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The explainability of Graph Neural Networks (GNNs) is critical to various GNNapplications, yet it remains a significant challenge. A convincing explanationshould be both necessary and sufficient simultaneously. However, existing GNNexplaining approaches focus on only one of the two aspects, necessity orsufficiency, or a heuristic trade-off between the two. Theoretically, theProbability of Necessity and Sufficiency (PNS) holds the potential to identifythe most necessary and sufficient explanation since it can mathematicallyquantify the necessity and sufficiency of an explanation. Nevertheless, thedifficulty of obtaining PNS due to non-monotonicity and the challenge ofcounterfactual estimation limit its wide use. To address thenon-identifiability of PNS, we resort to a lower bound of PNS that can beoptimized via counterfactual estimation, and propose a framework of Necessaryand Sufficient Explanation for GNN (NSEG) via optimizing that lower bound.Specifically, we depict the GNN as a structural causal model (SCM), andestimate the probability of counterfactual via the intervention under the SCM.Additionally, we leverage continuous masks with a sampling strategy to optimizethe lower bound to enhance the scalability. Empirical results demonstrate thatNSEG outperforms state-of-the-art methods, consistently generating the mostnecessary and sufficient explanations.

On the Probability of Necessity and Sufficiency of Explaining Graph Neural Networks: A Lower Bound Optimization Approach