets, the node features are extracted based DeepChem (Ramsundar et al.) 2019) and RDKit (Landrum). which includes atom type, degree, and many other chemical properties.
Since GAT was partially motivated to work on unseen data, consider two learning settings: transductive learning and inductive learning. the transductive learning setting, the model can access the features of all nodes in the graph. However, only a fraction of the nodes are labeled in the training phase and the model asked to predict the missing labels. In the inductive learning setting, we have two mutually exclusive sets nodes separately for training and testing. The model trained only on the features and labels of the nodes in the training set and is asked to predict the labels of the nodes in the testing set. A summary of the tasks and learning settings can be found in Table We leave more detailed information like dataset statistics, training/testing split and features in Appendix A.
4 METHODOLOGY
The introduction of multi-head attention into multi-layer GNNs poses many interesting questions, investigate five in this paper. Q1: In the GAT model, all nodes have different attention distributions on their incoming edges. How should we characterize the overall statistics of these learned attention distributions? Q2: How do attention distributions differ across different heads and layers? Q3: How does the choice of dataset, attention variant, and learning setting affect the learned attention? Q4:
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Table 2: Discrepancy between static attention and learned attention by the first head in the first layer
Cora
Citeseer
Pubmed
PPI
CEP
HIV
uniform learned
0.0083
0.0020
0.0059
0.5442 0.5791
0.1754
0.2376
GCN vs learned
0,1118
0.0796
0.1999
0.1759
0.2258
Is the statistics of the learned attention related to the intrinsic properties of the graph? Q5: How to transfer attention for further usage?
To answer Q1, we propose multiple metrics for characterizing attention distributions. For Q2, we examine the metrics different layers and compare the change of them over layers. To answer Q3, we run experiments to see how varying the dataset, attention variant and the learning setting impacts the learned attention. Previous works (Kipf& Welling 2017 Hamilton et al.. 2017 Veličković et al. 2018) only perform transductive learning on the citation networks and inductive learning on PPI. To fill in the gap, we perform transductive learning on PPI and inductive learning on the citation networks (see data processing strategy in Appendix B.1). We show that learning tasks are largely irrelevant To answer Q4, we propose a new task called Meta Graph Classification which asks the model to distinguish the type of the graphs by the characteristics of the attention distributions. For Q5, we transfer attention for graph sparsification and examine whether we can preserve enough task-related information with significant number of edges removed.
5
CHARACTERIZING ATTENTION
5.1 ANALYZING ATTENTION METRICS
Experiment settings Our attention study is completely based on the GAT architecture except that we try different types of attention mentioned in Section 3.1 We follow the experiment settings of the original authors whenever possible and perform a hyperparameter search otherwise. To make a fair comparison between attention variants, we use the same hidden size for each layer output across attention variants. The detailed settings can be found in Appendix B.2. Unless explicitly mentioned, we perform 100 random runs for Cora, Citeseer and Pubmed and 10 random runs for PPI, CEP and HIV. The test performance is mostly consistent across attention variants and is comparable to the original reported numbers, which we report Appendix B.4.
Learned attention v.s. static attention Attention-free GNNs employ static weights in updating node features. The first question then how does learned attention differ from static weights? the learned attention are almost the same the static ones, then there is point in performing costly attention computation and we do not need to proceed with the analysis. For any static attention (GCN uniform), quantify the discrepancy between it and learned attention for node i with EjeN(() leammed auuaiic J. With a range of [0, 1), the larger the value, the larger the discrepancy is. We average this value over all nodes in graphs. Taable[ssooe the discrepancy between the static and the learned attention for the first head with the 'concat" variant. Surprisingly, the discrepancy against the uniform attention is very small for the citation networks, suggesting that each node attends similarly to different neighbors.
Head-wise wiie and layer-wise differences Figure 2 visualizes the attention of node over its incoming edges in Cora and PPI, based on three heads in the last layer. We can find that different heads behave distinctively in the PP/ case and they are all uniform in the Cora case. We summarize the change of attention over heads and layers with several metrics. To quantify the variance between two head-wise distributions, compute the averaged L1 norm of the difference between the mean distribution over all heads and the learned distribution for each head.
