Few-shot graph link prediction with domain adaptation

1 INTRODUCTION . Real-world link prediction problems often deal with graphs with nodes from various domains . For example , we can separate the items from a product purchase network into multiple domains such as books , electronics , luxury items . A financial transaction network can have data coming from various countries . Different domains may have different distributions , but more importantly , domains are often not balanced in most real-world datasets . For example , we may expect to see hundreds of thousands of nodes for books but only a few hundred for luxury items . There might also be billions of financial transactions in developed countries but much fewer in developing countries . We observe a similar phenomenon even in academic graph datasets . For example , in the ogbn-arxiv ( Hu et al . ( 2020 ) ) dataset , which includes pre-prints from 40 computer science subjects , papers from the most popular subject take up to 16.13 % of all the nodes . In contrast , the least represented subject only has 0.02 % of the data . Training graph models with imbalanced data without precaution significantly downgrade model performances , especially for domains with fewer samples . The training process is overwhelmed by the popular domains that have abundant data . These popular domains act as noise , hampering the model to fit for small domains effectively . Despite the prevalence of this problem , current research on this topic is not adequate ( Zhao et al . ( 2021 ) ; Shi et al . ( 2020 ) ) . To improve model performance for small domains , we propose to train domain agnostic embedding that we can use for downstream tasks . The intuition is that if the latent representation learned by the graph model does not contain any domain-specific information , and therefore is domain-agnostic , it becomes more robust to domains with fewer data and even novel domains . We propose to use adversarial learning and force the graph model to learn domain agnostic embeddings . Our idea of building domain-agnostic features across domains aligns with the few-shot learning ( FSL ) problem in computer vision ( CV ) . FSL ( Fei-Fei et al . ( 2006 ) ; Fink ( 2005 ) ) refers to the type of ML algorithm that works with limited labeled data . A common FSL technique is to learn domainagnostic features first ( with/without external data ) and then let the model converge fast on to domains with limited shots ( Long & Wang ( 2015 ) ) . It is relatively easier for CV tasks because stacked convolutional neural networks used in CV usually learn features in a general-to-specific sequence . In contrast , as stacking graph neural networks ( GNN ) layers only explore larger neighborhoods , the same method couldn ’ t be applied directly to graph data . In addition , the CV literature has explored the idea of using adversarial learning to build better domain agnostic features . Domain-Adversarial Neural Network ( DANN ) ( Ganin et al . ( 2016 ) ) first introduces the idea of using a discriminator to align and separate semantic probability distributions . Further research ( Motiian et al . ( 2017 ) ) shows that it helps reduce the number of samples needed for each domain/class . Essentially , our work follows this path but further improves it with the enclosing subgraph idea to achieve better performances for graph link predictions . We find that the method of extracting enclosing subgraphs , proposed in the SEAL model Zhang & Chen ( 2018 ) suits well for our setting because it not only boosts the number of trainable samples but also gives them different topological features . It is similar to what “ cropping ” does for images . CV Research has shown that data augmentation method , such as cropping ( Qi et al . ( 2018 ) ; Zhang et al . ( 2018b ) ) , introduces invariances for models to capture and is effective for FSL problems . We summarize our main contributions as follows : • We propose the idea of learning domain-agnostic graph embedding to improve the quality of link prediction by using adversarial learning and enclosing subgraphs . • We define the concepts of “ shots ” for graph data , making it significantly easier to train , describe , and analyze graphs with multiple imbalanced domains . • We use t-SNE plots to demonstrate the effectiveness of our method . It also allows practitioners to decide the expected gain from our proposed ideas . 2 RELATED WORKS . 2.1 LINK PREDICTION . Link prediction focuses on predicting future or missing links between a pair of nodes . The current state-of-the-arts link prediction method is SEAL ( Zhang & Chen ( 2018 ) ) . The core idea of SEAL is to extract enclosing subgraphs for a set of sampled positive and negative links . To be more specific , a positive link is a pair of nodes connected with an edge , and a negative link is a pair of nodes that are not connected . An enclosing subgraph is the union set of h-hop neighboring nodes of the pair of nodes . This data-preprocessing step lets SEAL transform the link prediction problem to a ( sub ) graph classification problem . It uses graph neural networks ( GNNs ) to generate the graph embedding , which feeds into a classifier and generates new predictions . Other than SEAL , Graph Auto-Encoder ( GAE , Kipf & Welling ( 2016 ) ) is an alternative approach . GAE works by generating node embedding for all nodes and then model the link likelihood based on the concatenated node embedding vectors for any node pairs . However , recent researches have shown this method heavily relies on the assumption that similar nodes have similar connections . In addition to these GNN-based methods , traditional heuristic methods , such as common neighbor ( CN ) and Adamic-Adar ( AA ) , provide a completely different alternative . Given a pair of nodes , these non-parametric methods describe the topological structure of the surrounding neighborhood with a score . The advantage is that these methods could be used directly without training , but it ’ s challenging to utilize node/edge features with these methods . 