Graph Few-shot Learning with Task-specific Structures
Authors: Song Wang
Chen Chen
Jundong Li
Venue: NeurIPS 2022 Accept (2022)
Keywords: Graph Neural Networks
Few-shot Learning
Graph Mining

Abstract: Graph few-shot learning is of great importance among various graph learning tasks. Under the few-shot scenario, models are often required to conduct classification given limited labeled samples. Existing graph few-shot learning methods typically leverage Graph Neural Networks (GNNs) and perform classification across a series of meta-tasks. Nevertheless, these methods generally rely on the original graph (i.e., the graph that the meta-task is sampled from) to learn node representations. Consequently, the learned representations for the same nodes are identical in all meta-tasks. Since the class sets are different across meta-tasks, node representations should be task-specific to promote classification performance. Therefore, to adaptively learn node representations across meta-tasks, we propose a novel framework that learns a task-specific structure for each meta-task. To handle the variety of nodes across meta-tasks, we extract relevant nodes and learn task-specific structures based on node influence and mutual information. In this way, we can learn node representations with the task-specific structure tailored for each meta-task. We further conduct extensive experiments on five node classification datasets under both single- and multiple-graph settings to validate the superiority of our framework over the state-of-the-art baselines.

baselines. Our code is provided at Graph Few-shot Learning with
Task-specific Structures
Song Wang University of Virginia suSwv@virginia.co edu
Chen Chen University of Virginia zrh6du@virginia edu
Jundong Li
University of Virginia jundong@virginia. edu
Abstract
Graph few-shot learning is of great importance among various graph learning tasks. Under the few-shot scenario. models are often required to conduct classifi- cation given limited labeled samples. Existing graph few-shot learning methods typically leverage Graph Neural Networks (GNNs) and perform classification across a series of meta-tasks. Nevertheless, these methods generally rely on the original graph (i.e., the graph that the meta-task is sampled from) to learn node representations. Consequently, the graph structure used each meta-task is iden- tical. Since the class sets are different across meta-tasks, node representations should be learned task-specific manner to promote classification performance. Therefore, to adaptively learn node representations across meta-tasks, we propose a novel framework that learns a task-specific structure for each meta-task. To handle the variety of nodes across meta-taske specific structures based on node influence and mutual information. In this way, learn node representations with the task-speciff structure tailored for each meta-task. We further conduct extensive experiments on five node classification datasets under both single- and multiple-graph settings to validate the superior- ity of our framework over the http:///thtbbb.cooo//oWWWWWc
extract relevant nodes and learn task-
Introduction
Nowadays, graph-structured data is widely used in various real-world applications, such molec- ular property prediction [88. knowledge graph completion [4]. and recommender systems (45). More recently, Graph Neural Networks (GNNs) 43 44
32 have been proposed to learn node representations via information aggregation based on the given graph structure. Generally, these train models
graph with abundant la- beled samples [19]. However, in practice, it is often difficult to obtain sufficient labeled samples due the laborious labeling process [10]. Hence, there is a surge of research interests aiming performing graph learning with limited labeled samples as references, known as graph few-shot
methods adopt semi- superviee learning strategy
learning 48
Among various types of graph few-shot learning tasks, fasks, few-shot node classification is essential in real-world scenarios, including protein classification [3] and document categorization (0). To deal with the label deficiency issue in node classification, many recent works [6]
10 18 21 incorporate existing few-shot learning frameworks from other domains (33 27 into GNNs. Specifically, few-shot
classification during evaluation is conducted on a specific number of meta-test tasks. Each meta-test
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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task contains a small number of labeled nodes as references (i.e., support nodes) and several unlabeled nodes for classification (i.e., query nodes). To extract transferable knowledge from classes with abundant labeled nodes, the model is trained on series of meta-training tasks that are sampled from these disjoint classes but share similar structures with meta-test tasks. We refer to meta-training and meta-test tasks as meta-tasks. Note that few-shot node classification can be conducted on single graph (e.g., a citation network for author classification) or across multiple graphs (e.g.. a set of protein-protein interaction networks for protein property predictions). Here each meta-task is sampled from one single graph in both single-graph and multiple-graph settings, since each meta-test graph. Despite the success of recent studies on few-shot node classification, they mainly learn node representations from the original graph (i.e., the graph that the meta-task is sampled from). However, the original graph can be redundant and uninformative for specific meta-task as each meta-task only contains a small number of nodes. As a result, the learned node representations are not tailored for the meta-task (i.e., task-specific), which increases the difficulties of few-shot learning. Thus, instead of leveraging the same original graph for all meta-tasks, it is erucial to learn task-specific structure for each meta-task.
task is conducted
Intuitively, the task-specific structure should contain nodes in the meta-task along with other relevant nodes from the original graph. Moreover, the edge weights among these nodes should also be learned in a task-specific manner. Nevertheless, it remains a daunting problem to learn a task-specific structure for each meta-task task due to two challenges: (1) It is non-trive non-trivial to select relevant nodes for the task- speccfic structure. Particularly, this structure should contain nodes that are maximally relevant to the support nodes in the meta-task. Nevertheless, since each meta-task task consists of multiple support nodes, it is difficult to select nodes that are relevant to the entire support node set. (2) It is challenging to learn edge weights for the task-specific structure. The task-specific structure should maintain strong correlations for nodes in the same class, so that the learned node representations will be similar. Nonetheless, the support nodes in the same class could be distributed across the original graph, which increases the difficulty of enhancing such correlations for the task-specific structure learning.