K
K ΣΣ
mean
J.j. N()), Head-wise Variance = 2K V
a
mean
K
1=1 NEV JEV()))
To probe the concentration of attention, compute the maximum pairwise difference of them within one-hop neighborhoods max,).jeeWid Jas.ji a.,jzl To verify whether attention are on self loops when they are concentrated and GNNs degenerate MLPs, monitor the self-loop attention values.
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Concat
Dot product
General
Cora
3,500
0.50
0.50
Pubmed
0504
0.00
PPI
0.50
030
0.30
0:29
0.28
0.29
laper
layer
laycr
Tayer
kayer
CEP
0.50
0.0
0.50
0:2
0.35
0.25
Iayer
laper
Inyer
leper1
layer
Inger
Neyer
layer2
1aper3
max difference
Figure 1: Comparison of the learned attention across layers, datasets and attention variants
Head
Head 2
Head 3
Concat
Doe product
General
Cora
Figure 2:
Visualization of the learned attention of one node. Nodes are colored by labels and edges are colored by attention magnitude.
Figure 3: Attention learned by training on Cora and PPI with inductive and transductive settings respectively.
We average the metrics over all nodes and heads for a layer in each run, and compute the mean and standard deviation of the averaged metrics in all runs.
Varying settings and attention variants We leverage the defined metrics compare the learned attention by different attention variants. Figure [ visualizes the head-wise and layer-wise metrics for Cora, Pubmed, PPI and CEP. Independent of the attention variant, the learned attention change little across layers for Cora and Pubmed while the increasing max pairwise difference indicates that they get more concentrated with deeper layers for PPI and CEP. Besides, the attention does not get increasingly more concentrated on self loops while getting sharper over layers. We also experiment with different training settings on these graphs. Figure shows the learned attention when training inductively on Cora while training transductively on PPI. We observe the similar phenomenon of the learned attention, which eliminates the effect training settings.
5.2 META GRAPH CLASSIFICATION
Previous experiments suggest that the attention learned are highly traph-dependent and their charac- teristics can be predicted with proper knowledge of graph semantics. To verify it, attempt infer the graph types based on the attention learned. Specifically, we perform graph classification with attention-based features.

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Table 3: Graph Classification Accuracy
Concat
General
Dot product
All Layers
94.1 = 0.5%
95.6 = 0.7%
95.5 + 0.4%
First Layer
81.3 = 1.1%
88.4 = 0.8%
91.3 = 0.6%
Second Layer
83.5 + 0.6%
89.7 + 0.6%
85.3 + 0.5%
All layers
Layer
Layer
pubmed
citeseer
Figure 4: I-SNE visualization of 'concat' attention based features. From left right, the features are from all layers, the first, and the second layer, respectively. See Appendix B.5 for more results.
Synthetic dataset We construct a synthetic dataset for graph classification with two steps. First, we collect 480 samples/subgraphs of 20 to 30 nodes from each dataset. For HIV and CEP, we choose 480 graphs and construct balanced subset. For the rest four datasets, we sample 480 graphs for each using random walk as the case of inductive learning on citation networks. Second, we separately train 2-layer GAT for each collected dataset and compute the attention metrics for each layer. The mean and standard deviation of the metrics are then used as the graph features. We leave more details in Appendix B.5.
train logistic regression classifier for graph classification, where 20% the graphs are used for training and the rest used for testing. We have separately experimented with attention metrics from all layers, the first layer, and the second layer. The classification performance is reported in Table 3 with all experiments repeated for 10 times.
In the experiments, we find that more than 80% of the incorrect classifications happen within citation subgraphs. This is also verified by t-SNE (van der Maaten & Hinton| 2008 Pedregosa al.) 2011) visualization of the attention metrics in Figure|4 We can see that the attention metrics of citation networks tend to be indistinguishable while the attention metrics of other datasets better separated and clustered.
ATTENTION-BASER GRAPH SPARSIFICATION
6.1 PPI SPARSIFICATION FOR GAT PREDICTION
We consider two intuitive heuristics for attention based sparsification: 1) local top-k sparsification selects up to & incoming edges for each node with highest attention values; 2) global threshold sparsification selects edges whose attention value exceeds pre-specified threshold over the entire graph. Across all GNN layers, perform attention- baased sparsification for each attention head to get a subset edges. We then take the union of the edge subsets to get the edge set for the sparsified graph. All self loops selected to preserve the information of original node features.
