Neighbor2Seq: Deep Learning on Massive Graphs by Transforming Neighbors to Sequences

1 INTRODUCTION . Graph neural networks ( GNNs ) have shown effectiveness in many fields with rich relational structures , such as citation networks ( Kipf & Welling , 2016 ; Veličković et al. , 2018 ) , social networks ( Hamilton et al. , 2017 ) , drug discovery ( Gilmer et al. , 2017 ; Stokes et al. , 2020 ) , physical systems ( Battaglia et al. , 2016 ) , and point clouds ( Wang et al. , 2019 ) . Most current GNNs follow a message passing scheme ( Gilmer et al. , 2017 ; Battaglia et al. , 2018 ) , in which the representation of each node is recursively updated by aggregating the representations of its neighbors . Various GNNs ( Li et al. , 2016 ; Kipf & Welling , 2016 ; Veličković et al. , 2018 ; Xu et al. , 2019 ) mainly differ in the forms of aggregation functions . Real-world applications usually generate massive graphs , such as social networks . However , message passing methods have difficulties in handling such large graphs as the recursive message passing mechanism leads to prohibitive computation and memory requirements . To date , sampling methods ( Hamilton et al. , 2017 ; Ying et al. , 2018 ; Chen et al. , 2018a ; b ; Huang et al. , 2018 ; Zou et al. , 2019 ; Zeng et al. , 2020 ; Gao et al. , 2018 ; Chiang et al. , 2019 ; Zeng et al. , 2020 ) and precomputing methods ( Wu et al. , 2019 ; Rossi et al. , 2020 ; Bojchevski et al. , 2020 ) have been proposed to scale GNNs on large graphs . While the sampling methods can speed up training , they might result in redundancy , still incur high computational complexity , lead to loss of performance , or introduce bias ( see Section 2.2 ) . Generally , precomputing methods can scale to larger graphs as compared to sampling methods as recursive message passing is still required in sampling methods . In this work , we propose the Neighbor2Seq that transforms the hierarchical neighborhood of each node to a sequence in a precomputing step . After the Neighbor2Seq transformation , each node and its associated neighborhood tree are converted to an ordered sequence . Therefore , each node can be viewed as an independent sample and is no longer constrained by the topological structure . This novel transformation from graphs to grid-like data enables the use of mini-batch training for subsequent models . As a result , our models can be used on extremely large graphs , as long as the Neighbor2Seq step can be precomputed . As a radical departure from existing precomputing methods , we consider the hierarchical neighborhood of each node as an ordered sequence . The order information corresponds to hops between nodes . As a result of our Neighbor2Seq transformation , generic deep learning operations for gridlike data , such as convolution and attention , can be applied in subsequent models . In addition , our Neighbor2Seq can alleviate the over-squashing issue ( Alon & Yahav , 2020 ) suffered by current GNNs . Experimental results indicate that our proposed method can be used on a massive graph , where most current methods can not be applied . Furthermore , our method achieves superior performance as compared with previous sampling and precomputing methods . 2 ANALYSIS OF CURRENT GRAPH NEURAL NETWORK METHODS . We start by defining necessary notations . A graph is formally defined as G = ( V , E ) , where V is the set of nodes and E ⊆ V × V is the set of edges . We use n = |V | and m = |E| to denote the numbers of nodes and edges , respectively . The nodes are indexed from 1 to n. We consider a node feature matrix X ∈ Rn×d , where each row xi ∈ Rd is the d-dimensional feature vector associated with node i . The topology information of the graph is encoded in the adjacency matrix A ∈ Rn×n , whereA ( i , j ) = 1 if an edge exists between node i and node j , andA ( i , j ) = 0 otherwise . 2.1 GRAPH NEURAL NETWORKS VIA MESSAGE PASSING . There are two primary deep learning methods on graphs ( Bronstein et al . ) ; those are , spectral methods and spatial methods . The spectral method in Bruna et al . ( 2014 ) extends convolutional neural networks ( LeCun et al. , 1989 ) to the graph domain based on the spectrum of the graph Laplacian . The main limitation of spectral methods is the high complexity . ChebNet ( Defferrard et al. , 2016 ) and GCN ( Kipf & Welling , 2016 ) simplify the spectral methods and can be understood from the spatial perspective . In this work , we focus on the analysis of the current mainstream spatial methods . Generally , most existing spatial methods , such as ChebNet ( Defferrard et al. , 2016 ) , GCN ( Kipf & Welling , 2016 ) , GG-NN ( Li et al. , 2016 ) , GAT ( Veličković et al. , 2018 ) , and GIN ( Xu et al. , 2019 ) , can be understood from the message passing perspective ( Gilmer et al. , 2017 ; Battaglia et al. , 2018 ) . Specifically , we iteratively update node representations by aggregating representations from its immediate neighbors . These message passing methods have been shown to be effective in many fields . However , they have inherent difficulties when applied on large graphs due to their excessive computation and memory requirements , as described in Section 2.2 . 