To address these challenges, we propose novel Graph few-shot Learning framework w wITh Task- spEcific stRuctures GLITTER, which aims at effectively learning a spek-specific structure for each meta-task task in graph few-shot learning. Specifically. to reduce the irrelevant information from the original graph, we propose to select nodes via two strategies according to their overall node influence on support nodes in each meta-task. Moreover, we learn edge weights in the task-specifie structure based on node influence within classes and mutual information between query nodes and labels. With the learned task-specific structures, our framework can effectively learn node representations that tailored for each meta-task. In summary, the main contributions of our framework are as follows: (I) We selectively extract relevant nodes from the original graph and learn task-specific structure for each meta-task based on node influence and mutual information. (2) The proposed framework can handle graph few-shot learning under both single-graph and multiple- gultiple-graph settings. Differently. most existing works only focus on the single-gu -rapleggahhhhhe setting. (3) We conduct extensive experiments on five real-world datasets under aigle-graph and multiple-grape settings. The superior performance over the Bate-of-the--ae methods further validates the effectiveness of our framework.
2 Problem Formulation
Denote the set of input graphs as G = [GI,...,,Gyy (for the single-graph setting, | = 1), where M is the number of graphs. Here each graph can be represented as G = (V, X). where is the set of nodes, E is the set of edges, and X R|V[xd is a feature matrix with the i-th row vector (d- dimensional) representing the attribute of the i-th node. Under the prevalent meta-learning framework, the training process conducted on a series of meta-training tasks [T....., Tr where T is the number of meta- meta-training tasks. More specifically, T, = (St, 2.). where S; is the support set of Ti and consists of K labeled nodes for each of N classes (i.e.. SS,] NK) The corresponding label set of T; is Vr. where D4I = N. Y) is sampled from the whole training label set Verain. With S; as references, the model is required to classify nodes in the query set Q,, which contains Q unlabeled samples. Note that the actual labels of query nodes are from )i. After training, the model will be evaluated on a series of meta-test tasks, which follow a similar setting as meta-training tasks, except that the label set in each meta-test task is sampled from a distinct label set Vtest (i.e., Vtest n Verain 0). It is noteworthy that under the multiple-graph setting, meta-training and meta-test tasks can be sampled from different graphs, while each meta-task is sampled from one single graph.

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Scenario Multiple graphs
Scenario single graph
Node Influence
Optimize
Support Emb.
Classifier
Loss
Task-specific
Extracted Nodes
Structure
Optimize
O
O
GNN #
Meta-optimize
Classifier
Support
Query
Mutual Information
Loss
Relevant
Unlabeled
Optimize
Query Emb.
Figure I: The overall framework of GLITTER, We first extract relevant nodes based on two strategies: local sampling and common sampling. Then we learn the task-specific structure with the extracted nodes along with support and query nodes based on node influence and mutual information. The learned structure will be used to generate node representations with GNN. We further classify support nodes with a classifier, and the classification loss is used to optimize the GNN and the classifier. Finally, we meta-optimize the GNN and the classifier with the loss on query nodes.
3 Methodology
In this section, we introduce our framework that explores task-specific structures for different meta- tasks in graph few-shot learning. The detailed framework is illustrated in Figure []] We first elaborate on the process of selecting relevant nodes based on node influence construct the -pec-spec structure in each meta-task. Then we provide the detailed process of learning task-specific edge weights via maximizing node influence within classes and mutual information between query nodes and labels. Finally, we describe the meta-learning strategy used to optimize model parameters.
3.1 Selecting Nodes for Task-specific Structures
Given a meta-task T {S, 2). we first aim to extract relevant nodes that are helpful for T and construct a task-specific structure GT based on these nodes. In this way, we can reduce the impact of redundant information on the original graph and focus on meta-task T. Nevertheless, remains difficult to determine which nodes are useful for classification in T. The reason is that the support nodes in can be distributed across the original graph, which increases the difficulty of selecting nodes that are relevant to all these support nodes. Thus, we propose to leverage the concept of node influence select relevant nodes. Here we first define node influence based on 44 as follows:
Definition (Node Influence). The node influence from node U; to node Uj is defined as I(U;, Uy) |0h, /Oh,/Ohll []. where h; and h, are the output representations of Us and vj in GNN. respectively. /h is a Jacobian matrix, and the norm can be any specific subordinate norm.
According to Definition T large node influence denotes that the representation of node can be easily impacted by another node, thus rendering stronger correlations. Intuitively, we need to incorporate more nodes with large influence most crucial information that is useful for classification based on support nodes. To effectively select nodes with larger influence the support nodes, we consider important factors that affect node influence. The following theorem provides a universal pattern for node influence on support nodes:
the support nodes into Gy. In this way, GT can maintain the
Theorem 3.1. Consider the node influence from node Up the i-th class (i.e., C.) in a meta-task T. Denote the geometric mean of the node inftuence values to all support nodes in C₁ as Ic, (vk) =
V k [(VK, stg), where si,j is the j-th support node in C₁. Assume the node degrees randomly distributed with the mean value d. Then, E(log Ic, (vx)) 2 log d Σ SPD(VE, 8t.)/ K, where SPD(v Uk, 85.5) denotes the shortest path distance between Us and 85.j-
The proof is provided in Appendix A. Theorem 3.1 indicates that the lower bound of the node influence on a specific class is measured by its shortest path distances to all support nodes this class. Therefore, to effectively select nodes with large influence on a specific class, we can choose nodes with small average shortest path distances to support nodes of this class. Based on this theorem, propose two strategies, namely local sampling and common sampling, to select nodes for the
3

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task-specte speciki-eeiic structure Gr. In particular, we combine the selected nodes with support and query nodes in the meta-task (i.e., S and Q) to obtain the final node set V UVV.USUU Here Vi and Ve are the node sets extracted based on local sampling and common sampling, respectively. S and Q are the support set and the query set of T, respectively. Then we introduce the two strategies in detail.
