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: The paper provides DEGREE, which decomposes the feedforward propagation mechanism of a GNN to understand it. They give realistic decomposition techniques for those typically used layers in GNNs after presenting the key guidelines for developing decomposition-based explanations. They also devise an approach for providing subgraph-level explanation via agglomeration, which makes effective use of graph topology. DEGREE surpasses baselines in terms of fidelity and can capture important structures in graph data, according to experimental results.