Title: Graph Information Matters: Understanding Graph Filters from Interaction Probability. Abstract: Graph Neural Networks (GNNs) have received extensive affirmation for their promising performance in graph learning problems. Despite their various neural architectures, most are intrinsically graph filters that provide theoretical foundations for model explanations.  In particular, low-pass filters show superiority in label prediction in many benchmarks.  However, recent empirical research suggests that models with only low pass filters do not always perform well.  Although increasing attempts to understand graph filters,  it is unclear how a  particular graph affects the performance of different filters.  In this paper,  we carry out a comprehensive theoretical analysis of the synergy of graph structure and node features on graph filters’ behaviors in node classification, relying on the introduction of interaction probability and frequency distribution. We show that the homophily degree of graphs significantly affects the prediction error of graph filters.  Our theory provides a guideline for graph filter design in a data-driven manner. Since it is hard for a single graph filter to live up to this, we propose a general strategy for exploring a data-specified filter bank. Experimental results show that our model achieves consistent and significant performance improvements across all benchmarks.  Furthermore, we empirically validate our theoretical analysis and explain the behavior of baselines and our model.