Local Sampling. In this strategy, we extract the local neighbor nodes of support nodes within specific distance (i.e., neighborhood size). Intuitively, the neighbor nodes can maintain a small shortest path distance to specific support node. Therefore, by combining neighbor nodes of all support nodes in a class, we can obtain nodes with considerable node influence on this class without calculating the shortest path distances. Specifically, the extracted node set is denoted as
Vi = Ue(es, where (isss) = {uld(u, 0/) I h), and is the pre-defined neighborhood size.
Common Sampling. this strategy, we select nodes that maintain a small average distance to all nodes in the same class. In this way, the node influence on an entire class can be considered. Specifically, for each of N classes in T, we extract nodes with the smallest average distances to nodes in this support class. The overall extracted node set V. can be presented as follows:
V. = U argmin
Σ WOVIIN DEV'
Σ
where 8i,j is the j-th node of the i-th class in T. Here we extract C' nodes with the smallest sum of shortest path distances to nodes in each class. Then we aggregate these nodes into the final node set entire class. As result, the selected
Ve. In this way, can select nodes with large influence nodes will bear more crucial information for classifying a specific class. Note that since there are only N classes in T, the maximum size of V. is NC, i.c., [Pll S NC.
Edge Weight Functions. With the extracted node set VT, we intend to learn task-specific edge weights for Gr. Intuitively, although the original structural information is crucial for classification, can also be redundant for meta-task TT Therefore, we propose to construct the edges based on both node representations and the shortest path distance between two nodes. In this way, the model will learn to maintain and learn beneficial edges for T. Particularly, the edge weight starting from node 2; to node v; is denoted as a., = (a.,, + )/2, where aF,, and a²,, are learned by two functions that utilize node representations and structures as input, respectively.
Node representations as input.
og = exp (-118((.x5)) (i)))
6 (2Xx))
To (11(Wi))))l 1lo
(2)
where & is a non-linear activation function, and x, denotes the input feature vector of node Vie W1 E Rd, xd and W2 E Rd, xd are learnable parameters, where d, is the dimension size of W and W2xj. df is the edge weight between node v; and v learned from node representations. Such a design naturally satisfies that ai, E (0, 1]. Moreover, by introducing two weight matrices W1 and W2, the learned task task-specific structured will be directed graph to encode more information.
Structures as input.
di, Sigmoid (+b (SPD(ri, u;))):
where v is a learned function that outputs a scalar while utilizing the shortest path distance between os and vj (i.e., SPD(u;, 1))) on the original graph. In this way, we can preserve the structural information on the original graph by mapping the distance to scalar. For example, if so is learned as decreasing function regarding the input SPD (v,, Uy), the obtained lask-specific structure will result in stronger correlations among nodes that are close to each other the original graph.
3.2 Learning Task-specific Structures from Labeled Nodes
With the proposed functions for edge weights, we still need optimize these weights obtain the task-specific structure for T. In particular, we can leverage the label information inside labeled nodes (i.e., support nodes in each meta-task). Intuitively, the task-specific structure should ensure that the learned representations of nodes in the same class are similar, so that the classification of this class will be easier. According to Definition [] between nodes, which will increase the similarity between the learned representations. Therefore,
larger node influence represents stronger correlations

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we propose to enhance the node influence between support nodes in each class to optimize the task-specific structure. However, directly calculating the Jacobian matrix excessively large computational cost, especially when the representations are high-dimensional [34. Hence, instead of directly computing the node influence, we propose to enhance the node influence based on the following theorem:
can result in an
Theorem 3.2 (Node Influence within Classes), Consider the expectation of node inftuence from the j- th node v to all nodes its same class C G, where Ui E C. Assume that |C| K 2. The overall node influence is E (En,eCitey) (V; 1)) vz)) =EnEertos n- k- -h, 11- Emeri n-1 h
h he/l|b,l?, where n is the number of nodes in G.
The proof is deferred to Appendix B. Theorem 3.2 states that maximizing the node influence within one class is equivalent to maximizing the similarity within classes while minimizing similarity between classes. Specifically, the similarity is measured by the dot product between representations of v/ and other nodes. In this way, we can effectively optimize the edge weights by increasing the node influence within each class without significant computational cost. The overall loss in meta-task T can be formulated as follows:
CN
1
Σ
Σ
K-2 | hJ
h//
E NEVIC
Prl- h3 h h
K -
N
1=1 eyec,
(4)
where C₁ denotes the node set of the I-th support class in meta-task T, and |G| K.
3.3 Learning Task-specific Structures from Unlabeled Nodes
Mutual Information. Although we have leveraged the information in support nodes by enhancing node influence, the unlabeled nodes (i.e., query nodes) in GT remain unused. Thus, we further propose to utilize the query nodes to improve the learning of task-specific structures. Specifically, we leverage the concept of mutual information (MI) [12) by maximizing the MI between query nodes and their potential labels. Formally, the MI can be represented as follows:
LM = -((qq Ya)= = -(Y2) + H(YolXo)
1
(5)
Σ
Dj loggs
Pij log
O
iE
where :=
:= Ply: = j|x;) and p. Eigg Q). Q is the query node set in meta-task T. Here Xo and Yo are the features and labels of query nodes, respectively. In particular, by maximizing the mutual information between query nodes and their labels, the learned task-specific structure can effectively leverage the information from the query set.