Inspired by recent research on sampling based training of GNNs (Hamilton et al. 2017 Guag et al. 2018), consider two bascline random sparsification for comparison: 1) uniform neighbor sparsification uniformly samples at most & incoming edges for each node without replacement and compare against local top-k sparsification; 2) uniform graph-wise sparsification uniformly

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1.0
0.9
test scare
test 12 score
0.7
6.4
066
orginaa result neighbor
05
original result global
0.5
graph wise
0.5 edges
edges
Figure 5: Comparison of attention based sparsification against baseline random sparsification. The result of training unsparsified graphs included for reference. For local top-k sparsifi- cation we consider I k < 8. For global threshold sparsification, we consider threshold in (0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2, 0.1, 0.05, 0.03, 0.01, 0.005).
GAT Sparsifier Sg
GNN Predictor Ps
Figure 6: Sparsify graphs with GAT and predict sparsified graphs with another GNN
samples a proportion of edges over the entire graph(s) without replacement and we compare it against threshold sparsification.
We first experiment attention tiaeee graph sparsification with PPI, which tends to have sharp attention- based on our previous study and relatively dense (56944 nodes and 1644208 edges with self loops and bi-directional edges added). Despite similar sharp attention, molecules are not good candidates for the experiment as they are already sparse and on average each atom only has less than three neighbors. As illustrated in figure [6] after training GAT on PPI, perform cicention-bbsee sparsification and re-train a GAT on the sparsified training and validation graphs.
Figure 5 compares the results of attention- based sparsification with concat variant against random sparsification. 1) With a similar degree of sparsification, the attention- sparsification consistently performs better than the baseline in terms of test metrie and its variance across runs; 2) We reach a test accuracy comparable to the original result with only 40% ~u 50% edges left in the training and validation graphs
6.2 SPARSIFICATION WITH GENTLE ATTENTION
What if the attention are neither uniform nor sharp? Our previous study shows that the attention in Pubmed are not completely uniform in the last GAT layer with dot product and general attention variants so we use it as a testbed for the study. cases where attention are not very sharp, threshold sparsification is not very useful and consider only top-k sparsification.
Table 4 compares top-k sparsification with dot product attention against uniform neighbor sparsifica- tion. With similar proportion of edges left in the training and validation graphs, the top-k sparsification consistently outperforms uniform neighbor sparsification and achieves a performance comparable to that of training on the graphs.
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Table 4: Attention-based sparsification against random sparsification Pubmed
Sparsification
% edges left
Test score
None
100%
78.20 + 0.70%
local top-k (k=1)
57.37 + 0.78% 71.44 + 0.74%
77.57 + 0.12% 78.33 + 0.29%
local top-k (k=2)
uniform neighbor
13%
74.40 + 0.36%
uniform neighbor
72.72%
75.47 + 1.03%
Table 5: local top-1 sparsificatiton with light GATs of different sizes followed by GraphSAGE prediction on PPI. For reference, GraphSAGE can reach a test score of 0.9802 on unsparsified graphs and test score of 0.5431 on graphs sparsified by baseline methods with about 30% edges left.
# hidden
GAT test score
# GAT parameters
% edges left
GraphSAGE test score
4
0.5208
27574
29.55%
0.9700
16
0.7718
107686
30.06%
0.9776
64
0.9704
520294
30.95%
0.9802
6.3
LIGIT GAT SPARSIFICATION WITH GRAPHSAGE PREDICTION
For practical usage of speeding up the computation, we do not want to train two large GATs from scratch. We propose to first quickly train light GAT with lot fewer parameters for graph sparsification and then train large attention-free GNN for prediction on the sparsified graphs.
We explore this idea by varying the hidden sizes of attention heads GAT and use GraphSAGE for prediction PPI. The GraphSAGB has 3 layers and each layer has hidden size of 512. We consider the mean aggregator for it with skip connection added. The results are summarized in Table 5 Our experiments show that a GraphSAGE model can achieve good performance with training sparsified graphs as long as we sparsify test graphs. Surprisingly. while smaller hidden size in GAT does harm its own prediction performance, has little effect capturing important edges for GraphS AGE prediction even with top-/ sparsification. We note that the number of parameters in GraphSage and the baseline GAT is 1. 1.2M and 3.6M, respectively, i.e. 3-fold reduction.