2.2 FEW-SHOT LEARNING AND DOMAIN ADAPTATION . Few-shot learning is a special type of machine learning problem , which aims at obtaining good learning performances when supervised evidence information is limited ( Wang et al . ( 2020 ) ) . Common FSL methods include data augmentation ( cropping , rotating , generative models , etc ) and metalearning ( Sun et al . ( 2019 ) ) . Domain Adaptation is a type of Transfer Learning ( TL ) technique . It focuses on transferring learned knowledge from one domain to another while maintaining good performances . The core idea of domain adaptation is to learn a domain invariant representation , which is usually done by maximizing mutual information ( Hjelm et al . ( 2019 ) ) or training in an adversarial network ( Goodfellow et al . ( 2014 ) ) . Some recent works focus on the problem of few-shot learning with domain adaptation . The work from Motiian et al . ( 2017 ) also uses adversarial networks as the solution , but it introduced a 4-class confusion matrix classifier as the discriminator . The work from Zhao et al . ( 2020 ) explicitly added source/target per-class separation before doing embedding feature learning with domain adaptation . 2.3 ADVERSARIAL LEARNING . To learn domain agnostic models , we need to ensure that the learned graph embedding does not contain domain-specific information . To achieve this , a commonly used technique is adversarial learning , in which two models are trained against each other and play a minimax game based on game theory . For example , in the popular Generative Adversarial Network ( GAN ) ( Goodfellow et al . ( 2014 ) ) , adversarial learning helps generators build more realistic images by having a discriminator that identifies whether an image is synthetic or real . There have been prior works done on using adversarial training on graph-related tasks . The work from Wang et al . ( 2018 ) uses GAN to improve various graph-related tasks , such as link prediction and node property prediction , by generating “ fake ” graphs . The work from Lei et al . ( 2019 ) also uses GAN to solve the challenging temporal link prediction problem for weighted dynamic networks . 3 METHODOLOGY . 3.1 PROBLEM DEFINITION . We define the few-shot link prediction problem with imbalanced domains as follows : in a given graph space S , we have access to graphs from m ( m ∈ N , m ≥ 2 ) number of domains . Different domains may have variations in their marginal distributions . Among them domains , m−1 ( k ∈ N+ ) domains have k number of small graphs , which we define as graph “ shots ” . The last domain only has 1 shot . Our objective is to learn a link prediction model that leverages the domain differences to improve the performance in the domain with 1 shot of data . 3.2 NOTATIONS . In terms of graph-related notations , each graph is denoted by G = ( V , E , d ) , where V is the set of vertices , E is the set of edges and d is the type of domain , which could be a one-hot encoded vector with one extra digit representing any unseen domain . In the adjacency matrix A of G , Au , v = 1 if edge ( u , v ) ∈ E , and otherwise Au , v = 0 . For any node u , we use N ( u ) to denote the set of neighboring nodes of u . In addition , we use superscript as a shorthand to denote the domain class . For example , for a graph from domain i , we would denote it as Gi . 3.3 DOMAINS . For our setting , we have a total of m domains and each domain has one or more graphs or subgraphs Dk = { Gk1 , ... } mk=1 , where Dk denotes the kth domain and graph Gk1 is a realization of random variable Gk . The effective of our method compared to baseline is based on the assumption that the distribution of the random variable G is different across different domains p ( Gk ) 6= p ( Gk′ ) if k′ 6= k . 3.4 LINK PREDICTION . Here is the general form of any link prediction models : given ( u , v , G ) , where u and v are a pair of nodes in graph G , the likelihood of having a link between u and v , yu , v , is modeled by function f : ( u , v , G ) → yu , v . This function could be trained with a union set of positive and negative samples . Positive samples are sampled from existing edges while negative samples are built with node pairs with u and v′ , v′ /∈ N ( u ) . Based on the ideas from the SEAL framework , the first step is to extract an h-hop enclosing subgraph around edge ( u , v ) . We denote this subgraph as Su , v and the process of generating the enclosing subgraph as s ( u , v , G ) = Su , v . Function f could be rewritten as f : Su , v → yu , v . In general , f is composed with two separated functions g and c and could be written as f = g ◦ c . Function g generates the graph embedding vector g : Su , v → hSu , v . Function c takes in the graph embedding and generates the prediction ˆyu , v . Here we have the supervised loss function . Lcls ( g , c ) = E [ ` ( c ( g ( Su , v ) ) , yu , v ) ] = − 1 N ∑ yu , v · log ( c ( g ( Su , v ) ) ) + ( 1− yu , v ) · log ( 1− c ( g ( Su , v ) ) ) , where E is the statistical expectation of any loss function ` . In this case , ` is the standard binary cross-entropy loss . The input of the discriminator D is a pair of graph embedding hi and hj generated by g. As stated above , here we use superscript to denote the domain information . The objective is to determine whether these two subgraphs are from the same domain . Since we know from which graph G these subgraphs are generated , and we know the domain information of each G , we can easily get the ground truth di , j . The loss function here could be binary cross-entropy as well . An alternative design is to let the discriminator predict the exact domain that the graph embedding is from . We select the current one because it is easier to apply it to novel domains . Ldis ( D ) = E [ ` ( D ( g ( Siu , v ) , g ( S j u′ , v′ ) ) , d i , j ) ] Given the two loss functions above , we could get the total loss for the input of a pair of subgraphs from domain i and j . Since our objective is to make sure the graph embedding does not contain domain-specific information , we would like to minimize the supervised training loss while maximizing the discriminator ’ s loss . Ltotal ( g , c ) = E [ ∑m i=1 ` ( c ( g ( S i u , v ) ) , y i u , v ) − α · ∑m i=1 ∑m j=1 ` ( D ( g ( S i u , v ) , g ( S j u′ , v′ ) ) , d i , j ) ] , where α is hyperparameter to adjust the weight of the discrimination term compared to the supervised loss .

Summary: This paper proposes a method for few-shot graph link prediction by using a domain discriminator. The motivation is to learn domain-invariant graph-level representations. The authors also introduce the concept of “shot” in the graph. The proposed method outperforms baselines on three different datasets.