Transformed Markoy Chain. Although the strategy of maximizing MI between Xę and Yę has proven to be effective in few-shot learning. it remains non-trivial utilize such strategy for learning task-specific structures. Specifically, the crucial part to model the dependency of classification probabilities (i.e., P.j) on the learned task-specific structure. In this way. information to help optimize the task-specific structure. To estimate the classification probability, we propose to utilize node influence inferred from the task-specific structure in each meta-task. Intuitively, we can assume that the higher node influence between two nodes can represent a larger probability that they share the same class. As result, by estimating the node inffuence between a query node and support node, we can optimize the task-specific structure based on Eq. (5). However, it still remains challenging obtain an exact value of the node influence due to the potential computation cost of calculating the Jacobian matrix. Therefore, we propose to estimate node influence via the absorbing probability in a Markov chain (3.. Particularly, we create a Markov chain with absorbing states, where each state corresponds to node on GT. Here the support nodes in G- will be transformed into the absorbing states. Denoting the current adjacency matrix for GT as A, we obtain the transition matrix of the Markov chain from A as follows: (1) The transition probability from a support node to other nodes
leverage the mutual
(i.e., the absorbing state). (2) Each entry is
row-wise normalized to ensure that the sum of each row equals (i.e., the transition probabilities
starting from one node should sum up 1). Denoting the obtained transition probability matrix Á.
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Here we theoretically validate that the node influence between a query node and a support node can be effectively and efficiently estimated by the absorbing probability.
Theorem 3.3 (Absorbing Probabilities and Node Influence). Consider the Markov chain with transition matrix A derived from a graph G. Denote the probability of being absorbed in the j-th state (absorbing state) when starting from the i-th state as bt.j. Then, I [1,,, v2) bi.y where [(V,, Uz) is the node influence from the i-th node vi to the j-th node 1; on G.
The proof is provided in Appendix C. Theorem 3.3 provides that with the learned task task-specific structure in meta-task T, the node influence of a support node on a query node equals the absorbing probability starting from the query node to the support node in the Markov chain. In consequence, can leverage the property of Markov chains to provide an efficient approximation for node influence.
Theorem 3.4 (Approximation of Absorbing Probabilities). Denote states in the Markov chain, i.e., t = |VT| (S) where [Vr] is the node set in GT. Denote Q E Rtxt as the transition probability matrix for non-absorbing states in the Markov chain. Estimating b with by.j Ei-= Ak. Em=o( (Q*) i,k' then the absolute error is upper bounded as [bc., ba, - C/tem1(tt 1), where controls the upper bound, and C' is a constant.
the number of itn-absorbing
The proof is provided in Appendix D. Theorem 3.4 indicates that the estimation error of node influence is upper bounded by C/ 1). which is controlled by m. Therefore, we can adjust the value of m fit in different application scenarios. With Theorem 3.3 and Theorem 3.4 we can estimate the value of the classification probability as follows:
P(u = j|(x) = Σ A
[(")) h==0
(6)
As a result, we can optimize the objective in Eq. (5) according to the estimated node influence. In this way, we can adjust edge weights to reach state that tailored for this meta-task T and beneficial for the following classification of query nodes.
3.4 Meta Optimization
To effectively accumulate learned meta-knowledge across variety of meta-tasks, we adopt the prevalent meta-optimization strategy based on MAML [4]. Generally, MAML aims at learning the unique update scheme based each meta-task for fast adaptation novel meta-tasks. Specifically, for each tasa- we first perform several update steps to learn task-specific structures based on
05 1) a C." where Og denotes the parameters used to learn (()
the two structure losses: S
task-specific structures, and C L N + M- During each update step, we leverage a GNN to learn node representations from the current task task- speeifii structure, followed by an MLP layer to classify a V ((1)
(4) support where Suupport = EKES E3 Uk.j log Pk.. is the cross-entropy loss, and a is the base learning rate. Here Yk., E {0, 1} and yk.r = if the k-th node in S (i.e., the support set) belongs to the j-th support class in meta-task T: otherwise yk.j 0. Pk. (5)
the support nodes: o( = _ (
c(i)
the j-the entry of the class distribution vector P
(i) calculated by P
(8) Softmax
)).
(1) where h is extracted from the learned node representations H GNN(A (0), X). Here A (+) is the edge weights during the i-th step. 0G denotes the parameters of the GNN and the MLP classifier. After repeating these steps for 1) times, the loss on the query set will be used for the meta-update:
GG = 0 - 31 V ((n)
and 0g = 0in)) V2 c(?). where 31 and 82 are the meta-learning rates, and Lquery is the cross-entropy loss calculated on query nodes. For the meta-test tasks, will use the final uddated parameters to evaluate the
Experiments
Experimental settings. The training process of our framework is conducted on series of meta-training tasks. After training, the model will be evaluated on a specific number of meta- test tasks. Here we introduce four few-shot node classification settings for the experiments. (1) Shared graphs with disjoint labels. Under this setting, the meta-training and meta-test tasks are sam- pled from the same graph set. The labels in meta-test tasks are disjoint from those in meta-training

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tasks. (2) Disjoint graphs with shared labels. Under this setting, the meta meta-training training tasks and meta-test tasks are from disjoint graph sets. In other words, the graphs meta-test tasks will be new graphs unseen during training. The labels are shared during training and test, which slightly decreases the difficulty. (3) Disjoint graphs with disjoint labels. Under this setting. both the graph set and the label set of meta-test tasks are disjoint from those in meta-training tasks. That being said, the model is re- quired to handle new labels on graphs unseen during training, which renders the highest difficulty. (4) Single graph with disjoint labels. This setting is widely adopted in existing graph few-shot learning methods 36] 48. Under this setting, all nodes will be sampled from the same graph. Note that setting (1) and setting (2) are proposed in G-Meta [8], while setting (3) is new setting introduced in this paper. Detailed experimental and parameter settings are provided in Appendix E.