7 CONCLUSIONS AND DISCUSSIONS
In this work, we propose an analytical paradigm that can summarize the characteristics of multi-head attention learned over graphs and compare them against topology- -based static attention. This allows a deeper understanding of attention beyond simply comparing the performance of models and motivates further use of attention like transfer learning. In addition to the attention- sparsification we explored, we believe below several interesting directions that underexplored:
Theory. Many efforts have performed to theoretically understand and explain GNNs, particularly their connection to kernel methods and Weisfeiler-Lehmen tests (Morris et al. 2019 Xu et
2019 Maron al. 2019), but few of them have considered attention.
Interpretability. The use of attention can add interpretabiiltyy This is particularly valued in risk-sensitive scenarios like medicine. Several efforts have been made in the chemistry community (Ryu et al. 2018 Preuer et al.) 2019 Xiong et al. 2019).
Unsupervised learning. Unsupervised representation learning is an important approach when the training of a model is expensive and the labeled data scarce. This is partic- ularly the case for biological networks and molecular graphs due the need of wet-lab experiments. The NLP community has witnessed the success of unsupervised learning with attention (Radford et al. 2019) and we might expect the same for GNNs. /Wihhua Hul (2019) demonstrates the effectiveness training GNNs with unsupervised learning for chemistry and biology, but did not employ attention.
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A DATASETS
A.1 DATASET SUMMARY
Table and summarize the statistics about the raw graph datasets. When computing the number of edges and node degrees, have not considered self loops. Also when model directed graphs as undirected graphs, the number of edges will get doubled. For transductive learning, the number of edges is considered be the same for training/validationnesst as all edges may be involved in message passing.
Table 6: Statistics and properties for single graph datasets
Datasets
Properties
Cora
Citeseer
Pubmed
# node feats
1433
3703
500
# classes/labels
7
3
# graphs
1
# nodes
2708
3327
19717
# isolates
0
48
0
# edges
5278
4552
44324
# mean degree
1.95
1.37
2.25
# train nodes
140 (5.2%)
120 (3.6%)
60 (0.3%)
# train edges
5278
4552
44324
# val nodes
300 (11.1%)
500 (15.0%)
500 (2.5%)
# val edges
5278
4552
44324
# test nodes
1000 (36.9%)
1000 (30. (.0.1))
1000 (5.1%)
# test edges
5278
4552
44324
Table 7: Statistics and properties for multi-graph datasets
Datasets
Properties
PPI
CEP
HIV
# node feats
50
58
75
classes/label size
121 (binary multi-label)
I (regression)
1 (binary)
# graphs
24
29978
41913
# nodes
56944
829135
1069968
# isolates
256
0
2351
edges
793632
1000952
1151942
# average degree
13.9
1.2
1.1
# train graphs
20
17986 (60.0%)
33530 (80.0%)
train nodes
44906 (78.9%)
497311 (60.0%)
846823 (79.1 %)
# train edges
613184 (77.3%)
600365 (60.0%)
906666 (78.7 %)
# val graphs
2 6514 (11.4%)
5995 (20.0%)
4191 (10.0%)
# val nodes
165916 (20.0%)
117216 11.0%)
val edges
99460 (12.5%)
200269 (20.0%)
128751 (11.2%)
test graphs
2
5997 (20.0%)
4192 (10.0%)
# test nodes
5524 (9.7%)
165908 (20.0%)
105929 (9.9%)
# test edges
80988 (10.2%)
200318 (20.0%)
116525 (10.1%)
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A.2 ADDITIONAL DETAILS
Graph Construction For citation networks, the nodes correspond to documents and the edges correspond to citations between pairs of documents. For PPI, the nodes represent proteins and the edges represent physical interactions between them. For molecules, the nodes correspond to atoms and the edges correspond to chemical bonds.
