DEGREE: Decomposition Based Explanation for Graph Neural Networks

1 INTRODUCTION . Graph Neural Networks ( GNNs ) play an important role in modeling data with complex relational information ( Zhou et al. , 2018 ) , which is crucial in applications such as social networking ( Fan et al. , 2019 ) , advertising recommendation ( Liu et al. , 2019 ) , drug generation ( Liu et al. , 2020 ) , and agent interaction ( Casas et al. , 2019 ) . However , GNN suffers from its black-box nature and lacks a faithful explanation of its predictions . Recently , several approaches have been proposed to explain GNNs . Some of them leverage gradient or surrogate models to approximate the local model around the target instance ( Huang et al. , 2020 ; Baldassarre & Azizpour , 2019 ; Pope et al. , 2019 ) . Some other methods borrow the idea from perturbation based explanation ( Ying et al. , 2019 ; Luo et al. , 2020 ; Lucic et al. , 2021 ) , under the assumption that removing the vital information from input would significantly reduce output confidence . However , approximation based methods do not guarantee the fidelity of the explanation obtained , as Rudin ( 2019 ) states that a surrogate that mimics the original model possibly employs distinct features . On the other hand , perturbation based approaches may trigger adversarial nature of the depth model . Chang et al . ( 2018 ) reported this phenomenon where masking some parts of the input image introduces unnatural artifacts . Additionally , additive feature attribution methods ( Vu & Thai , 2020 ; Lundberg & Lee , 2017 ) such as gradient based methods and GNNExplainer only provide a single heatmap or subgraph as explanation . The nodes in graph are usually semantically individual and we need a fine-grained explanation to the relationships between them . For example , in organic chemistry , the same functional group combined with different structures can exhibit very different properties . To solve the above problems , we propose DEGREE ( Decomposition based Explanation for GRaph nEural nEtworks ) , which measures the contribution of components in the input graph to the GNN prediction . Specifically , we first summarize the intuition behind the Context Decomposition ( CD ) algorithm ( Murdoch et al. , 2018 ) and propose that the information flow in GNN ’ s message propagation mechanism is decomposable . Then we design the decomposition schemes for the most commonly used layers and operations in GNNs , so as to isolate the information flow from distinct node groups . Furthermore , we explore the subgraph-level explanation via an aggregation algorithm that utilizes DEGREE and the structural information of the input graph to construct a series of subgraph sets as the explanation . DEGREE guarantees explanation fidelity by directly analyzing GNN feed-forward propagation , instead of relying on input perturbation or the use of alternative models . DEGREE is non-additive and can therefore uncover non-linear relationships between nodes . We quantitatively and qualitatively evaluate the DEGREE on both synthetic and real-world datasets to validate the effectiveness of our method . The contributions of this work are summarized as follows : • We propose a new explanation method ( DEGREE ) for GNNs , from the perspective of decomposition . By elucidating the feed-forward propagation mechanism within GNN , DEGREE allows capturing the contribution of individual components of the input graph to the final prediction . • We propose an aggregation algorithm that provides a series of subgraphs as explanation in order to mine the complex interactions between graph components . We combine the property of the message propagation mechanism to further reduce the computation . • We evaluate DEGREE on both synthetic and real-world datasets . The quantitative experiments show that our method could provide faithful explanations . The qualitative experiments indicate that our method may capture the interaction between graph components . 2 RELATED WORK . Despite the great success in various applications , the black-box nature of deep models has long been criticized . Explainable Artificial Intelligence ( XAI ) tries to bridge the gap by understanding the internal mechanism of deep models ( Du et al. , 2019 ) . Meanwhile , the need to tackle non-Euclidean data , such as geometric information , social networks , has given rise to the development of GNNs . Similar to the tasks on image or text data , GNNs focus on node classification ( Henaff et al. , 2015 ) , graph classification ( Xu et al. , 2018 ; Zhang et al. , 2018 ) , and link prediction ( Zhang & Chen , 2018 ; Cai & Ji , 2020 ) . Message passing mechanism allows the information to flow from one node to another along edges , and empowers GNNs with convolutional capabilities for graph data . While the explainability in image and text domains is widely studied ( Shrikumar et al. , 2017 ; Sundararajan et al. , 2017 ; Simonyan et al. , 2013 ) , the explainability of GNN is on the rise . First , some recent work adapts the interpretation methods used for traditional CNNs to GNNs ( Baldassarre & Azizpour , 2019 ; Pope et al. , 2019 ) . They employ gradient values to investigate the contribution of node features to the final prediction . However , these methods ignore the topological information , which is a crucial property of graph data . Second , some methods trace the model prediction back to the input space in a backpropagation manner layer by layer ( Schwarzenberg et al. , 2019 ; Schnake et al. , 2020 ) . Third , some methods define a perturbation-based interpretation whereby they perturb node , edge , or node features and identify the components that affect the prediction most . Specifically , GNNExplainer and PGExplainer ( Ying et al. , 2019 ) maximize the mutual information between perturbed input and original input graph to identify the important features . Causal Screening ( Wang et al. , 2021 ) searches for the important subgraph by monitoring the mutual