  Spectral Graph Neural Networks (GNNs) with various graph filters have
received extensive affirmation due to their promising performance in graph
learning problems. However, it is known that GNNs do not always perform well.
Although graph filters provide theoretical foundations for model explanations,
it is unclear when a spectral GNN will fail. In this paper, focusing on node
classification problems, we conduct a theoretical analysis of spectral GNNs
performance by investigating their prediction error. With the aid of graph
indicators including homophily degree and response efficiency we proposed, we
establish a comprehensive understanding of complex relationships between graph
structure, node labels, and graph filters. We indicate that graph filters with
low response efficiency on label difference are prone to fail. To enhance GNNs
performance, we provide a provably better strategy for filter design from our
theoretical analysis - using data-driven filter banks, and propose simple
models for empirical validation. Experimental results show consistency with our
theoretical results and support our strategy.