Node featurization and labeling For citation networks, nodes have bag-of-words features and labels for the topic of documents. For PPI, the node features include positional gene sets, motif gene sets and immunological signatures and the node labels are gene ontology (Hamilton et al. 2017), collected from the Molecular Signatures Database (Subramanian et al.) 2005). For the CHP dataset, the node features consist of one hot encodings of atom type, node degree, the total numher of hydrogens attached to it, the number of implicit hydrogens attached to it, and its aromaticity indicator. For the HIV dataset, in addition to those features we also consider the formal charge of the atom, the number of radical electrons of the atom and the atom's 's hybridization.
Dataset splits For Cora, Citeseer, Pubmed, and PPI, we consider a deterministic dataset split for training, validation and test. For CEP, randomly split the dataset in each run, where approximately the proportion of graphs used for training, validation and test is separately 60%, 20% and 20%. The statistics of the CEP dataset in table is obtained in one random run. For the HIV dataset, we use the scaffold split (Wu et al. 2018 Li et al. 2017). which structurally separates molecules into training, validation and test subsets and poses a greater challenge for generalization.
Imbalanced dataset The HIV dataset is highly imbalanced, with only 1487 compounds are positive, constitute approximately 3.5% of the dataset.
B EXPERIMENT SETTINGS
B. VARYING LEARNING SETTING FOR CITATION NETWORKS AND PPI
Transductive Learning on PPI To perform transductive learning on PPI, we sample two mutually exclusive subsets of the nodes as the training set and validation set for each graph, leaving the rest as the test set. We experiment on two splitting settings. In the first setting, we sample about 5% nodes for training and 18% nodes for validation, similar to the splitting ratio of the transductive learning setting on Cora. In the second setting, we sample 79% nodes for training and 11% nodes for validation, similar to the case of inductive learning on PPI.
Inductive Learning on Citation Networks To perform inductive learning on citation networks, we first sample 120 graphs of 100 nodes for each dataset. We use random walk based sampling algorithm described in Algorithm [ which by the study of (Leskovec & Faloutsos] (2066) performs best in preserving the properties of static graphs. Separately, 60%, 20%, 20% of the graphs are used for training, validation and test.
B.2 HYPERPARAMETERS FOR ATTENTION STUDY
For the attention study, we consider the hyperparameters below:
Transductive learning on Cora and Citescer: 2-layer GAT with 8 heads in the first layer and head in the second layer, 8 hidden units for each head in the first layer, dropout of 0.6, no residual connection, a learning rate of 0.005, L2 regularization with coefficient 0.0005, cross entropy loss
Transductive learning Pubmed: 2-layer GAT with 8 heads in both layers, 8 hidden units for each head in the first layer, a dropout of 0.6, no residual connection, a learning rate of 0.01, I.2 regularization with coefficient 0.001, cross entropy loss
Inductive/Transe Transductive learning PPI: L2 regularization, no dropout, 3-layer GAT with residual connections added for the last two layers; a learning rate of 0,005 for concat attention and learning rate of 00.001 for other attention variants; the number of attention
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Algorithm 1 Random Walk Sampling
Require: G = (V, the original graph, g.size 100 the target subgraph size
1: step =0
2: start ~ UV))
Uniformly choose a starting node.
3: Vau {start) Esub {(start, start)}
4: src = start
5: while Vuubl < g size and slep 100 size do
6: step = slep +
7: back Bernoulli(015).
Return the starting point with probability 0.15.
8:
if' back then
9:
src start
10:
else
dst Unse(( E E})
12:
Vauh Voub Uddo)
Eaub Eab U((dsl, dol), (sre, dol), (dot, sre)}
14:
dst Return (Vsub, Exub)
15:
heads in the three layers is separately 8, 8, 6 for general attention and 4, 4, 6 for other attention variants; 128 hidden units for each head in the first two layers for general attention and 256 hidden units for each head in other attention variants use batch size 1 for general attention and a batch size 2 for other attention variants; binary cross entropy loss
batch size 24, 3-layer GAT with separately 4, 4, 6 attention heads, residual connection is added for the last two layers, hidden units per head for the first two layers, learning rate of 0.005, dropout of 0.6; L2 regularization with coefficient 0.001 for pubmed and 0.0005 for the rest two citation networks; cross entropy loss
Inductive learning on Cora, Citeseer, Pubmed:
CEP: batch size of 512, 3-layer GAT where each layer has heads and each head has 32 hidden units, dropout of 0.O, residual connection is added for the last two layers, learning rate of 0.001, no L2 regularization; smooth L1 loss
HIV: a batch size of 64, 2-layer GAT where each layer has heads and each head has 32 hidden units, dropout 0.0, residual connection is added for the last two layers, no L.2 regularization, an initial learning rate of 0.0005, a decay of learning rate by 0.99 after each epoch; weighted focal loss with - (wyy( - p)7 log + (1 u)p" log (1 - p)), where y = and negative samples/ positive samples
An early stop is performed if the validation score hasn't 't been improved for 100 epochs.