2.2 GRAPH NEURAL NETWORKS ON LARGE GRAPHS . The above message passing methods are often trained in full batch . This requires the whole graph , i.e. , all the node representations and edge connections , to be in memory to allow recursive message passing on the whole graph . Usually , the number of neighbors grows very rapidly with the increase of receptive field . Hence , these methods can not be applied directly on large-scale graphs due to the prohibitive requirements on computation and memory . To enable deep learning on large graphs , two families of methods have been proposed ; those are methods based on sampling and precomputing . To circumvent the recursive expansion of neighbors across layers , sampling methods apply GNNs on a sampled subset of nodes with mini-batch training . Sampling methods can be further divided into three categories . First , node-wise sampling methods perform message passing for each node in its sampled neighborhood . This strategy is first proposed in GraphSAGE ( Hamilton et al. , 2017 ) , where neighbors are randomly sampled . This is extended by PinSAGE ( Ying et al. , 2018 ) , which selects neighbors based on random walks . VR-GCN ( Chen et al. , 2018a ) further proposes to use variance reduction techniques to obtain a convergence guarantee . Although these node-wise sampling methods can reduce computation , the remaining computation is still very expensive and some redundancy might have been introduced , as described in Huang et al . ( 2018 ) . Second , layer-wise sampling methods sample a fixed number of nodes for each layer . In particular , FastGCN ( Chen et al. , 2018b ) samples a fixed number of nodes for each layer independently based on the degree of each node . AS-GCN ( Huang et al. , 2018 ) and LADIES ( Zou et al. , 2019 ) introduce between-layer dependencies during sampling , thus alleviating the loss of information . Layer-wise sampling methods can avoid the redundancy introduced by node-wise sampling methods . However , the expensive sampling algorithms that aim to ensure performance may themselves incur high computational cost , as pointed out in Zeng et al . ( 2020 ) . Third , graph-wise sampling methods build mini-batches on sampled subgraphs . Specifically , LGCN ( Gao et al. , 2018 ) proposes to leverage mini-batch training on subgraphs selected by Breadth-First-Search algorithms . ClusterGCN ( Chiang et al. , 2019 ) conducts mini-batch training on sampled subgraphs that are obtained by a graph clustering algorithm . GraphSAINT ( Zeng et al. , 2020 ) proposes to derive subgraphs by importance-sampling and introduces normalization techniques to eliminate biases . These graph-wise sampling methods usually have high efficiency . The main limitation is that the nodes in a sampled subgraph are usually clustered together . This implies that two distant nodes in the original graph usually can not be feeded into the GNNs in the same mini-batch during training , potentially leading to bias in the trained models . The second family of methods for enabling GNNs training on large graphs are based on procomputing . Specifically , SGC ( Wu et al. , 2019 ) removes the non-linearity between GCN layers , resulting in a simplification as Y = softmax ( ÂLXW ) . In this formulation , Â = D̃− 1 2 ÃD̃− 1 2 is the sym- metrically normalized adjacency matrix , Ã = A + I is the adjacency matrix with self-loops , D̃ is the corresponding diagonal node degree matrix with D̃ ( i , i ) = ∑ j Ã ( i , j ) , L is the size of receptive field ( i.e. , the number of considered neighboring hops ) , which is the same as a L-layer GCN , Y is the output of the softmax classifier . Since there is no learnable parameters in ÂLX , this term can be precomputed as a feature pre-processing step . Similarly , SIGN ( Rossi et al. , 2020 ) applies an inception-like model to the precomputed features Â ` X for ` ∈ { 1 , · · · , L } , where L is the predefined size of receptive field . Instead of precomputing the smoothing features as in SGC and SIGN , PPRGo ( Bojchevski et al. , 2020 ) extends the idea of PPNP ( Klicpera et al. , 2018 ) by approximately precomputing the personalized PageRank ( Page et al. , 1999 ) matrix , thereby enabling model training on large graphs using mini-batches . Generally , the precomputing methods can scale to larger graphs than sampling methods because the latter still needs to perform the recursive message passing during training . Differing from these precomputing methods , we consider the hierarchical neighborhood of each node as an ordered sequence , thus retaining the useful information about hops between nodes and enabling subsequent powerful and efficient operations . 3 THE PROPOSED NEIGHBOR2SEQ METHOD AND ANALYSIS . In this section , we describe our proposed method , known as Neighbor2Seq , which transforms the hierarchical neighborhood of each node into an ordered sequence , thus enabling the subsequent use of general deep learning operations . We analyze the scalability of our method ( See Section 3.5 ) and describe how our method can alleviate the over-squashing issue suffered by current message passing methods ( See Section 3.6 ) . 3.1 OVERVIEW . As described in Section 2.1 , existing message passing methods recursively update each node ’ s representation by aggregating information from its immediate neighbors . Hence , what these methods aim at capturing for each node is essentially its corresponding hierarchical neighborhood , i.e. , the neighborhood tree rooted at current node , as illustrated in Figure 1 ( b ) . In this work , we attempt to go beyond the message passing scheme to overcome the limitations mentioned in Section 2 . We propose to capture the information of this hierarchical neighborhood by transforming it into an ordered sequence , instead of recursively squashing it into a fixed-length vector . Our proposed method is composed of three steps . First , we transform a neighborhood to a sequence for each node . Second , we apply a normalization technique to the derived sequence features . Third , we use general deep learning operations , i.e. , convolution and attention , to learn from these sequence features and then make predictions for nodes . In the following , we describe these three steps in detail .

Summary: This paper proposed a simple graph neural network architecture that is easy to scale up and perform stochastic training. Instead of performing message passing as commonly used GNN, this paper first performs weighted combinations of node features per each hop of the neighbors of a center node, and then performs either CNN or attention mechanism to aggregate the features and obtain center node embedding. Since the feature aggregation can be performed offline, and the computation can easily be decomposed and stochastic training is straightforward, the method can easily scale up to graphs with 10M nodes. Experiments on median size or large size graphs show the comparable or better performance than alternatives.