Datasets. To evaluate the performance of GLITTER on the few-shot node classification task, we conduct experiments on five prevalent real-world graph datasets, two for the multiple-grape setting and three for the single-graph setting. The detailed statistics are provided in Table T, For the multiple- graph setting, we use the following two datasets: (1) Tissue-PPI [49] consists 24
interaction (PPI) networks obtained from different types of tissues. Specifically, node features are gene signatures, and labels are gene ontology functions. This dataset consists of ten binary classification tasks. (2) Fold-PPI [8] contains of 144 tissue PPI networks, where the node labels are protein structures in SCOP database M. Moreover, the node features are conjoint triad protein descriptors [26]. There are 29 classes in this dataset. For the single-gras graph setting, leverage three datasets: (1) DBLP 30) is a citation network with nodes and edges representing papers and citation relations, respectively. The node features are based the paper abstracts, and the labels are determined by the paper venues. This dataset consists of 137 classes. (2) Cora-full (2) is also a citation network which contains 70 classes. (3) ogbn-arxiv [17] is citation network that consists of 20 classes and significantly more nodes than other datasets.
Table 1: Statistics of five node classification datasets.
Dataset
Graphs
# Nodes
# Edges
# Features
# Class
Tissue-PPI
24
51,194
1,350,412
50
10
Fold-PPI
144
274,606
3,666,563
512
29
DBLP
1
40,672
288,270
7,202
137
Cora-full
1
19,793
65,311
8,710
70
-gbn-arxiv
1
169,343
1,166,243
128
40
Baseline methods. (1) KNN 13 trains GNN based on all samples in training graphs to learn node representations. Then it classifies query samples according to the labels in the support set of each meta-task. (2) ProtoNet 27 learns the prototype of each class by averaging its node representations. Then it classifies query samples based on their distances to the prototypes. (3) MAML (14 adopts the meta-learning strategy and learns node representations via GNNs. (4) Meta-GNN combines MAML with Simple Graph Convolution (SGC) [40]. (5) G-Meta [8] learns node representations from extracted local subgraphs and further adopts MAML for meta-learning.
(6) GPN [10] learns prototypes based on node importance and classifies query nodes according to their distances to the learned prototypes. (7) RALE (2I) Iearns node dependencies based on node locations the graph. It is noteworthy that Meta-GNN, GPN, and RALE only work for the single-graph setting, while other baseline methods can be applied to both single-graph and multiple-gra graph settings.
4.1 Main Results
Table 2 and Table present the performance comparison of our framework GLITTER and all baselines on few-shot node classification under the multiple-graph setting and the single-graph setting, respectively. Specifically, for the multiple-graph task, we conduct experiments under three settings (i.e., shared graphs with disjoint labels, disjoint graphs with shared labels, and disjoint graphs with disjoint labels). For the single-rrphh task, we choose two different few-shot settings to obtain a more comprehensive comparison: 5-way 3-shot (i.e. N 5 and K 3) and 10-way 3-shot (i.e.. NN = 10 and K- = 3). For the evaluation metric, we utilize the average classification accuracy over 10 repetitions. From Table 2 and Table 3 we can obtain the follow observations: (1) Our framework GLITTER outperforms all other baselines in all multiple-graph and single-graph datasets. GLITTER also consistently achieves the best results under different settings, which validates the superiority of GLITTER on sew-shot node classification problems. (2) When the multiple-grayy
7

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setting is conducted with disjoint graphs, GLITTER has the least performance drop compared with other baselines. This because the learned task-specific structure tailored for each meta-task and is thus more capable in the multiple-grayy
setting, where the structures in meta-tasks across graphs can vary greatly. (3) The performance improvement of GLITTER over other baselines is relatively larger on Fold-PPI and DBLP. This is because these datasets consist of more classes than other datasets under the same setting. Since the classes in different meta-tasks are more diversely distributed, the task-specific structures learned by GLITTER will provide better performance in this situation. (4) When the value of N increases (i.e., larger class set in each meta-task), the performance of all baselines drops significantly due to the higher classification difficulty. Nevertheless, GLITTER consistently outperforms the other baselines. The reason is that a larger class set in a meta-task will require more specific representations for classification due to the variety of classes.
Table 2: The few-shot node classification results (accuracy in %) under the calltiple-grahl setting.
Dataset
Tissue-PPi
Fold-PPI
Graph Setting
Shared Disjoint
Disjoint Shared
Disjoint Disjoint
Shared
Disjoint Shared
Disjoint Disjoint
Label Setting
Disjoint
KNN
3..1
6..6
56.613322
40.0
50.212..
38.6
ProtoNet 27
59.0
61.8=2.7
55.7=3.2
37.4
46.5+:2.0
34.7:13.1
MAML 14
63.8+41.3
68.4=3.7
56.0#1.0
41.0#3.0
49.3422.8
42.5+44.4
G-Meta 18
64.6
69.9=4.1
58.1+22.4
46.1
54.7:+4.3
50.3+:.2
GLITTER
69.7+33.9
73.1=1.8
60.3+33.1
53.3+:3.6
61.34:2.9
54.7:1400
Table 3: The few-shot node classification results (accuracy in %) under the single-grapl, setting,
Dataset
DBLP
Cora-full
ogbn-arxiv
Setting
5-way 3-shot
10-way 3-shot
5-way 3-shot
10-way 3-shot
5-way 3-shot 10-way 3-shot
KNNN 13
63.4+3.3
54.1
56.942.1
42.4+2.7
45.4+3.3
35.7+2.4
ProtoNet 27
58.5+2.6
51.3+3.3
49,6+2.2
41.4+:1.9
42.9+3.0
34.6#33.