B.3 GRAPH-LEVEL PREDICTION
Based on node features updated with GNN, we can also perform graph-level prediction. First, graph representation can be obtained with:
hG = Σ Sigmoid (g (h!)) ReLU(f (hb)
NEV
where is the number of GNN layers, RT + R and f R + Bn are two linear layers with bias added. In all cases we consider ng 128. A graph-level prediction is then computed with 3-layer MLP where all hidden sizes are equal to ng and a ReLU activation is applied after each of the first linear layers.
With the formulation of score(h., hy) (h.) Bh,, the general attention variant requires lot more parameters than the other two attention variants with same number of hidden units, which can result in an out of memory error. As work around, use a larger number of heads for this variant with smaller hidden size per head that the final output size of layers does change.
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Table 8: GAT performance with three attention rypes
Datasets
Reference
Concat
General
Dot product
Cora
83.0 = 0.7%
83.0 0.7%
84.2 + 0.5%
84.0 + 0.5%
Citeseer
72.5 + 0.7%
72.5 0.7%
71.4 0.8%
Pubmed
79.0 + 0.3%
79.0 + 0.3%
78.6 0.0%
78.2 0.7%
PPI
0.973 + 0.00
0.973 + 0.00
0.982 0.00
0.975 = 0.00
PPI trans 5%
0.476 0.03
0.565 + 0.01
0.524 0.01
PPI trans 79%
+
0.936 0.01
0.947 0.000
Cora inductive
1.7%
88.1
8
Citeseer inductive
84.1
84.8 = 1.3%
Pubmed inductive
8
8.25
86.0 + 0.8%
CEP
0.66 + 0.12 (Ryu y et al. 0.776 (Li et al. 2017)
2018
0.43 + 0.02 0.02
0.39 + 0.02
0.44 + 0.02
HIV
0.746 + 0.02
0.760 + 0.01
0.758 = 0.02
B.4 TEST PERFORMANCE ACROSS ATTENTION VARIANTS
We evaluate test performance using different metrics for different datasets accuracy for Cora, Citeseer, Pubmed, micro-averaged F1 score for PPI, mean absolute error for CEP and for HIV. See table for a summary the prediction performance, where different attention variants mostly have similar performance. The reference numbers from Veličković al. (2018) unless stated otherwise. For Cora, Citeseer, Pubmed and PPI, we include the original results of GATs for reference. For the rest datasets, include the best performance of previous work for reference whenever applicable, but some models do not involve attention mechanism.
score
B.5 GRAPH CLASSIFICATION
Dataset construction 1) As PPI has multiple graphs, we sample the same number of subgraphs from each original graph. 2) For CEP dataset, we first sort the whole dataset based on the photovoltaic efficiency and split it into 96 buckets. We sample graphs from each bucket where 3 graphs are used for training, one graph is used for validation and one graph is used for test. 3) As the HIV dataset intrinsically imbalanced, we construct a balanced subset by sampling a same number of positive samples and negative samples. Also since we considering special dataset split, the training, validation and test subset is separately constructed from the training, validation and test set. 4) For Cora, Citeseer, Pubmed and PPI, we construct a whole dataset of subgraphs first and then perform random split to get the training, validation and test set with a splitting ratio of 60%:20%:00%
Graph feature extraction For graph feature extraction, we train 2-layer GAT with heads for each layer and 64 hidden units for each head. We use batch size of 16 and perform an early stop if the validation score no longer improves for epochs. We use the same loss functions for each dataset explained in B.2 We also perform hyperparameter search for learning rate, dropout, L2 regularization coefficient and whether to perform a residual connection. The selected hyperparameters
follows:
Cora: a dropout of 0.1, residual connection is added for the second layer, no L2 regular- ization, learning rate of 0.01 for concal attention and a learning rate of 0.005 for other attention variants
Citeseer: a dropout of 0.1, residual connection is added for the second layer, a learning rate of 0.01, no L2 regularization
Pubmed: a dropout of 0.1, residual connection is added for the second layer, no L2 regular- ization, a learning rate of 0.005 for general attention and a learning rate of 0.01 for other attention variants
PPI: no dropout, residual connection is added for the second layer, no L2 regularization, learning rate of 0.005 for concal attention and a learning rate of 0.01 for other attention variants
The original work only has har plot and we contacted the authors for the numbers.