information from a cause-effect standpoint . CF-GNNExplainer ( Lucic et al. , 2021 ) proposes to generate counterfactual explanations by finding the minimal number of edges to be removed such that the prediction changes . In addition , XGNN ( Yuan et al. , 2020 ) builds a model-level explanation for GNNs by generating a prototype graph that can maximize the prediction . Moreover , due to the discrete and topological nature of graph data , XGNN defines graph generation as a reinforcement learning task instead of gradient ascent optimization . Many previous explanation methods for GNNs suffer from adversarial triggering issues , faithfulness issues and additive assumptions . To this end , we propose a decomposition based explanation for GNNs ( DEGREE ) to remedy these problems . DEGREE enables to track the contribution of the components from the input graph to the final prediction by decomposing a trained GNN . Thus , DEGREE guarantees the integrity of the input and eliminates the adversarial triggering issue of the perturbation-based approach . Since no surrogate models are used , DEGREE guarantees its faithfulness . Meanwhile , by integrating the decomposition to the normal layer , DEGREE does not have any additional training process . 3 DEGREE : DECOMPOSITION BASED EXPLANATION FOR GNNS . In this section , we introduce the details of the proposed explanation method . First , we introduce the notations and problem definition . Then , we discuss the general idea of decomposition based explanation . Finally , we develop concrete decomposition schemes for different GNNs layers 3.1 PROBLEM DEFINITION . We first introduce the notations used in this work . Given a graph G = ( V , E ) , where V is the set of nodes and E is the set of edges between nodes . The adjacency matrix of G is denoted as A ∈ RN×N , where N = |V| is the number of nodes , so V = { v1 , v2 , ... , vN } . The nodes are associated with features , and the feature matrix is denoted as X ∈ RN×F , where F is the feature dimension . In this work , we focus on explaining GNN-based classification models . Let f denote the target GNN model . f computes the likelihood of a node or graph belonging to the target class , where f : G 7→ R|V| or f : G 7→ R , for node or graph classification respectively . The goal of explanation is to find the most important subgraph in G given f ( G ) , which requires measuring the contribution of all possible subgraphs and find the ones with high contribution scores . However , there are two challenges to be addressed . ( 1 ) Given any subgraph of interest , how to estimate its contribution without breaking up the input graph ? ( 2 ) The number of all subgraphs in G is usually very large , so how to choose candidate subgraphs for improving explanation efficiency ? We tackle the first challenge in the following part of Sec 3 and solve the second one in Sec 4 . 3.2 DECOMPOSITION BASED EXPLANATION . In general , a prediction model f contains multiple layers Lt , t ∈ { 1 , . . . , T } : f ( X ) = LT ◦ LT−1 ◦ · · · ◦ L2 ◦ L1 ( X ) . ( 1 ) Let X [ t ] denotes the input of Lt , so X [ t + 1 ] = Lt ( X [ t ] ) and X [ 1 ] = X . The symbol ◦ denotes function composition . Here X [ t ] ∈ RN×Ft is the embedding matrix at t-th layer , where Ft is the latent dimension . The embedding vector of the i-th node at t-th layer is denoted as Xi [ t ] . The core idea of decomposition based explanation is that , given a target node group ( or subgraph ) of interest , we estimate its contribution score to model prediction merely through feed-forward propagation . We call the information propagated from the target group as target portion , and the rest of information is called background portion . In the following , we use γ and β to denote target portion and background portion , respectively . Let m ∈ RN and mn ∈ { 0 , 1 } , where mn = 1 means vn belongs to the target group and otherwise mn = 0 . Then , the decomposition is initialized from the layer of node features , where the target portion and background portion of the input feature matrix are : Xγ = diag ( m ) X and Xβ = ( I−diag ( m ) ) X , respectively . In a neural network , different parts of input are entangled in the feedforward process into latent representations , which poses challenges for explanation . Suppose the target and background portion in X [ t ] are known from prior layer , we could explain the model if we can still distinguish the information flows of the two portions inside Lt. That is , at layer Lt , suppose its input can be decomposed as X [ t ] = Xγ [ t ] +Xβ [ t ] , the following relations need to hold for explanation : LDt ( X γ [ t ] , Xβ [ t ] ) = Γ ( Xγ [ t ] , Xβ [ t ] ) ︸ ︷︷ ︸ Xγ [ t+1 ] , B ( Xγ [ t ] , Xβ [ t ] ) ︸ ︷︷ ︸ Xβ [ t+1 ] ( 2 ) Lt ( X [ t ] ) = X [ t+ 1 ] = X γ [ t+ 1 ] + Xβ [ t+ 1 ] , ( 3 ) where LDt ( · , · ) denotes the decomposed version of layer Lt. Γ ( · , · ) and B ( · , · ) corresponds to the contribution of the target and the background portion to layer Lt. Xγ [ t+ 1 ] and Xβ [ t+ 1 ] denotes the target and background portion of X [ t + 1 ] as the input to the next layer . The decomposition above goes from the input , through all intermediate layers , to the final prediction . If a target node group or subgraph is important , then it should contributes to most of the prediction , meaning that Γ ( Xγ [ T ] , Xβ [ T ] ) ≈ f ( X ) .

Summary: This paper proposes a decomposition-based explanation method for graph neural networks. The motivation of this paper is that existing works based on approximation and perturbation suffer from drawbacks. To address the issue of existing works, the authors directly decompose the influence of node groups in the forward pass. The decomposition rules are designed for GCN and GAT. Further, to efficiently select subgraph groups from all possible combinations, the authors propose a greedy approach to search for maximally influential node sets. Experiments on synthetic and real-world datasets verify the improvements over existing works.