MAML 14
6..4+4.5
52.343.5
54,0+4.4
38.1+3.7
43.0#3.1
34.1#3.4
Meta-GNi 48
66.4112.8
58.0+2.6
59.742.7
42.5+3.6
48.8
35.0::4.8
G-Meta 18
73.1+3.4
57.5+5.0
63.7+4.0
47.6+4.9
50.8+4.8
38.2+4.9
GPN 10
75.4
64.0+3.8
62.143.4
47.3+4.4
55.343.1
36.3+4.6
RALE 21
74.9+3.2
65.1
63.414.0
45.21:3.6
54.713.9
38.415.1
GLITTER
79.3+3.1
69.5+5.2
66.2+4.4
52.0+2.6
58.5+4.7
44.2+3.8
4.2
Ablation Study
In this section, we perform series of ablations studies to evaluate the effectiveness of different components in our framework GLITTER. The results are presented in Figure 2 Specifically, we compare proposed framework GLITTER with three degenerate versions: (1) GLITTER without sampling relevant nodes (i.e., the task-specific structure only consists of support nodes and query nodes), denoted as GLITTER'S; (2) GLITTER without enhancing the node influence within classes, denoted as GLITTERIN; (3) GLITTER without maximizing the mutual information between query nodes and their potential labels, denoted GLITTERAM. From Figure 2 we can obtain several meaningful observations. First, our framework consistently outperforms all variants with certain components removed, which demonstrates that each module in GLITTER plays a crucial role in learning task-specific structures. Second, removing the relevant nodes deteriorates the performance under the cittippleegrphh setting with disjoint graphs more than that on shared graphs. The reason is meta- weeta-knowledee from training graphs can be uninformative for novel graph during evaluation. As a result, the relevant nodes on the novel graph become more crucial for learning -ask-specific structures. Third, the performance decreases differently by removing the node influence module or the mutual information module. Specifically, removing node influence generally causes a more significant drop when N increases. This is due to that larger class set in each meta-task requires more distinct node representations in different classes. Therefore, enhancing the node influence can provide better performance by enlarging correlations within classes.

---

80
70
Test Accuracy (%)
GLITTER GLITTER'S GLITTERNN
Test Test Accuracy (%)
GLITTER
GLITTER'S GLITTERNN
70-
60
GLITTERIM
GLITTERMM
60
L
50
50
40
shared graphs
disjoint graphs shared labels
disjoint graphs disjoint labels
shared graphs disjoint labels
disjoint graphs shared labels
disjoint graphs disjoint labels
disjoint labels
(a) Tissue-PPI
(b) Fold-PPI
85
75
GLITTER
GLITTER
Test Accuracy (%)
75
GLITTER'S
Test Accuracy (%)
65
Th
GLITTER'S
GLITTER'N GLITTER:M
GLITTERAN
GLITTER/M
55
U
45
45
35
DBLP
Cora-full
ogbn-arxiv
DBLP
Cora-full
ogbn-arxiv
(c) S-way
(d) 10-way
Figure 2: The ablation study of GLITTER on both single-graph and multiple-graph settings.
5
Related Work
Few-shot Learning on Graphs. Few-shot learning aims at effectively solving tasks with scarce labeled samples references. The general approach is learn meta-knowledge from tasks with abundant labeled samples and generalize it to new task with few labeled samples. Few-shot learning frameworks generally adopt the meta- meta-learnig strategy, which means both training and test processes will be conducted on a series of learning tasks. Existing few-shot learning methods can be broadly divided two categories: (1) Metric-based approaches, which aim learning a metric function
between the query set and the support set for classification R3 20
28
27 (2) Optimization-buaa
approaches, which focus on updating model parameters according to gradients calculated on
support set in each meta-task [22 25 14 233] Recently, many works propose to incorporate existing
few-shot learning frameworks into graph few-shot learning problems [21 36 42 39
For example, GPN [10] leverages the importance of nodes and combines Prototypical Networks 227 to solve few-shot node classification problem. G-Meta [8] extracts local subgraphs for nodes for fast adaptations across multiple graphs. TENT 337 proposes to mitigate the adverse impact of task variance with proposed lode-level and class-level adaptations.
Graph Neural Networks. Recently, many researchers have focused using Graph Neural Networks (GNNs) learn effective representation for nodes on graphs (5)
4 46 & IT. With GNNs, models will gradually learn comprehensive node representations based on the specific downstream tasks [16]. In particular, GNNs learn node representations from neighbor nodes via an information aggregation mechanism. As result, GNNs can incorporate both structural information and node feature information into the representation learning process. For example, Graph Convolutional Networks (GCNs) [19] leverage first-order approximation to learn node representations with graph convolution layers in a simplified manner. Graph Attention Networks (GATs) [32 aim to learn adjustable weights for neighboring nodes via the attention mechanism to aggregate features of neighboring nodes in an adaptive manner.
44 41
Conclusion
In this paper, we propose to learn task-specific structures for each meta-task in graph few-shot learning, so that the learned node representations will be tailored for each meta-task. To handle the

---

variety of nodes among meta-tasks, we propose a novel framework, GLITTER, to extract important nodes and learn task -asc-specific structures based on node influence and mutual information. As result, our framework can adaptively learn node representations to promote classification performance in each meta-task. We conduct extensive experiments on five node classification datasets under both the single-graph and multiple-grann settings, and the results validate the superiority of our framework over other state-of-the-an baselines.