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CEP: dropout, learning rate of 0.005; residual connection is added for the second layer only with general attention; L2 regularization with coefficient 0.001 is used except for concar attention
HIV: a residual connection is added for the second layer, no L.2 regularization is used; dropout of 0.6 is used only for concul attention; a learning rate of 0.005 used for dot product attention and a learning rate of 0.01 is used for the rest attention variants
t-SNE visualization of attention metrics In the text we included the t-SNE visualization of attention based features for concal attention only. Here include the results for all three variants for comparison in figure 7 and 9 the attention metries for citation subgraphs get blurred while those for the datasets are better separated and clustered.
Across all attention variants, we observe similar pattern that
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All layers
Layer
Layer
cora
puomed
ppi
Figure 7: t-SNE visualization of concat attention based features. From left to right, the features are separately from all layers, the first layer and the second layer.
dteseer
ppi
Figure 8: t-SNE visualization of dol product attention based features. From left to right, the features separately from all layers, the first layer and the second layer.
cora
pubmed
CEP
Gleswer
HV
Figure 9; t-SNE visualization of general attention based features. From left to right, the features separately from all layers, the first layer and the second layer.
17


Reviews:
[{"review_id": "rklO62vCYH", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper analyzes attention in graph neural networks. It makes two major claims:\n(1) Datasets have a strong influence on the effects of attention. The attention in citation networks are more uniform, but they behave differently in protein or molecule graphs. \n(2) With attention-based graph sparsification, it is possible to remove a large portion of edges while maintaining the performance.\n\nI have some concerns about this paper: (1) the analysis lacks theoretical insights and does not seem to be very useful in practice; (2) the proposed method for graph sparsification lacks novelty and the experiments are not thorough to validate its usefulness; (3) the writing of this paper is messy, missing many details.\n\nIn the analysis part (section 5), the choices of probing metrics seem arbitrary and lack theoretical insights. The authors used the L1 norm, but it seems not appropriate for the tasks here, e.g. KL divergence is preferred to measure the distributional discrepancy, entropy for concentration etc. Many important details are missing or not clear, for example, in Table 2, which head/layer is used for computing the attention, and what does “GCN vs learned” mean? The maximum pairwise difference is not clearly defined. The meta graph classification (section 5.2) only considers a synthetic dataset. Overall, I feel the analysis didn’t present too many interesting observations, and I cannot see too much potential value in applications (even for the graph sparsification task in this paper, its correlation with the analysis is quite weak).\n\nIn section 6, it explores whether it is possible to remove part of the edges from the graph while maintaining the classification performance. It is an interesting task, but the method proposed in this paper is not realistic and lacks novelty. In 6.1 and 6.2, it needs to train a GAT first to get the attention scores, then remove edges according to attention scores and train another GAT. In this way, it doesn’t reduce the computational requirement, as it still trains a full model to get the attention. Only in 6.3 it presents a realistic setting, where the attention scores are derived from a small GAT, and train another GNN on the sparsified graph. But the paper didn’t explain why it is possible to get reliable attention scores with a small GAT, and the experiment is only on one dataset. Does it apply to other datasets (citation network, molecules) and settings (transductive, inductive)? So far the experiments are not enough to be considered as a valid contribution.", "extra_scores": {"experience_assessment": "I have published one or two papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "N/A"}, "extra_text": {}}, {"review_id": "HkeCr9okcS", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper carries out several kinds of analysis on the GAT networks of Velickovic (2018), which augment GNN updates with multihead self attention. Three standard attention types are compared, on several different datasets, and differences between uniform attention and learned attention are reported. An experiment is carried out where low-attention edges are pruned.\n\nWhile understanding the value of attention is important, this paper leaves many questions open. First, since the graphs studied in this paper are, if not generally sparse to begin with at least they only include connections that are meaningful, the sparsification experiment is a bit hard to understand. One particular extension would improve things: adding random edges (can the model learn to prune them out?), but learning sparse attention (see e.g., Maruf et al., 2019) rather than thresholding seems to be a reasonable point of comparison.\n\nOverall this paper would be more valuable if a clear and concise recommendation could be given regarding how to use or understand attention; but the lack of a consistent pattern of results makes any obvious narrative hard to support. I would encourage the authors to continue this line of work so that it can be used to provided guidance to those who would like to make more effective use of GNNs.", "extra_scores": {"experience_assessment": "I have published one or two papers in this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "N/A"}, "extra_text": {}}, {"review_id": "HJlML8uhtr", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "This paper presents an empirical study of the attention mechanism in the graph attention networks (GAT).  The study reveals that the attention patterns largely depend on the dataset, on some datasets they are sharp, but on others the attention patterns are almost uniform and not so different from the uniform aggregation weights in GNNs that does not have attention.  The authors further tried to utilize these findings and attempted to do attention-based graph sparsification, and showed that they can get a similar level of performance with only a fraction of the edges in the original graph if they do the sparsification based on the attention weights.\n\nGiven the popularity of the GAT model in the graph neural networks literature and the effectiveness of the attention mechanism in many deep learning architectures, the empirical study presented in this paper focused on attention is valuable.  The experiments are clearly motivated and executed, which I appreciate.\n\nAs this is an empirical paper, one (maybe) problem with it is that the findings presented aren’t that surprising in hindsight - of course the attention patterns should be data dependent, and doing attention-based graph sparsification seems like an obvious thing that should work.  The results on dataset-dependent attention patterns may have told us more about the datasets rather than the GAT model.\n\nThere are a few presentation issues that need clarification:\n- sec 5.1: it is not clear from the text what \\alpha^{static} is.  As mentioned earlier there are multiple possible static attention weights (GCN vs GraphSAGE).\n- sec 5.1: I found it strange to normalize the discrepancy score by 2|V|.  The number of terms in the sum should be 2|E| where |E| is the number of edges as each edge is counted twice.  Normalizing by 2|V| does not guarantee the score is in [0, 1] as claimed in the paper.\n- sec 5.1: “Besides, these attention do not get concentrated on self loops based on relatively stable values.” -- I don’t see why having stable attention values can show there’s no concentration on self-loops.\n- Figure 5 left: looks like the curves can’t reach the right end, this means 1 <= k <= 8 is probably not a good range.\n- Table 5 is a bit confusing.  From the text my understanding is that a GAT is trained first, and then do sparsification, and then train another GraphSAGE on the sparsified graph to do prediction.  It’s not immediately clear why the GAT performance in Table 5 is so bad, while the GraphSAGE performance is just way better.  After reading this a few times I realized a much smaller GAT is used (with much smaller hidden size) while the GraphSAGE model is always using a large hidden size.  I think this part needs some improvement.\n\nOverall I liked the empirical study and think the community can benefit from this paper.", "extra_scores": {"experience_assessment": "I have published in this field for several years.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper thoroughly.", "review_assessment:_checking_correctness_of_experiments": "I carefully checked the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "N/A"}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper suggests that datasets have a strong influence on the effects of attention in graph neural networks and explores the possibility of transferring attention for graph sparsification, suggesting that attention-based sparsification retains enough information to obtain good performance while reducing computational and storage costs. 

Unfortunately I cannot recommend acceptance for this paper in its present form. Some concerns raised by the reviewers are: the analysis lacks theoretical insights and does not seem to be very useful in practice; the proposed method for graph sparsification lacks novelty; the experiments are not thorough to validate its usefulness. I encourage the authors to address these concerns in an eventual resubmission.

Decision: Reject