Acknowledgement
This work is supported by the National Science Foundation under grants IIS-2006844, IIS-2144209, IIS-2223769, CNS-2154962, and BCS-2228534, the JP Morgan Chase Faculty Research Award, the Cisco Faculty Research Award, and Jefferson Lab subcontract JSA-22-D0311.
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12


Reviews:
[{"review_id": "hw-wD4Hj0o-", "reviewer": "Reviewer_AUxb", "summary": "This paper proposes a method to select relevant nodes and learn desirable edge weights for each few-shot node classification task, so that the input graph structure is tailored for each task individually. Specifically, the method has three stages: 1) selecting nodes based on proximity to the support set nodes; 2) optimizing edge weights function such that the resulting node influence between support nodes in the same class is maximized; 3) optimizing edge weights function such that the mutual information n between query nodes and all labels in the task is maximized. Over the produced graphs, the authors then use a meta-optimization method based on MAML to learn and perform the few-shot tasks. Experiment results over diverse settings demonstrate the effectiveness of the method over various baseline methods.", "questions": "- Is the training of edge weight function using the two objectives completed before the meta-optimization? Or how are they blended together?\n- It would be beneficial to confirm the performance gains with statistical tests due to the high variance.\n- What is computational overhead with this method over using MAML on the full graphs directly? ", "limitations": "The limitations are adequately addressed in the paper other than the ones mentioned above. ", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "# Strengths\n- The proposed method is novel and well-motivated from practical and theoretical perspectives. However, it is worth noting that similar ideas have been explored and somewhat standardized in the link prediction literature following SEAL (https://proceedings.neurips.cc/paper/2018/file/53f0d7c537d99b3824f0f99d62ea2428-Paper.pdf), where a specific subgraph is selected for predicting the existence of an edge even in the standard (non few-shot) setting. Yet to my knowledge, the most popular methods of this kind simply select an enclosing neighborhood of the two end nodes based on simple heuristics (e.g. intersection of k hop neighborhoods). Therefore, this work is still more novel and sophisticated in the specific methods.\n- The method can serve as a simple drop-in preprocessing step for various other methods, and it can potentially benefit tasks even outside few-shot setting or node classification setting. \n- The experiment is comprehensive and demonstrates empirical gains.\n- The paper is clearly written overall.\n\n# Weaknesses\n- The experiment results have significant variance, making it less clear how significant the results are. It would be beneficial to confirm the gains with statistical tests.\n- It seems computationally intensive to precompute and extract subgraphs for each task. \n ", "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": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "P0C--Yl561j", "reviewer": "Reviewer_PD5y", "summary": "The authors have proposed a new mechanism to learn **task-specific** structure for each meta-task under the problem setup of few-shot node representation learning. The intuition behind the idea is clear in that the node representations should be learned in compliance with the structure of the meta-task being considered. As the authors mention, its evident that the entire graph is not necessarily useful since it may contain a substantial amount of redundant information, especially since each meta-task usually has a small number of nodes (this may not be the case when the graph at hand is small, nevertheless, most of the real world are much larger compared to the size considered in the experiments). Furthermore, the authors suggest that the edge weights over the task-specific structure should also be learned in a task-specific manner. The proposed methodology seeks to answer two questions in particular - (i) how to select relevant nodes for the task specific structure and (ii) how to compute the edge weights for the task specific structure. To answer the first question, the authors derive an approximation based on the “influence” of one node embedding to another and give a lower bound to the expected value of this influence across the nodes in a class in terms of the “shortest path distance” (SPD) amongst the nodes. Based on the SPD criterion, the authors have proposed two simple strategies to sample the nodes - local and common sampling. To answer the second question, the authors compute the edge weights as a combination of two different terms obtained using - node representations and SPD between the two given nodes. To further assist the learning process of the task-specific structures, the authors propose 2 loss functions based on the - (i) label information of the support set nodes and (ii) mutual information between the query nodes and their estimated / potential labels. For the first one, a loss function that enforces the maximization of the similarity within classes along with the minimization of the similarity between classes, is proposed (eq 4 of the main paper). For the second one, the maximization of mutual information is suggested. The optimization of the model is performed via MAML. Experiments on multiple popular benchmarks and comparison against state of the art baselines demonstrates that the proposed approach indeed performs very well. Although, there are several concerns regarding the theoretical analysis, as elaborated below.\t\t\t\t", "questions": "1. The authors have explicitly focused on selecting nodes for learning task-specific structures, conditioned on the support set. While this is a perfectly sound method, why haven’t the authors considered the same for the query set nodes? Even if a single node is present in the query set, the same proposed mechanisms might help expand the subgraph around the query set node to tackle bottlenecks, particularly oversquashing, as well as densify the graph in certain cases (on average), to prevent under reaching from the nodes in the set $\\mathcal{V_T}$ \\ $\\mathcal{Q}$. Moreover, the first limitation mentioned by the authors in appendix (lines 177-180) can potentially be addressed via a similar mechanism. It will be interesting to hear the authors’ opinion about this point. \n2. What specific GNN architecture has been used to obtain node representations in the proposed model?\n3. Can the authors provide more details on how the hyperparameters are selected? Are these searched over some space or selected based on previous works?\n4. The improvements suggested in section F.3 (appendix), although simple, can be utilized in cases where the scalability is not an issue. Although the statement - “Nevertheless, such a strategy is not suitable for the multiple-graph setting, since learning GNNs across different graphs can lead to suboptimal performance” - lines 204-206 in appendix, isn’t compelling. Why do the authors mention that learning one, or infact multiple GNNs can be suboptimal? These GNNs can learn the original graph structures and provide useful auxiliary information.\n", "limitations": "I commend the authors for explicitly stating some of the main limitations of the approach. A question regarding the first limitation (lines 177-180) has also been raised above. I strongly suggest the authors to explicitly elaborate the GNN architecture that has been used in the experiments, as there could be potential concerns regarding the architecture used. I have **serious** concerns regarding the theoretical analysis, as mentioned above. Lastly, I don’t see any serious negative societal impacts of the work. \n", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "1. The empirical evidence of the efficacy of the proposed method is strong. All the results in tables 2 and 3 indicate that the model outperforms the baselines by substantial margins on both - multi graph and single graph settings.\n2. The result of the ablations in figure 2 also explicitly demonstrate the utility of each of the 3 main components of the proposed method. The explanations provided in lines 322-332 of the main paper are also inline with the demonstrated results. \n3. The authors haven’t provided sensitivity analyses for many important hyperparameters, such as - the value of $C$ in common sampling, value of $h$ in local sampling, value of $m$ in eq 6 etc. I believe such analyses are important to further interpret the proposed solution better.\n4. I believe that the authors should also provide some qualitative results along with the quantitative evidence. For eg - constructing t-SNE (or any other) plots showing the class wise separation over the embeddings of the penultimate layer can be a useful analysis of the learned embeddings. This can be done for the proposed method and the best performing baseline. Similarly, some analysis of the “average size of the task-specific graph structure” will be useful. Additionally, some plots for the “performance-runtime” concerning the proposed method and the best performing baseline will provide an interesting tradeoff analysis.\n5. Notational problems in eq 6 of the main paper - $j$ is the class label.\n6. Definition 1 (in the main paper) as well as the proof of Lemma A.1 (appendix) have multiple notational problems: (i) The layer indexing for $h_i$ and $h_j$ are missing. From the proof, it's evident that the authors have considered layer recursion upto the shortest path length between $i$ and $j$; (ii) $d$ and $\\bar{d}$ are the same. $\\bar{d}$ is not defined in the statement of Lemma A.1\n7. Eq 6 in the appendix hasn’t used Jensen's inequality properly in the first comparison. \n8. Multiple problems in proof of theorem 3.2, following the concerns above about the layer indexing and the entire proof sketch thereof. The assumption that the expected value of each $I_{ij} = \\frac{1}{n-1}$ since the sum of probabilities of all paths starting from $v_{i}$ equals 1, is extremely loose.\n9. It will be useful to compute the value of the constant $C$ in theorem 3.4, since it seems to follow the order of $t^{h}$ (computing its upper bound might be easier than the actual value)\n", "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": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "BxqWdmS65XK", "reviewer": "Reviewer_zmmr", "summary": "Considering graph few-shot learning, the paper proposes a method called GLITTER to learn node representations across different meta-tasks. \nIn particular, GLITTER selects nodes according to their overall node influence on support nodes, learns edge weights in the task-specific structure based on node influence within classes and mutual information between query nodes and labels. The authors believe that these make the learned structures tasks-specific and consequently improve the node representations. \n\n", "questions": "The key problem is that the authors fails to convince the readers why existing methods cannot learn task-specific structures. The authors should provide a clear definition for their believed task-specific structure. \n\nIndeed, in meta-learning, after the inner loop fine-tuning, the model and the node embeddings naturally become task-specific. In this sense, existing meta-learning works can learn task-specific structures. \nOn line 5-6, the authors write that \"these methods generally rely on the original graph (i.e., the graph that the meta-task is sampled from) to learn node representations\". On line 39, the authors write that \"Here each meta-task is often sampled from one single graph\". \nThen, why this problem exists in multiple-graph setting? If the problem is that one should adjust the provided graph adaptively, why not using GAT [1] or recent graph structure learning methods [2-5]? Any difference or relevance w.r.t them? \n\n[1] Graph attention networks, ICLR, 2018 \n\n[2] Few-shot learning with graph neural networks, ICLR, 2018\n\n[3] Learning to propagate labels: Transductive propagation network for few-shot learning, ICLR, 2018\n\n[4] Property-Aware Relation Networks for Few-Shot Molecular Property Prediction, NeurIPS, 2021\n\n[5] Deep graph structure learning for robust representations: A survey, 2021\n\n", "limitations": "The authors did not discuss any limitations nor potential negative societal impact of their work. ", "rating": 5, "confidence": 4, "soundness": 1, "presentation": 2, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper considers broader graph few-shot learning applications following GMeta. \nThe results show clear improvement. \n\nHowever, this paper is confusing due to unclear motivation and claim. \nPlease see my detailed comments in questions. \n", "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": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "6mB6KYwKxnl", "reviewer": "Reviewer_SEHi", "summary": "This paper proposes an improved MAML algorithm that tailors node representation to specific meta-tasks. The result is a significant performance improvement across meta-tasks.", "questions": "How does the node influence in the paper relate to the field of graph influence maximilation? It seems that the definition, stripping the GNNs from it, is quite similar to the one of Kempe et al., 2003.\n\nWhat is the computational complexity of the distance computations, during training and for a new task?", "limitations": "Limitations and societal impact are addressed in the paper.\n", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "[S] The problem is well-motivated and relevant to the research community. The proposed algorithm seems interesting and novel, while the approach intuitive.\n\n[S] The performance improvements observed are significant compared to the baselines.\n\n[W] The terminology is rather hard to follow in the beginning, especially before reading the introduction to the problem.\n\n(comment) The attached code is extremely counterintuitive and messy, even for research code. I had a hard time trying to clarify some computational aspects of the paper for myself.", "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": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept